The quadratic shortest path problem: Theory and computations

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Tilburg University

The quadratic shortest path problem

Hu, Hao

DOI: 10.26116/center-lis-1920 Publication date: 2019 Document Version

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Link to publication in Tilburg University Research Portal

Citation for published version (APA):

Hu, H. (2019). The quadratic shortest path problem: Theory and computations. CentER, Center for Economic Research. https://doi.org/10.26116/center-lis-1920

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The Quadratic Shortest Path Problem

-Theory and Computations

Proefschrift

ter verkrijging van de graad van doctor aan Tilburg University op gezag van prof. dr. G.M. Duijsters, als tijdelijk waarnemer van de functie rec-tor magnificus en uit dien hoofde vervangend voorzitter van het college voor promoties, in het openbaar te verdedigen ten overstaan van een door het college voor promoties aangewezen commissie in de Aula van de Universiteit op

vrijdag 13 september 2019 om 10.00 uur door

Hao Hu

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Promotiecommissie:

Promotores: prof. dr. ir. Renata Sotirov prof. dr. ir. Edwin van Dam Overige leden: prof. dr. Etienne de Klerk

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Acknowledgements

I would not be wrong to say that countless people have helped me during these three years with my research. In the beginning, it looked like a huge obstacle for me to overcome but as time went on, there were many who supported me and made the path easier for me to walk down. That’s why I would like to thank them all in the best-abridged way possible.

First of all, I would like to acknowledge the support of my advisor Renata Sotirov. I cannot thank you enough for the guidance that you offered me whenever I was downcast and you showed me the light at the end of the tunnel when, at one point, I thought was unachievable. I can easily remember the times when I was hopeless and I didn’t think that it was possible to continue but you poured common sense into me and opened my mind. If I have completed my project, then you are one of the essential pillars that allowed me to do so. Then there is my second advisor Edwin van Dam, who encouraged me to pursue this field. I can still remember the time that I spent with you as a bachelor student and your continued motivation.

I would like to extend my gratitude to the other members of my committee, as well: Etienne de Klerk, Christoph Helmberg, Ren´e Peeters, Juan Vera Lizcano and Henry Wolkowicz. There is no doubt; I faced many hurdles while completing the project. It was their cooperation that helped in noticing the gaps I was leaving while covering the guidelines. With their help, it became possible to improve my project. I will never forget the advice and feedback that you offered to me. That is why I would like to thank you. A special thank you goes to Henry Wolkowicz, who provided me with the opportunity to visit Waterloo University in Canada during the second year of my PhD.

There are some notable individuals that I just can’t forget and I often find myself thinking about the time that I have spent with them. And these are all my friends Riley Badenbroek, Daniel Brosch, Stefan ten Eikelder, Olga Kuryatnikova, Gou Mei, Ernst Roos, Stefan Sremac, Zhehao Sun, Oliver Wichert, Zixiao Xu, Peng Zhang and Trevor Zhen, whom I always counted on whenever I found myself in a difficulty. There were times when I was unable to choose the right sources or was stuck in one particular section and they helped me by providing their feedback and giving me their spare time. Such moments cannot be relived and that’s why these moments are so precious to me. Now, it’s all about the most important figures, my family. I don’t even know the words that will be capable of expressing my gratitude to my parents, Hongwu Hu and Ping Fang, who always stood by my side. They were the constant reminder that gave me the will to continue onwards. I am also deeply indebted to my girlfriend, Danqi Xu, for her endless love. Besides them, I would like to dedicate this thesis to my grandmother. I will never forget her persistent support and the delicate times that I spent with her. Even though she is not here with me now, I know that she would be very happy to see me graduate as a PhD student.

You guys all are very important and precious to me. Thank you again. Hao Hu

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List of Figures

2.1 Example graph G. . . 16

2.2 The auxiliary graph G0 of G. . . 16

2.3 Simplified K₄∗ with s = v1, t = v4. . . 21

5.1 Lower bounds for the QSPP. . . 76

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List of Tables

1.1 The complexity results of the QSPP . . . 3

3.1 _{grid1 instances: bounds, optimal values and time (s)} . . . 41

3.2 _{grid3 instances: bounds, optimal values and time (s)} . . . 41

3.3 _{par-k instances: bounds, optimal values and time (s)} . . . 42

3.4 _{tour instances: bounds, optimal values and time (s)} . . . 42

3.5 instances with negative weights: bounds, optimal values and time (s) . . . 43

4.1 fsos 1 and fsos 0 1 for the QSPP . . . 51

4.2 Notation with respect to target vertex v. . . 59

5.1 SDP bounds for the QSPP instances on G20,20. . . 71

5.2 grid1-square: bounds and optimal values . . . 82

5.3 _{grid1-square: running times and iterations} . . . 83

5.4 _{grid2-square: bounds, running times, iterations . . . .} 84

5.5 _{grid2-long: bounds, running times, iterations . . . .} 84

5.6 _{grid2-wide: bounds, running times, iterations . . . .} 84

5.7 _{grid3-square: bounds, running times, iterations . . . .} 84

5.10 grid4: bounds, running times, iterations . . . 85

5.8 _{grid3-long: bounds, running times, iterations . . . .} 85

5.9 _{grid3-wide: bounds, running times, iterations . . . .} 85

5.11 par-k: bounds . . . 86

5.12 par-k: running times, iterations . . . 86

5.13 tour: bounds, running times, iterations . . . 86

5.14 grid5: bounds, running times, iterations . . . 87

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List of Algorithms

1 _{The QSPP linearization algorithm on grid graphs . . . .} 30

2 _{The reformulation-based bound} . . . 37

3 _{The Generalized Gilmore-Lawler bound . . . .} 54

4 _{The QSPP linearization algorithm on DAGs . . . .} 60

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List of Symbols

Rn n dimensional Euclidean vector space MT _{the transpose of a matrix M}

Sn

the set of all n × n symmetric matrices {M ∈ Rn×n_{|M = M}T_}

M 0 ( 0) M ∈ Sn is positive (semi)definite M ≺ 0 ( 0) M ∈ Sn is negative (semi)definite

Sn

+ the set of positive semidefinite matrices

Sn

++ the set of positive definite matrices

Sn

− the set of negative semidefinite matrices

tr(M ) _{the trace of M ∈ R}n×n _isPn

i=1Mii

hM1, M2i the trace inner product trace(M1TM2)

vec : Rm×n_{→ R}mn _{the vec operator stacks the columns of the matrix M}

M1⊗ M2 the Kronecker product of M1 and M2

diag(M ) ∈ Rn the vector that contains all the diagonal entries of M ∈ Rn×n Diag(v) ∈ Rn×n the diagonal matrix with (i, i)-th entry equals vi for v ∈ Rn

Πn the set of n × n permutation matrices

Jn the n × n dimensional all-ones matrix

In the n × n dimensional identity matrix

ei the i-th unit vector

Eij the n × n matrix eieTj

0n the all-zeros vector of length n

¯

S the complement of a set S

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Chapter 1

Introduction

1.1 Background

The study of optimal transportation and resource allocation is an important subject in operations research. The transportation plan is obtained from advanced analytical methods, and optimal or near-optimal solutions can be used to reduce the waste of money, time, and effort in the organiza-tion. Major advances were made in the field of operations research during World War II. After the war, the models of operations research were employed in a wide range of industries, including logis-tics, finance, airlines, and government. Operations research adapts techniques from mathematical modeling, statistical analysis, and mathematical optimization to develop models that can be used to analyze and arrive at superior solutions to complex decision-making problems. In this thesis, we focus on the mathematical optimization approach, which typically includes an objective function that is maximized or minimized subject to a set of constraints.

Mathematical Programming is probably the most powerful model in operations research. For example, many real-world problems can be formulated as linear programming (LP) problems, and we are able solve very large LP problems due to the fast advances in algorithms and computer hardware. Since the early 1990’s, it is recognized that interior-point methods (i.p.m.) are able to solve convex optimization problems with efficiently computable self-concordant barrier functions in polynomial time. This allows us to solve semidefinite programming problems efficiently, just as its special case LP problems. Consequently, semidefinite optimization became the next milestone in mathematical programming. Since the beginning of the twenty-first century, there also has been a surge of interest in the interplay between semidefinite programming and algebraic geometry through semidefinite matrices and polynomial optimization problems.

