Linear programming insights into solvable cases of the quadratic assignment problem

Adams, Warren P.; Waddell, Lucas

doi:10.1016/j.disopt.2014.07.001

Cited by 16 publications

(12 citation statements)

References 34 publications

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“…We construct the graph G ′ from G, see Figure 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 the shortest V (1,1) -V (5,5) path in G ′ , whose cost is ǫ. However this V (1,1) -V (5,5) path does not correspond to a path in G, but to a walk.…”

Section: The Adjacent Qspp Restricted To Dagsmentioning

confidence: 99%

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Special cases of the quadratic shortest path problem

Sotirov

2017

J Comb Optim

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The quadratic shortest path problem (QSPP) is the problem of finding a path with prespecified start vertex s and end vertex t in a digraph such that the sum of weights of arcs and the sum of interaction costs over all pairs of arcs on the path is minimized. We first consider a variant of the QSPP known as the adjacent QSPP. It was recently proven that the adjacent QSPP on cyclic digraphs cannot be approximated unless P = NP. Here, we give a simple proof for the same result. 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. Similarly, it is shown 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. We also characterize linearizable QSPP instances on complete symmetric digraphs with more than four vertices. Finally, we derive an algorithm that examines whether a QSPP instance on the directed grid graph G pq ( p, q ≥ 2) is linearizable. The complexity of this algorithm is O( p 3 q 2 + p 2 q 3 ).

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Section: The Adjacent Qspp Restricted To Dagsmentioning

confidence: 99%

“…for every s-t path P in G. We call c ′ a linearization vector of the QSPP instance. Linearizable instances for the quadratic assignment problem were considered in e.g., [1,6], and linearizable instances for the quadratic minimum spanning tree problem in [2].…”

Section: Polynomially Solvable Cases Of the Qsppmentioning

confidence: 99%

Special cases of the quadratic shortest path problem

Sotirov

2017

J Comb Optim

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“…The linearization problem for the quadratic assignment problem (QAP) was considered by Kabadi and Punnen [13], Adams and Wadell [1], and C ¸ela et al [4]. The special case of Koopmans-Beckman QAP linearization problem was studied by Punnen and Kabadi [24] and C ¸ela et al [4].…”

Section: Introductionmentioning

confidence: 99%

A characterization of linearizable instances of the quadratic minimum spanning tree problem

Ćustić¹,

Punnen²

2015

Preprint

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We investigate special cases of the quadratic minimum spanning tree problem (QMSTP) on a graph G = (V, E) that can be solved as a linear minimum spanning tree problem. Characterization of such problems on graphs with special properties are given. This include complete graphs, complete bipartite graphs, cactuses among others. Our characterization can be verified in O(|E| 2 ) time. In the case of complete graphs and when the cost matrix is given in factored form, we show that our characterization can be verified in O(|E|) time. Related open problems are also indicated.

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“…However, some specific cases of QAP are easy to solve [8,9]. Many exact and heuristic methods have been developed to solve different cases of QAP.…”

Section: Introductionmentioning

confidence: 99%

Robust quadratic assignment problem with budgeted uncertain flows

Feizollahi

Feyzollahi

2015

Operations Research Perspectives

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h i g h l i g h t s• A robust approach for quadratic assignment problem (RQAP) with budgeted uncertainty. • An exact and two heuristic methods to solve RQAP.• Extensive experiments to show performance of methods and quality of solutions.• RQAP can be solved significantly faster than minmax regret QAP. • RQAP has adjustable conservativeness while minmax regret QAP has not. a b s t r a c tWe consider a generalization of the classical quadratic assignment problem, where material flows between facilities are uncertain, and belong to a budgeted uncertainty set. The objective is to find a robust solution under all possible scenarios in the given uncertainty set. We present an exact quadratic formulation as a robust counterpart and develop an equivalent mixed integer programming model for it. To solve the proposed model for large-scale instances, we also develop two different heuristics based on 2-Opt local search and tabu search algorithms. We discuss performance of these methods and the quality of robust solutions through extensive computational experiments.

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Linear programming insights into solvable cases of the quadratic assignment problem

Cited by 16 publications

References 34 publications

Special cases of the quadratic shortest path problem

Special cases of the quadratic shortest path problem

A characterization of linearizable instances of the quadratic minimum spanning tree problem

Robust quadratic assignment problem with budgeted uncertain flows

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