The Wiener maximum quadratic assignment problem

Çela, Eranda; Schmuck, Nina Sabine; Wimer, Shmuel; Woeginger, Gerhard J.

doi:10.1016/j.disopt.2011.02.002

Cited by 35 publications

(26 citation statements)

References 11 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%

“…is a v 1 1 -u 2 l path of cost zero. Conversely, a s 1 -t 2 path P ′ in G ′ 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 (5). We take the following ordered sets of vertices P 1 = (v 1 , v 2 , .…”

Section: Now We Define the Cost Functions Cmentioning

confidence: 99%

“…A matrix M ∈ R m×m is called a symmetric product matrix if M = aa T for some vector a ∈ R m . A nonnegative product matrix with integer values plays a role in the Wiener maximum QAP, see [5].…”

Section: Special Cost Matricesmentioning

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%

Section: Now We Define the Cost Functions Cmentioning

confidence: 99%

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Special cases of the quadratic shortest path problem

Sotirov

2017

J Comb Optim

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“…This problem is similar to the Wiener-type QAP introduced in [10], and is NP-complete, since it is a generalization of the classic partition problem (Setting q = 2, n = 2 + 2k, µ 1 = µ 2 = 0, d 1 = d 2 = k makes a partition problem for 2k elements. )…”

Section: Upper-bound Estimatementioning

confidence: 99%

Maximizing Wiener index for trees with given vertex weight and degree sequences

Goubko

2018

Applied Mathematics and Computation

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The Wiener index is maximized over the set of trees with the given vertex weight and degree sequences. This model covers the traditional "unweighed" Wiener index, the terminal Wiener index, and the vertex distance index. It is shown that there exists an optimal caterpillar. If weights of internal vertices increase in their degrees, then an optimal caterpillar exists with weights of internal vertices on its backbone monotonously increasing from some central point to the ends of the backbone, and the same is true for pendent vertices. A tight upper bound of the Wiener index value is proposed and an efficient greedy heuristics is developed that approximates well the optimal index value. Finally, a branch and bound algorithm is built and tested for the exact solution of this NP-complete problem.Keywords: Wiener index for graph with weighted vertices, upper-bound estimate, greedy algorithm, optimal caterpillar 2010 MSC: 05C05, 05C12, 05C22, 05C35, 90C09, 90C35, 90C57 NomenclatureThis section introduces the basic graph-theoretic notation. The vertex set and the edge set of a simple connected undirected graph G are denoted with V (G) and E(G) respectively, and the degree (i.e., the number of incident edges) of vertex v ∈ V (G) in graph G is denoted with d G (v). Let W (G) be the set of pendent vertices (those having degree one) of graph G, and let $

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“…In general, the QAP is extremely difficult to solve and hard to approximate. One branch of research on the QAP concentrates on the algorithmic behavior of strongly structured special cases; see for instance Burkard & al [4], Deineko & Woeginger [17], Ç ela & al [12], Ç ela, Deineko & Woeginger [9], or Laurent and Seminaroti [24] for typical results in this direction. In our paper we follow recent developments and represent several new results in this exciting area of research.…”

Section: Introductionmentioning

confidence: 99%

New special cases of the Quadratic Assignment Problem with diagonally structured coefficient matrices

Çela

Deı̆neko

Woeginger

2018

European Journal of Operational Research

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We consider new polynomially solvable cases of the well-known Quadratic Assignment Problem involving coefficient matrices with a special diagonal structure. By combining the new special cases with polynomially solvable special cases known in the literature we obtain a new and larger class of polynomially solvable special cases of the QAP where one of the two coefficient matrices involved is a Robinson matrix with an additional structural property: this matrix can be represented as a conic combination of cut matrices in a certain normal form. The other matrix is a conic combination of a monotone anti-Monge matrix and a down-benevolent Toeplitz matrix. We consider the recognition problem for the special class of Robinson matrices mentioned above and show that it can be solved in polynomial time.

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The Wiener maximum quadratic assignment problem

Cited by 35 publications

References 11 publications

Special cases of the quadratic shortest path problem

Special cases of the quadratic shortest path problem

Maximizing Wiener index for trees with given vertex weight and degree sequences

New special cases of the Quadratic Assignment Problem with diagonally structured coefficient matrices

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