Nowadays, transportation infrastructure plays a critical role in a country’s development and its cost benefits are evident from America to China. For example, the Interstate Highway System in America or the High-speed rail in China accounts for a significant factor in both nations’ progress. But the costs of building and maintaining transportation infrastructure are also tremendous. Thus, it is also important to have an intelligent route planning system, which improves the effectiveness and the efficiency of transportation infrastructures. The major challenge in route planning is that many side constraints have to be taken into consideration, such as uncertainties of travel time. The quadratic shortest path problem can be used to model a variety of real-life transportation problems and in this thesis, we provide both theoretical and computational insights into this problem.

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1.2 The binary quadratic program

A binary quadratic program (BQP) is a discrete optimization problem with binary variables, quadratic objective function and linear constraints. The BQP can be used to model a wide range of combinatorial optimization problems, including the quadratic assignment problem (QAP), the quadratic shortest path problem (QSPP), the graph partitioning problem, the max-cut problem, and clustering problems. In general, binary quadratic programs are NP-hard which means that it can be a formidable computational task to obtain an optimal solution. Therefore one can try to compute near-optimal solutions to cope with its computational intractability.

Given a quadratic cost matrix Q of size m × m, the binary quadratic program can be formulated as

min

x∈Kx

T_Qx, _(1.1)

where K is the set of feasible binary vectors, i.e.,

K := {x ∈ Rm| Bx = b, x ∈ {0, 1}m_}, _(1.2)

for some B ∈ Rn×mand b ∈ Rn. Note that it is also possible to have a more generalized model with inequalities in (1.2). Here, we focus on two special cases of the binary quadratic problem, namely, the quadratic shortest path problem and the quadratic assignment problem. Both problems are of the form (1.1) with feasible set (1.2). We describe these problems in detail in the next two sections.

1.2.1 The quadratic shortest path problem

Given a directed (multi) graph and a cost matrix, the quadratic shortest path problem (QSPP) is the problem of finding a path between two given vertices such that the sum of quadratic costs over all pairs of arcs on the path is minimized. The quadratic costs are sometimes also referred as interaction costs. Let us formally introduce the QSPP. Let G = (V, A) be a directed graph with n vertices and m arcs. The two distinguished vertices are the source vertex s and the target vertex t in G. A path is a sequence of distinct vertices (v1, . . . , vk) such that (vi, vi+1) ∈ A for i = 1, . . . , k − 1.

An s-t path is a path P = (v1, v2, . . . , vk) from the source vertex s = v1to the target vertex t = vk.

Let the quadratic cost between two distinct arcs e and f be qef, and the linear cost of an arc e

be qe,e. Let Q = (qe,f) ∈ Rm×mbe the cost matrix. The entries of Q do not have to be nonnegative,

unless specified otherwise. The quadratic shortest path problem is given by:

minimizexT_{Qx | x ∈ P ,} _(1.3)

where P is the set of characteristic vectors of all s-t paths in G. We also study special cases of the QSPP. For example, the adjacent QSPP is a variant of the QSPP where the interaction costs of all non-adjacent arcs are equal to zero. The set P above is of the form (1.2). Indeed, one can take the incidence matrix as B ∈ Rn×m and define the vector b ∈ Rm such that bs= 1, bt= −1 and bv= 0

otherwise.

The complexity

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1.2. THE BINARY QUADRATIC PROGRAM 3

the problem called the adjacent QSPP, see Rostami et al. [80]. In Theorem 2.3.2, we will give an alternative proof for the same result using a simple reduction from the arc-disjoint paths problem. Rostami et al. [80] also show that the QSPP is APX-hard even when the objective function is convex.

On the other hand, it is known that the QSPP can be solved efficiently for some special graphs or cost matrices. Rostami et al. [79] claim that the adjacent QSPP can be solved in polynomial time for any graphs and this is not true. In Theorem 2.3.2, we corrected the claim and provided a polynomial time algorithm for the adjacent QSPP on directed acyclic graphs. In Proposition 2.4.6, we show that the QSPP can be efficiently solved if the cost matrix is a nonnegative symmetric product matrix. In Proposition 2.4.3, we show that if the cost matrix is a sum matrix and every s-t path in the graph has constant length. Both product matrices and sum matrices are important classes of cost matrices for binary quadratic programs. For example, the sum matrices are studied for quadratic assignment problems in [19]. In Theorem 4.7.8, we present one of our main results, which shows that the linearization problem of the QSPP on directed acyclic graphs can be detected in polynomial-time. (The definition of the linearization problem is given in Section 1.3.) This also results in a class of QSPPs that can be solved efficiently. This is the first non-trivial result for the linearization problem of the QSPP. If Q is a diagonal matrix, then the QSPP is equivalent to a linear shortest path problem which can be solved efficiently. We also note that the longest path problem is NP-hard in general.

In Table 1.1, we summarize the NP-hardness results for the QSPP under different settings. Problem General graphs Acyclic graphs

QSPP NP-hard NP-hard convex QSPP NP-hard NP-hard adjacent QSPP NP-hard P

Table 1.1: The complexity results of the QSPP

Etienne de Klerk pointed to us a proof of the NP-hardness of the convex QSPP on directed acyclic graphs. We present this proof below. Recall that in the PARTITION problem, we are given a vector α ∈ Zn_{, and the task is to decide if there exists a subset S ⊆ {1, . . . , n} such that}

P

i∈Sαi=Pi6∈Sαi. The PARTITION problem is one the first six basic NP-complete problems in

Garey and Johnson [39]. The next result provides a reduction from the PARTITION problem to the convex QSPP.

Proposition 1.2.1. The decision version of the convex QSPP is NP-hard.

Proof. Given an instance of the PARTITION problem with the input vector α ∈ Rn_{, we construct}

a convex QSPP instance as follows. Let G be a directed graph with vertices v1, . . . , vn+1, and arcs

ei = (vi, vi+1) and fi = (vi, vi+1) for i = 1, . . . , n. (Note that there are 2 distinct arcs from vi to

vi+1.) The source vertex is v1and the target vertex is vn+1. The interaction costs between the arcs

are given by qei,ej = qfi,fj = αiαj, and qei,fj = −αiαj for every i, j = 1, . . . , n. If we order the arcs

as e1, . . . , en, f1, . . . , fn, then the quadratic cost matrix has the form Q = (−αα ) (−αα )T 0.

Let x be the characteristic vector of an optimal s-t path. Since exactly one of the arcs ei and

fi is included in any s-t path, the subset S = {i | xei = 1} and its complement ¯S = {i | xfi = 1}

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path in G and the partition of {1, . . . , n}. The optimal value of the constructed convex QSPP is zero if and only if

xTQx = xT(−αα ) (−αα ) T x = ((−αα ) T x)2= 0 ⇐⇒ (−αα ) T x = 0 ⇐⇒ X i∈S αi= X i6∈S αi.

The last equivalence follows from xei+ xfi = 1. Thus the PARTITION problem is a YES instance

if and only if the optimal value of the constructed convex QSPP is zero.

We would like to point out an error in Sen et al. [84]. It is claimed in [84] that there is a polynomial-time algorithm for the QSPP when the quadratic cost matrix is positive definite and the so-called cycle covariance assumption is satisfied. We verified numerically that the algorithm does not return the optimal solution. Furthermore it also contradicts to the complexity result in Rostami et al. [80]. Indeed, the authors in [80] provide a polynomial-time reduction to show the convex QSPP is NP-hard, and the quadratic cost matrix constructed in their proof is positive definite and satisfies the cycle covariance assumption.

The applications

Although the QSPP was only recently introduced, it arises in many different applications. Sen et al. [84] investigate the route-planning problems in which the choice of a route is based on the mean as well as the variance of the path travel-time. Sivakumar and Batta [86] consider the variance-constrained shortest path problem. This problem is used to model the situation when the travel cost of a link is a random variable and it is correlated to the travel cost of other links. For example, it can be found in a transportation problem involving liquefied-gas hazardous materials. The quadratic shortest path problem can also be used to model this application. Nie and Wu [69] study the problem of finding most reliable a priori shortest path in a stochastic and time-dependent network, which is also related to the QSPP. The QSPP also plays a role in network protocols. Murakami and Kim [67] point out that solving the quadratic shortest path problem is useful to the study of different restoration schemes for self-healing ATM networks. Gourv`es et al. [42] consider the QSPP on undirected edge-colored graphs with nonnegative reload costs. The edge-colored graphs are for example used to model cargo transportation and large communication networks, see Galbiati [36], Wirth and Steffan [92]. The QSPP can be also applied in satellite network designs as discussed in Gamvros [38].

On solving the QSPP

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1.2. THE BINARY QUADRATIC PROGRAM 5

instance on 15 × 15 grid graph can be solved within 25 seconds. This is a dramatic improvement over previous results. (The computers used in Rostami et al. [80] and Chapter 5 have similar configuration.)

1.2.2 The quadratic assignment problem

The quadratic assignment problem is an NP-hard discrete optimization problems, introduced by Koopmans and Beckmann [55]. The QAP is known as a generic model for various real-life problems, see e.g., [14]. In the survey paper [8], some of the prominent applications of the QAP are presented and the authors justify that the QAP is important. Since the QSPP is closely related to the QAP, it is important to compare our results for the QSPP to the existing results for the QAP. The traveling salesman problem is a special case of the quadratic assignment problem, and therefore the QAP is NP-hard in the strong sense. Assume there are n facilities and n locations. The flow between each pair of facilities, say i, k, is fik. The distance between each pair of locations, say j, l, is djl. (The

distance djl can also be interpreted as the transportation cost per unit between two locations.) The

problem is to assign all facilities to different locations such that the sum of the distances multiplied by the corresponding flows is minimized. The quadratic assignment problem is given by:

min    X i,j,k,l fikdjlxijxkl: X = (xij), X ∈ Πn    ,

where Πnis the set of n × n permutation matrices. Let F = (fik) be the flow matrix, and D = (djl)

the distance matrix. If x = vec(X) ∈ Rn2, then the objective function can be written as xT(F ⊗D)x.

On solving the SDP relaxation of the QAP

Although SDP relaxations have proven to be incredibly strong for many hard optimization problems, it is a challenging task to solve semidefinite programs of medium size. Zhao et al. [94] introduced one of the strongest and computational tractable SDP relaxation for the QAP. This SDP relaxation cannot be solved directly by interior point methods for non-trivial instances (n ≥ 15), see Ferreira et al. [31], Rendl and Sotirov [77].

On the positive side, Oliveira et al. [70] proposed a novel approach which exploits the alternating direction method of multipliers and facial reduction techniques to solve the SDP relaxation for the QAP from [94]. Their approach results in a first order method with inexpensive iterations, and it exploits the special structure of the problem. This yields currently the best bound solved for large instances in the literature. We also note that the idea of projecting onto a Slater feasible face has been used in the context of QAP in Hadley et al. [43].

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1.3 The linearization problem of the BQP

If there exists a cost vector c such that xT_{Qx = c}T_{x for every x in the feasible set K, then the}

binary quadratic optimization problem (1.1) is said to be linearizable. The cost vector c is called a linearization vector of Q. If the BQP is clear from the context, we simply say that the quadratic cost matrix Q is linearizable if the underlying binary quadratic program is linearizable. If Q is linearizable, then (1.1) can be equivalently formulated as the following linear optimization problem:

min

x∈Kc

T_x. _(1.4)

The obtained linear optimization problem could be much easier to solve. For instance, it is well-known that the Hungarian algorithm can be used to solve the linear assignment problem efficiently. The linearization problem of a binary quadratic problem asks whether Q is linearizable, and provide its linearization vector c if it exists. The linearization problem is investigated in the context of many discrete optimization problems.

Kabadi and Punnen [54] provide a polynomial-time algorithm to solve the linearization problem of the quadratic assignment problem in general form. The linearization problem for the QAP in Koopmans-Beckmann form is studied in [73]. The linearizable special cases of the QAP are studied in [1, 19]. ´Custi´c et al. [23] consider the linearization problem for the bilinear assignment problem, and show that it can be solved in polynomial time. The linearization problem for the quadratic minimum spanning tree problem was studied in [22]. Punnen et al. [75] provide a characterization of linearizable cost matrices for the quadratic traveling salesman problem. Proposition 2.5.4 presents necessary conditions for an instance of the QSPP on complete digraphs to be linearizable. If the complete digraph has only four vertices, these conditions are also sufficient, see Proposition 2.5.5. Those are the only known results on the linearization problem of the QSPP on complete graphs at this moment. As far as we know, Theorem 4.7.8 provides the first polynomial time algorithm to solve the linearization problem of the quadratic shortest path problem on directed acyclic graphs. This is one of our main results and it is a generalization of the linearization algorithm in [50] for grid graphs.

There is relatively few research concerning applications of the linearization problem. Punnen and Pandey [74] show that an equivalent representations of a binary quadratic optimization problem can be derived by exploiting its linearization problem. They also demonstrate that equivalent repre-sentations could result in different lower bound values for various lower bounding schemes. Theorem 4.6.1 is one of our main results, it shows the linearization problem can be used to prove that the first level reformulation linearization technique bound dominates the Generalized Gilmore-Lawler bound for any binary quadratic problems. This result was only known for quadratic assignment problems before, and it also brings insight into the connection between Gilmore-Lawler bound and the linearization problem.

1.4 Semidefinite programming

Semidefinite programming (SDP) is an important tool in discrete optimization. The quality of the bounds for discrete optimization problems obtained via semidefinite programming is remarkable, see Laurent [59].

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1.4. SEMIDEFINITE PROGRAMMING 7

of a semidefinite program is given in (1.5). inf

X hC, Xi

s.t. A(X) = b X 0,

(1.5)

where the linear oprator A(·) : Sn→ Rm _{is defined as}

A(X) := hA1, Xi, . . . , hAm, Xi T

. The adjoint operator A∗_{: R}m_{→ S}n _{is given by A}∗_{(y) =}Pm

i=1yiAi. The dual problem of (1.5) is

sup y,S bT_y s.t. A∗_{(y) + S = C,} S 0. (1.6)

The primal program (1.5) is called strictly feasible if there exists a feasible matrix X 0. The dual program (1.6) is strictly feasible if there exists a vector y satisfying C − A∗(y) 0.

For a given primal problem (1.5) with optimal supremum value v, the dual program is often an equivalent problem with the same optimal value v as infimum; this provides a recipe to upper or lower bound v. Duality theory provides a relation between the primal program and the dual program.

Theorem 1.4.1. Assume we have a pair of primal and dual programs (1.5) and (1.6). Let p∗ be the supremum of the primal (1.5) and d∗ the infimum of the dual (1.6).

1. (weak duality) If X is feasible for the primal program and (y, S) is feasible for the dual program, then

hC, Xi ≥ bTy. In particular, p∗ _{≥ d}∗_.

2. (complementary slackness) If there exists a primal feasible solution X attaining the supre-mum p∗, and a dual feasible solution (y, S) attaining the infimum d∗such that p∗= d∗. Then

hC − A∗_{(y), Xi = 0.}

3. (optimality condition) Suppose that X is a feasible solution of the primal program, and (y, S) is a feasible solution of the dual program, such that the complementary slackness con-dition hC − A∗(y), Xi = 0 holds. Then the primal program attains its supremum at X, and that the dual program attains its infimum at (y, S).

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We note that there are several differences between linear programming and semidefinite pro-gramming. For example, there is no duality gap between the primal and dual programs in linear programming; and the supremum/infimum may not be attained in semidefinite programming. A constraint qualification (CQ) is a technical condition for strong duality, see Jeyakumar [53]. For example, strict feasibility is a CQ and it provides a sufficient condition for strong duality. Facial reduction, which was proposed by Borwein and Wolkowicz, is a conceptual method to obtain strict feasibility for an instance of semidefinite program, see [28]. We also note that strict feasibility is not a necessary condition for strong duality. The standard approach to come up with a semidefinite relaxation for 0-1 problems is introduced in Lov´asz and Schrijver [65], Shor [85]. There are a lot of excellent books about semidefinite programming, its (duality) theory and applications, e.g., Anjos and Lasserre [5], Ben-Tal and Nemirovski [7], Boyd and Vandenberghe [11], Laurent and Vallentin [61], Wolkowicz et al. [93].

1.5 The alternating direction method of multipliers

In this section, we review a method called the alternating direction method of multipliers (ADMM) for solving semidefinite programming problems. Although interior-point-methods can be used to solve SDP problems in polynomial-time with arbitrary accuracy, the i.p.m. becomes impractical for large-scale SDPs as it requires a lot of computation and storage. In contrast, the alternating direction method of multipliers is often much cheaper in computation, and it could also be the only practical choice for solving large instances.

We follow the notation in Wen et al. [91]. For convenience, we write the dual problem (1.6) as a minimization problem, i.e.,

min y,S −b T_y s.t. A∗_{(y) + S = C,} S 0. (1.7)

The augmented Lagrangian of (1.7) corresponding to the linear constraints is given by:

L(X, y, S) = −bT_{y + hX, A}∗_{(y) + S − Ci +}β

2||A

∗_{(y) + S − C||}2_,

where X ∈ Sn and β > 0 is a positive constant.

Given the dual slack variables S0 and the primal variable X0, the alternating direction method

of multipliers uses the augmented Lagrangian L(X, y, S) and performs the following updates at k-th iteration: yk+1= arg min y∈RmL(Xk, y, Sk), Sk+1= arg min S0L(Xk, yk+1, S), Xk+1= Xk+ β(A∗(yk+1) + Sk+1− C).

At k-th iteration, we are given the dual slack variables Sk and the primal variable Xk. The

ADMM algorithm above first minimizes the augmented Lagrangian function with respect to y for given Xk and Sk, and then with respect to S for given Xk and yk+1, and finally it updates X.

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1.6. CONTENTS OF THE THESIS 9

(1.7), when the primal program is strictly feasible and the matrix A := vec(A1), . . . , vec(Am)

T ∈ Rm×n

2

has full row rank. This result is proven by Wen et al. [91] by using a fix-point argument, and the authors also implemented the algorithm to compute lower bounds for frequency assign-ment, maximum stable set and binary integer quadratic programming problems. In general, the ADMM converges under the assumption: the functions involved are convex, closed, proper and the unaugmented Lagrangian −bTy + hX, A∗(y) + S − Ci has a saddle point, see Boyd et al. [12].

1.6 Contents of the thesis

Throughout this thesis, we explore both theoretical and computational aspects of the quadratic shortest path problem. In the second and the fourth chapters, we investigate the computational complexity and polynomial-time solvable special cases of the problem, respectively. The third and the fifth chapters focus on various lower bounding schemes for the QSPP. The rest of the thesis is based on the following publications and preprints with only minor adjustment.

1. H.Hu and R.Sotirov. Special cases of the quadratic shortest path problem. Journal of Com-binatorial Optimization, 35(3): 754–777, 2017.

2. H.Hu and R.Sotirov. A lower bound based on the linearization problem of the quadratic shortest path problem. arXiv:1802.02426v1

3. H.Hu and R.Sotirov. The linearization problem of binary quadratic problems and its appli-cations. arXiv:1802.02426v2

4. H.Hu and R.Sotirov. On solving the quadratic shortest path problem. INFORMS Journal on Computing, 2019. (To appear).

The author of the thesis also has a paper in the topic of extension complexity listed below. Since the problem is studied from a different perspective there, we do not include it in this thesis. The interested reader is referred to [48].

1. H.Hu and M.Laurent. On the linear extension complexity of stable set polytopes for perfect graphs. European Journal of Combinatorics, 2018.

We give a short overview of each chapter below. The rest of the thesis is divided into five self-contained chapters.

1. In the second chapter, we prove that the adjacent QSPP on cyclic digraphs cannot be approx-imated to a constant factor in polynomial time (unless P = NP). We also show that if the quadratic cost matrix is a symmetric weak sum matrix and all s-t paths have the same length, then an optimal solution for the QSPP can be obtained by solving the corresponding instance of the shortest path problem. We also show that the QSPP with a symmetric product cost matrix is solvable in polynomial time.

Further, we provide sufficient and necessary conditions for a QSPP instance on a complete symmetric digraph with four vertices to be linearizable when Q is nonnegative. We also characterize linearizable QSPP instances on complete symmetric digraphs with more than four vertices for nonnegative cost matrices. Finally, we derive an algorithm that examines whether a QSPP instance on the directed grid graph Gpq (p, q ≥ 2) is linearizable. The

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2. In the third chapter, we study an application of the linearization problem for the quadratic shortest path problem on directed acyclic graphs. By exploiting the linearization problem, we introduce a family of lower bounds for the QSPP on acyclic digraphs. The strongest lower bound from this family of bounds is the optimal solution of a linear programming problem. To the best of our knowledge, this is the first study in which the linearization problem is exploited to compute bounds for the corresponding optimization problem. Numerical results show that our approach provides the best known linear programming bound for the QSPP. We also present a lower bound for the QSPP that is derived from a sequence of problem reformulations, and prove finite convergence of that sequence. This lower bound belongs to our family of linear bounds, and requires less computational effort than the best bound from the family.

3. In the fourth chapter, we provide several applications of the linearization problem for binary quadratic problems. The key results in this chapter are generalization of some results in the previous two chapters. For example, the linearization algorithm for directed grid graphs is generalized to directed acyclic graphs. The linearization-based bounds for the QSPP on directed acyclic graphs in [49] is adapted to any binary quadratic program in a specific form. We first construct a hierarchy of semidefinite programming relaxations for binary quadratic problems based on the sum-of-squares relaxations and the linearization problem. Then, we propose a new lower bounding scheme, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be nonnegative on the feasible set. We prove that the Generalized Gilmore-Lawler bound for binary quadratic problems is a special case of one of the linearization-based bounds. Consequently, we show that the Generalized Gilmore-Lawler bound is dominated by the bound obtained from the first level reformulation linearization technique. Finally, we provide a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives the characterization of the set of linearizable matrices for the quadratic shortest path problem.

4. In the fifth chapter, we derive several semidefinite programming relaxations for the quadratic shortest path problem with a matrix variable of order m + 1, where m is the number of arcs in the graph. We use the alternating direction method of multipliers to solve the semidefinite programming relaxations. Numerical results show that our bounds are currently the strongest bounds for the quadratic shortest path problem.

We also present computational results on solving the quadratic shortest path problem using a branch and bound algorithm. Our algorithm computes a semidefinite programming bound in each node of the search tree, and solves instances with up to 1300 arcs in less than an hour. 5. In the last chapter, we provide a simplification of the SDP relaxation for the QSPP in the

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Chapter 2

Special cases of the quadratic

shortest path problem

2.1 Introduction

The shortest path problem (SPP) is the problem of finding a path between two vertices in a directed graph such that the total weight of the arcs on the path is minimized. The quadratic shortest path problem (QSPP) is the problem of finding a path between two vertices in a directed graph such that the total weight of the arcs and the sum of interaction costs over all pairs of arcs on the path is minimized.

The SPP is a well-studied combinatorial optimization problem, that can be solved in polynomial time if there do not exist negative cycles in the considered graph. There exist several efficient algorithms for solving the shortest path problem, e.g., the Dijkstra algorithm (Dijkstra [27]), and the Floyd–Warshall algorithm (Floyd [32], Warshall [90]). The SPP can be applied to various problems such as transportation planning, network protocols, plant and facility layout, robotics, VLSI design etc. On the other hand, there are not many results on the quadratic shortest path problem. In the recent paper by Rostami et al. [80] it is proven that the general QSPP cannot be approximated unless P=NP. The same result is proven for the adjacent QSPP (AQSPP), that is a variant of the QSPP. In the AQSPP interaction costs of all non-adjacent pairs of arcs are equal to zero, see Rostami et al. [80]. However, the adjacent QSPP is solvable in polynomial time for acyclic graphs and series-parallel graphs, see Rostami et al. [80].

Although the QSPP was only recently introduced, several variants of the SPP that are related to the QSPP were studied in Sen et al. [84], Sivakumar and Batta [86]. In particular Sivakumar and Batta [86] consider a variance-constrained shortest path, and Sen et al. [84] a route-planning model in which the choice of a route is based on the mean as well as the variance of the path travel-time. The QSPP is also related to the reliable shortest path problem, see e.g., Nie and Wu [69]. The QSPP appears in a study on network protocols. Namely, Murakami and Kim [67] study different restoration schemes of survivable asynchronous transfer mode networks that can be formulated as the QSPP. For a detailed overview on applications of the QSPP see Rostami et al. [80].

Buchheim and Traversi [13], and Rostami et al. [80] present several approaches to solve the gen-eral QSPP. In particular, Buchheim and Traversi [13] consider separable underestimators that can be

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exploited for solving binary quadratic programming problems, including the QSPP. Several lower bounding approaches for the QSPP, including a Glimore-Lawler-type bound and reformulation-based bound are presented in Rostami et al. [80]. In this chapter we do not investigate computa-tional aspects for solving the QSPP in general.

Main results and outline.

In Section 2.2, we formulate the quadratic shortest path problem as an integer programming prob-lem. Complexity results for the general and adjacent QSPP are given in Section 2.3. In particular, in Section 2.3.1 we derive a new polynomial-time reduction from the well known quadratic assign-ment problem (QAP) to the QSPP. Our reduction differs from the one given by Rostami et al. [79]. Namely, our approach results in an instance for the QSPP with n2 _{arcs, while the reduction from}

Rostami et al. [79] derives an instance with O(n3_{) arcs. Here, n is the order of the data matrices in}

the quadratic assignment problem. The here presented polynomial-time reduction from the QAP, in combination with the library of the QAP Burkard et al. [16], provides a source of difficult QSPP test instances.

In Section 2.3.2, we describe the polynomial-time algorithm for solving the adjacent QSPP from Rostami et al. [79]. We also show that the algorithm fails for the adjacent QSPP considered on directed cyclic graphs, while it performs well on directed acyclic graphs (DAGs). Further, we provide a polynomial-time reduction from the 2-arc-disjoint paths problem, that is known to be NP-complete, to the adjacent QSPP considered on a directed cyclic graph. Our proof of inapprox-imability is considerably simpler than the proof from Rostami et al. [80].

In Section 2.4, we consider special cases of the QSPP. We first consider linearizable instances; that is, when there exists a corresponding instance of the SPP such that the associated costs for both problems are equal for every feasible path. It is easy to see that the QSPP considered on a directed cycle is linearizable. Here, we also show that a QSPP instance on a digraph for which every s-t path has the same length and whose quadratic cost matrix is a symmetric weak sum matrix is linearizable. Finally, we prove that a solution of the QSPP whose quadratic cost matrix with linear costs included on the diagonal is a nonnegative symmetric product matrix can be obtained by solving the corresponding SPP.

We provide sufficient and necessary conditions for an instance of the QSPP on a complete digraph with four nodes to be linearizable in Section 2.5. In the same section we give several properties of linearizable QSPP instance on complete digraphs with more than four nodes.

In Section 2.6, we present an algorithm that examines whether a QSPP instance on the directed grid graph Gp,q(p, q ≥ 2) is linearizable. If the instance is linearizable, then our algorithm provides

the corresponding linearization vector. The complexity of the algorithm is O(p3_q2_{+ p}2_q3_).

2.2 Problem formulation

Let G = (V, A) be a directed graph with vertex set V (|V | = n) and arc set A (|A| = m). A walk is defined as an ordered set of vertices (v1, . . . , vk), k > 1 such that (vi, vi+1) ∈ A for i = 1, . . . , k − 1.

The length of a walk equals to the number of visited arcs. A walk is called a path if it does not contain repeated vertices. Given a source vertex s ∈ V and a target vertex t ∈ V , an s-t path is a path P = (v1, v2, . . . , vk) such that v1= s and vk= t.

The quadratic cost of an s-t path P is calculated as follows. We are given a nonnegative vector c ∈ Rm

+ indexed by the arc set A, and a nonnegative symmetric matrix of order m Q = (qe,f) with

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2.3. COMPLEXITY RESULTS FOR THE GENERAL AND ADJACENT QSPP 13

to non-negative cost matrices here. An arc e ∈ P has the linear cost (weight) ce, and a pair of arcs

e, f ∈ P the interaction cost 2qef. The total cost of the s-t path P is given by

C(P, c, Q) = X e,f ∈P qef + X e∈P ce. (2.1)

If Q is a zero-matrix, then the cost of an s-t path P is denoted by C(P, c). We assume that the graph G does not contain a directed cycle whose total cost is zero.

Let us introduce the quadratic shortest path problem in a formal way. Let P be an s-t path, and x a binary vector of length m such that xij is one if the arc (i, j) ∈ A is on the s-t path P and

zero otherwise. Now, the quadratic cost of the s-t path P with the characteristic vector x, is given by X (i,j),(k,l)∈A qij,klxijxkl+ X (i,j)∈A cijxij = xTQx + cTx.

Given a vertex i ∈ V , the set of predecessor and successor vertices of i are denoted by δ−(i) := {j ∈ V | (j, i) ∈ A} and δ+_{(i) := {j ∈ V | (i, j) ∈ A}, respectively. The path polyhedron is defined}

as follows: Pst(G) := {x ∈ Rm| X j∈δ+_(i) xij− X j∈δ−_(i) xji= bi ∀i ∈ V, 0 ≤ x ≤ 1}, (2.2)

where b is a vector of length n such that bi= 1 if i = s, bi= −1 if i = t, and bi= 0 if i ∈ V \{s, t}.

Now the QSPP can be modeled as the following quadratic integer programming problem: minimize xTQx + cTx

subject to x ∈ Pst(G) ∩ {0, 1}m.

(2.3)

Since there are no cycles of cost zero in G, the optimal solution of (2.3) is guaranteed to be an s-t path. If Q is a zero-matrix, then the problem reduces to the shortest path problem. Due to the flow conservation law, one can remove one of the equations in Pst(G).

A QSPP instance I (resp. SPP instance I0) can be specified by the tuple I = (G, s, t, c, Q) (resp. I0 = (G, s, t, c)). Note that we use both, e and (i, j), to denote an arc e = (i, j). Sometimes one is more convenient than another.

2.3 Complexity results for the general and adjacent QSPP

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2.3.1 The general QSPP

In Rostami et al. [79] it is proven that the QSPP is NP-hard by providing a polynomial-time reduc-tion from the QAP. The size of so constructed QSPP instance in Rostami et al. [79] is considerably larger than the size of the input QAP instance. In particular, if a QAP instance consists of n facilities and n locations, then the constructed QSPP instance as described in Rostami et al. [79] has n2_{+ 2 vertices and n}3_{− 2n}2_{+ 3n arcs.}

We present here another polynomial-time reduction from the QAP to the QSPP. Our reduction yields a QSPP instance with n + 1 vertices and n2arcs, where n is the order of the data matrices in the QAP. This enables us to derive QSPP test instances of reasonable sizes, from the QAP instances given in the QAP library Burkard et al. [16]. Note that the QAP library contains solutions and/or bounds for many QAP instances, and is therefore a source of test instances for the QSPP.

The quadratic assignment problem is the following optimization problem:

min    X i,j,k,l aikbjlxijxkl+ X i,j cijxij : X = (xij), X ∈ Πn    ,

where A = (aik), B = (bjl) are given symmetric n × n matrices, C = (cij) ∈ Rn×n, and Πn is the

set of n × n permutation matrices.

The quadratic assignment problem has the following interpretation. Suppose that there are n facilities and n locations. The flow between each pair of facilities, say i, k, and the distance between each pair of locations, say j, l, are given by aik and bjl, respectively. The cost of placing a facility

i to location j is cij. The QAP problem is to find an assignment of facilities to locations such

that the sum of the distances multiplied by the corresponding flows together with the total cost is minimized.

Let us now allow a directed multigraph in the definition of the QSPP problem. A multigraph is a graph which is permitted to have multiple arcs between any pair of vertices. We can now prove the following theorem.

Theorem 2.3.1. There exists a polynomial time reduction from QAP to QSPP, i.e., QAP ∝ QSP P .

Proof. Suppose we are given a QAP instance with n × n input matrices A, B, C. Assume without loss of generality that all diagonal entries of A and B are equal to zero. (Note that one could “shift” the diagonal elements appropriately to the linear cost matrix C.) We construct a QSPP instance on the graph G = (V, A) whose vertex and arc sets are defined as follows:

V := {wj: j = 1, . . . , n + 1},

A := {(wj, wj+1)i: i, j = 1, . . . , n},

where the superscript i indicates the ith arc between vertices wjand wj+1. The starting and ending

vertices are s = w1 and t = wn+1, respectively.

The linear cost of the arc e is defined as

ce:=

(

cij if e = (wj, wj+1)i

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The interaction cost between the pair of arcs e = (wj, wj+1)i and f = (wl, wl+1)k is defined:

qe,f :=

(

aik· bjl if j 6= l

M if j = l and i 6= k,

where M is a big number. All the other pairs of arcs have zero interaction costs.

If we have a feasible QAP instance, say facility i is mapped into location π(i), then we take the feasible QSPP instance with arcs (wπ(i), wπ(i)+1)i (i = 1, . . . , n). By construction, those two

instances have the same objective value. Conversely, every s-t path in G with objective value less than M corresponds to the assignment for the QAP with the same objective value.

It follows from the above construction that for a given input instance of the QAP with n × n data matrices, one can construct a QSPP instance with n + 1 vertices and n2 _{arcs. We note that}

the polynomial reduction from Theorem 2.3.1 is valid also for the undirected version of the QSPP.

2.3.2 The adjacent QSPP restricted to DAGs

The adjacent QSPP is a variant of the QSPP, where interaction costs of all non-adjacent pairs of arcs are equal to zero. In other words, only the interaction cost of the form qij,kl with j = k and

i 6= l, or with i = l and j 6= k can have nonzero value. A polynomial time algorithm that solves instances of the adjacent QSPP on directed acyclic graphs is presented in Rostami et al. [79]. It is actually stated in Rostami et al. [79] that the proposed algorithm finds an optimal solution for the AQSPP on any digraph in polynomial time, which turns out to be true only for directed acyclic graphs.

In this section we first describe the approach from Rostami et al. [79], and then provide an example to show that the algorithm fails if the graph under consideration is not acyclic.

Let G be a directed acyclic graph and I = (G, s, t, c, Q) an instance of the AQSPP. We construct the auxiliary graph G0 = (V0, A0) from G = (V, A) in the following way:

V0 := {V(s,s), V(t,t)} ∪ {Ve| e ∈ A}, A0:= {(V(i,j), V(j,l)) | i 6= l}, (2.4)

where V(s,s)and V(t,t) represent vertices s and t, respectively. The costs of the arcs in the graph G0

are given as follows

c0_(V_e_,V f)=      cf if e = (s, s) 0 if f = (t, t) cf+ 2qe,f otherwise.

Now, the auxiliary instance I0 of I is the following SPP instance I0 = (G0, V(s,s), V(t,t), c0).

The following theorem shows that the optimal s-t path for the AQSPP instance I on a directed acyclic graph can be obtained by solving the SPP instance I0.

Theorem 2.3.2. Let G be a directed acyclic graph, I = (G, s, t, c, Q) an AQSPP instance, and I0 _{= (G}0_{, V}

(s,s), V(t,t), c0) the auxiliary instance of I. Then, an optimal solution of I can be obtained

by solving the SPP for I0.

Proof. (See also Rostami et al. [79].) We first show that any s-t path in G corresponds to a V(s,s)-V(t,t) path in G0, and vice-versa. For ease of notation we set v1 := s and vk := t. Let

P = (v1, v2, . . . , vk) be a v1-vk path in G. Then it is not difficult to verify that

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is a V(v1,v1)-V(vk,vk)path in the graph G

0_{. The cost of the path P}0 _{is given by}Pk−1

i=1 c(vi,vi+1)+ 2 ·

Pk−2

i=1 q(vi,vi+1),(vi+1,vi+2), which is exactly the cost of the path P .

Conversely, let P0 = (V(v1,v1), V(v1,v2), V(v2,v3), . . . , V(vk−1,vk), V(vk,vk)) be a V(v1,v1)-V(vk,vk) path

in G0. Take the ordered set of vertices P = (v1, v2, . . . , vk). Let us verify that P is a walk that does

not contain repeated vertices. From the definition of V0 and A0, see (2.4), it follows that vi ∈ V

for all i, and (vi, vi+1) ∈ A for i = 1, . . . , k − 1. It remains now to verify that there do not exist k, l

(k 6= l) for which vk = vl. Indeed, since G is acyclic this is not possible. Thus P is an s-t path in

G whose total cost equals to the linear cost of P0.

Note that if G is not acyclic, then there may exist a V(s,s)-V(t,t) path in the auxiliary graph for

which does not exist a corresponding s-t path in G. Let us give an example.

Example 2.3.3. Consider a QSPP instance on the directed graph G from Figure 2.1. The costs are given as follows. Set c(3,4) = for some 0 < < 1 and q(1,2),(2,5)= 1. All other linear and

interaction costs are zero. Set for the source and target vertex s = 1 and t = 5, respectively. Clearly, we have a well defined AQSPP instance. Moreover, P = (1, 2, 5) is the unique s-t path in G, whose cost is two.

1 2

3 4

5 Figure 2.1: Example graph G.

We construct the graph G0 from G, see Figure 2.2. It is not difficult to verify that (V(1,1), V(1,2), V(2,3), V(3,4), V(4,2), V(2,5), V(5,5))

is a shortest V(1,1)-V(5,5) path in G0, whose cost is , for which no corresponding s-t path exists in

G.

V(1,1) V(1,2)

V(2,3) V(3,4) V(4,2)

V(2,5) V(5,5)

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2.3.3 The general adjacent QSPPs

In this section, we prove that the adjacent QSPP that is not restricted to DAGs cannot be approx-imated unless P=NP. In particular, we show that the 2-arc-disjoint paths problem polynomially transforms to the AQSPP. Rostami et al. [80] provide a polynomial-time reduction from 3SAT to the AQSPP. Our reduction is considerably shorter (and simpler) than the reduction from Rostami et al. [80].

The k-arc-disjoint paths problem is defined as follows: Let G = (V, A) be a directed graph and (s1, t1), . . . , (sk, tk) pairs of vertices in G. The k-arc-disjoint paths problem asks for pairwise

arc-disjoint paths P1, . . . , Pk where Pi is an si-ti path (i = 1, . . . , k).

An instance of the k-arc-disjoint paths problem can be specified via the tuple I = (G, (s1, t1), . . . , (sk, tk)).

Fortune et al. [33] prove that the k-arcs-disjoint paths problem is NP-complete even for k = 2. We use this to prove the main result in this section.

Theorem 2.3.4. The adjacent QSPP on a cyclic digraph cannot be approximated within a constant factor unless P=NP.

Proof. We provide a polynomial-time reduction from the 2-arc-disjoint paths problem to the adja-cent QSPP. Let I = (G, (s, t), (¯s, ¯t)) such that s 6= ¯s and t 6= ¯t be an instance of the 2-arc-disjoint paths problem on the graph G = (V, A).

We construct a directed graph G0= (V0, A0), where the vertex set is given as follows V0 =v1_{, v}2_{| v ∈ V } ∪ {N}

uv| (u, v) ∈ A .

In particular, for each vertex v ∈ V there are two vertices v1_{, v}2_{in V}0_{, and for each arc (u, v) ∈ A}

there is the vertex Nuv ∈ V0. The arc set A0 is given by

A0 =(ui_{, N}

uv), (Nuv, vi) | (u, v) ∈ A, i = 1, 2} ∪ {(t1, ¯s2) ,

i.e., for each arc (u, v) ∈ A there are two pairs of arcs (ui_{, N}

uv), (Nuv, vi) (i = 1, 2), and additionally

the arc (t1_{, ¯}_s2_).

Now we define the cost functions c0 : A0 → R+ and Q0 : A0× A0 → R+, Q0 = (qef0 ) as follows.

The linear cost of each arc in A0 is zero, i.e., c0(e) = 0 for every e ∈ A0. For every two pairs of arcs (ui_{, N}

uv) and (Nuv, vi), (i = 1, 2) the interaction cost is:

q_(u0 i_,N

uv)(Nuv,vj)=

(

0 if i = j 1 if i 6= j.

The interaction costs of all other pairs of arcs are zero. Clearly, the non-adjacent arcs have zero interaction costs. Now, the constructed AQSPP instance is I0= (G0, s1_{, ¯}_t2_{, c}0_{, Q}0_).

It remains to show that I is a yes-instance of the 2-arc-disjoint paths problem on G if and only if I0 has an s1_-¯_t2 _{path of cost zero. Now if I is a yes-instance, then there exists an s-t path}

P1= (v1, v2, . . . , vk) of length, say, k−1 with v1:= s and vk:= t, and a ¯s-¯t path P2= (u1, u2, . . . , ul)

of length, say, l − 1 with u1 := ¯s and ul := ¯t, such that P1 and P2 are arc-disjoint. Clearly, the

following ordered set of vertices

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is a v1

1-u2l path of cost zero.

Conversely, an s1_-¯_t2 _{path P}0 _{in G}0 _{of cost zero has to consist of a sequence of vertices with}

superindex one followed by a sequence of vertices with superindex two, as specified in (2.5). We take the following ordered sets of vertices P1 = (v1, v2, . . . , vk) and P2 = (u1, u2, . . . , ul) in G. It

is not difficult to verify that P1 and P2 are two paths. It remains to show that P1 and P2 are

arc-disjoint. Let us assume this is not the case. Then there exists an arc (q, w) ∈ A visited by both paths P1 and P2. From the construction of P0, it follows that the arcs (q1, Nq,w), (Nq,w, w1),

(q2, Nq,w) and (Nq,w, w2) are visited by the path P0. This means that the vertex Nq,w is visited

twice in the s1-¯t2 path P0 in G0, which contradicts to the fact that P0 is a walk that does not contain repeated vertices.

Finally, the inapproximability result follows since the objective value of any feasible solution of the constructed AQSPP is either zero or at least two. Indeed, if the AQSPP can be approximated to a constant factor in polynomial time, then the reduction above provides us a polynomial time algorithm for the 2-arc-disjoint paths problem.

2.4 Polynomially solvable cases of the QSPP

In this section we investigate polynomially solvable cases of the quadratic shortest path problem. Here we prove, among other things, that all QSPP instances on a directed cycle can be solved by solving an appropriate instance of the SPP. In Section 2.4.1 we consider special cost matrices such as sum and product matrices.

An instance of the QSPP is said to be linearizable if there exists a corresponding instance of the SPP with the cost vector c0≥ 0 such that associated costs are equal i.e.,

C(P, c, Q) = C(P, c0),

for every s-t path P in G. We call c0 a linearization vector of the QSPP instance. Linearizable instances for the quadratic assignment problem were considered in e.g., Adams and Waddell [1], Cela et al. [19], and linearizable instances for the quadratic minimum spanning tree problem in ´Custi´c and Punnen [22].

We start this section by proving several basic results.

Lemma 2.4.1. If the QSPP instance I = (G, s, t, c, Q) is linearizable, and d is a vector such that c + d ≥ 0, then the QSPP instance I0 = (G, s, t, c + d, Q) is also linearizable.

Proof. Since I is linearizable, there exists a linear cost vector c0 such thatP

e,f ∈Pqef+Pe∈Pce=

P

e∈Pc 0

efor every s-t path P in G. Let c00:= c0+ d, then we have

C(P, c + d, Q) = X e,f ∈P qef + X e∈P (ce+ de) = X e∈P c0_e+ de= X e∈P c00_e= C(P, c00)

for every s-t path P . Thus I0 is also linearizable.

Lemma 2.4.2. If two QSPP instance I1= (G, s, t, c1, Q1) and I2= (G, s, t, c2, Q2) are

lineariz-able, then the QSPP instance I3= (G, s, t, α1c1+ α2c2, α1Q1+ α2Q2) is also linearizable for all

nonnegative scalars α1, α2.

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2.4. POLYNOMIALLY SOLVABLE CASES OF THE QSPP 19

An instance of the QSPP may be linearizable if the underlying graph has special structure and/or the corresponding quadratic cost matrix has special properties. Let us give a class of the QSPP instances that is linearizable for any pair of cost matrices (Q, c). The directed cycle C_n∗ of order n is a graph with the vertex set {v1, . . . , vn} and arc set {(vi, vi+1) | i = 1, . . . , n} where

addition is modulo n. It is not difficult to verify that any QSPP instance on C_n∗ is linearizable, as there is only one path.

Directed cycles are not the only digraphs on which QSPP instances are linearizable. However, they seem to be easiest cases. In the following sections we present necessary conditions for which a QSPP instance on a directed complete graph with more than four nodes is linearizable. We also show that every instance on a tournament with four vertices is linearizable.

2.4.1 Special cost matrices

If the interaction cost matrix has a special structure, then the associated QSPP instance may be solved efficiently. We consider here two types of cost matrices for which QSPP instances are linearizable.

We say that a matrix M ∈ Rm×n

is a sum matrix generated by vectors a ∈ Rm

and b ∈ Rn _if

Mi,j = ai+ bj for every i = 1, . . . , m and j = 1, . . . , n. A matrix is called a weak sum matrix if

the condition above is not required for the diagonal entries. It is not difficult to show that if a sum matrix M of order n is symmetric, then there exists a vector a ∈ Rn _{such that M}

i,j = ai+ aj for

all i, j = 1, . . . , n. Similarly, if the weak sum matrix M of order n is symmetric, then there exists a vector a ∈ Rn such that Mi,j= ai+ aj for all i, j = 1, . . . , n, i 6= j. Recognition of a (weak) sum

matrix can be done efficiently. Sum matrices are also considered in the context of the QAP in Cela et al. [19], and the quadratic minimum spanning tree problem in ´Custi´c and Punnen [22].

The following result shows that symmetric weak sum matrices provide linearizable QSPP in-stances on graphs where all s-t paths have the same length.

Proposition 2.4.3. Let I = (G, s, t, c, Q) be a QSPP instance. If every s-t path in G has the same length and Q is a symmetric weak sum matrix, then I is linearizable.

Proof. Suppose that every s-t path in G has length L. Assume without loss of generality, the diagonal entries of Q are zeros. Since Q is a symmetric weak sum matrix, there exists a vector a ∈ Rm_{such that q}

e,f = ae+ af for all e, f = 1, . . . , m, e 6= f . Let P be an s-t path in G with the

arc set {ei: i = 1, . . . , L}. Then the cost of P is:

C(P, c, Q) = L X i=1 L X j=1 qei,ej + L X i=1 cei= L X i=1 L X j=1,i6=j (aei+ aej) + L X i=1 cei = 2(L − 1) L X i=1 aei+ L X i=1 cei= L X i=1 2(L − 1)aei+ cei.

Now, define the linear cost c0_e := 2(L − 1)ae+ ce for every arc e = 1, . . . , m. Thus, C(P, c, Q) =

C(P, c0) for every s-t path P in G.

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Example 2.4.4. The directed grid graph Gp,q = (V, A) is defined as follows. The set of vertices

and the set of arcs are given as follows:

V = {vi,j | 1 ≤ i ≤ p, 1 ≤ j ≤ q} ,

A = {(vi,j, vi+1,j) | 1 ≤ i ≤ p − 1, 1 ≤ j ≤ q}

∪ {(vi,j, vi,j+1) | 1 ≤ i ≤ p, 1 ≤ j ≤ q − 1} .

(2.6)

Note that |V | = pq and |A| = 2pq − p − q. If s = (1, 1) and t = (p, q), then every s-t path has length p + q − 2.

Example 2.4.5. The directed hypercube graph Hn is defined as follows. There is a vertex for each

binary string of length n, there is an arc (u, v) if the vertices u and v differ in exactly one bit position and the binary value of u is less than the binary value of v. Note that Hn has 2n vertices,

2n−1n arcs. If s is an all-zeros string and t is an all-ones string, then every s-t path has length n. Let us consider another special cost matrix. A matrix M ∈ Rm×m_{is called a symmetric product}

matrix if M = aaT

for some vector a ∈ Rm_{. A nonnegative product matrix with integer values}

plays a role in the Wiener maximum QAP, see Cela et al. [18].

If Q + Diag(c) is a positive semidefinite matrix, then the tuple (G, s, t, c, Q) is a convex QSPP instance. Here, the ’Diag’ operator maps a m-vector to the diagonal matrix, by placing the vector on the diagonal of the m × m matrix. In Rostami et al. [80], it is shown that the convex QSPP is APX-hard, but can be approximated within a factor of |V |. The following result shows that one can solve the QSPP efficiently whenever Q + Diag(c) is a symmetric product matrix, thus positive semidefinite matrix of rank one.

Proposition 2.4.6. Let I = (G, s, t, c, Q) be a QSPP instance. If Q + Diag(c) is a nonnegative symmetric product matrix, then I is solvable in O(m + n log n) time.

Proof. Since Q + Diag(c) is a nonnegative symmetric product matrix, there exists a vector a ∈ Rm +

such that Q + Diag(c) = aaT_{. Let x ∈ {0, 1}}m _{be the characteristic vector of an s-t path P in G.}

The cost of this path satisfies

C(P, c, Q) = xT(Q + Diag(c))x = xTaaTx = (xTa)2. Let us define the linear cost vector c0 ∈ Rm

+ by taking c0e = ae for every e. Thus, we have

C(P, c, Q) = C(P, c0)2 for every s-t path P in G, and the complexity of solving the shortest path problem by Dijkstra’s algorithm is O(m + n log n).

2.5 The QSPP on complete digraphs

In this section we analyze the QSPP on the complete symmetric digraph K_n∗, that is a digraph in which every pair of vertices is connected by a bidirectional edge. It is trivial to solve the QSPP on K_n∗ for n ≤ 3. Here, we provide sufficient and necessary conditions for a QSPP instance on K₄∗ to be linearizable. We also provide several properties of linearizable QSPP instances on K_n∗ with n ≥ 5.

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2.5. THE QSPP ON COMPLETE DIGRAPHS 21

arcs from K₄∗results in the simplified graph, see Figure 2.3. Note that removing the arc (s, t) does not change the linearizability of an instance. Further, we assume that the cost vector c is the all-zero vector (see Lemma 2.4.1), and that interaction costs of pairs of arcs that can not be together included in any s-t path is zero. For example, the interaction cost of arcs (v1, v2) and (v3, v2) in

graph from Figure 2.3 is zero.

v1

v2

v3

v4

Figure 2.3: Simplified K₄∗ with s = v1, t = v4.

Let I = (K_n∗, s, t, c, Q) (n ≥ 4) be an instance of the QSPP. We classify s-t paths by their lengths for that instance. This classification leads us to necessary and/or sufficient conditions for a QSPP instance to be linearizable.

Let Pk denotes the set of s-t paths of length k for k ∈ {2, . . . , n − 1}. The total cost of all

s-t paths of length k is CPk = P_{P ∈P}_kC(P, c, Q). The number of s-t paths of the length k is

|Pk| = n−2_k−1 · (k − 1)!. In what follows, we show that CPk is bounded from above by CPk+1 for

k = 3, . . . , n − 2, and several other related results.

Proposition 2.5.1. Let I = (K_n∗, s, t, c, Q) be a QSPP instance and n ≥ 4. Then the average cost of all s-t paths of the length k is not greater than the average cost of all s-t paths of the length k + 1, i.e., 1 |Pk| CPk≤ 1 |Pk+1| CPk+1, for k = 3, . . . , n − 2.

Proof. We will derive an expression for 1

|Pk|CPk (k ≥ 2) in terms of the interaction costs. Then,

the claim follows from the fact that the expression is an increasing function for k ≥ 3.

Given an arc e = (i, j), we define h(e) = i to be the head vertex i, and t(e) = j to be the tail vertex j. Let

H = {e ∈ A | h(e) = s or t(e) = t} (2.7) be the set of arcs either leaving s or entering t. Let S = {{e, f } | t(e) = h(f ) or h(e) = t(f )} be the set of distinct unordered pairs of adjacent arcs. Based on the sets H and S, we define the following sets of distinct unordered pairs of arcs:

T1= {{e, f } ∈ S | e ∈ H and f ∈ H}, T2= {{e, f } /∈ S | e ∈ H and f ∈ H},

T3= {{e, f } ∈ S | e ∈ H and f /∈ H}, T4= {{e, f } /∈ S | e ∈ H and f /∈ H},

T5= {{e, f } ∈ S | e /∈ H and f /∈ H}, T6= {{e, f } /∈ S | e /∈ H and f /∈ H}.

Clearly, H and its complement partition the arc set, and T1, . . . , T6 partition the arc pairs in Kn∗.

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si= 2 ·P_{{e,f }∈T}_iqef for i = 1, . . . , 6. Note that P6_i=1si= uTQu, where u ∈ Rm is the vector of

all-ones.

It holds that every pair of arcs {e, f } ∈ Ti (i ∈ {1, . . . , 6}), which is included in at least one s-t

path, is contained in the same number, denoted by ti,k, of s-t paths of length k. In particular we

have t1,k= ( 1 if k = 2 0 otherwise, t2,k= ( _n−4 k−3 · (k − 3)! if k ≥ 3 0 otherwise, t3,k= ( _n−4 k−3 · (k − 3)! if k ≥ 3 0 otherwise, t4,k= ( _n−5 k−4 · (k − 3)! if k ≥ 4 0 otherwise, t5,k= ( _n−5 k−4 · (k − 3)! if k ≥ 4 0 otherwise, t6,k= ( _n−6 k−5 · (k − 3)! if k ≥ 5 0 otherwise. For k ≥ 2, the average cost of all s-t paths of length k can be written as:

1 |Pk| CPk = 1 |Pk| 6 X i=1 ti,k· si (2.8) = 1 n − 2· s1· 1{k=2}+ 1 (n − 2)(n − 3)· (s2+ s3) · 1{k≥3} + (k − 3) (n − 2)(n − 3)(n − 4)· (s4+ s5) · 1{k≥4} + (k − 3)(k − 4) (n − 2)(n − 3)(n − 4)(n − 5) · s6· 1{k≥5}.

Here, 1A is the indicator function defined as 1x= 1 if condition x is true, and zero otherwise. It is

clear that _|P1

k|CPk is an increasing function in k ≥ 3 and this finishes the proof.

Note that from (2.8) it follows that one can easily compute CPkfor k ≥ 2. As direct consequences

of the previous proposition, we have the following two results.

Corollary 2.5.2. a) Let I = (K_n∗, s, t, c, Q) be a QSPP instance and n ≥ 4. Then CPk≤

1

n − k − 1CPk+1, for k = 3, . . . , n − 2.

b) Let I = (K_n∗, s, t, c, Q) be a QSPP instance and n ≥ 7. Then CPk ≤ (n − k) ·

k − 3

k − 5· CPk−1 for k = 6, . . . , n − 1.

Proof. The first part follows directly from Proposition 2.5.1. To show the second part, note that ti,k in the proof of Proposition 2.5.1 satisfy _tt2,k