A linear time algorithm for the Koopmans–Beckmann QAP linearization and related problems
TLDR
It is shown that Bookhold’s condition is also necessary for linearizability of symmetric Koopmans–Beckmann QAP.About:
This article is published in Discrete Optimization.The article was published on 2013-08-01 and is currently open access. It has received 32 citations till now. The article focuses on the topics: Generalized assignment problem & Quadratic assignment problem.read more
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Neural Graph Matching Network: Learning Lawler's Quadratic Assignment Problem with Extension to Hypergraph and Multiple-graph Matching
TL;DR: This paper presents a QAP network directly learning with the affinity matrix (equivalently the association graph) whereby the matching problem is translated into a vertex classification task, and is the first network to directly learn with the general Lawlers QAP.
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Special cases of the quadratic shortest path problem
Hao Hu,Renata Sotirov +1 more
TL;DR: In this article, it was shown that the quadratic shortest path problem can be solved in polynomial time on complete symmetric digraphs with more than four vertices.
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A characterization of linearizable instances of the quadratic minimum spanning tree problem
Ante Ćustić,Abraham P. Punnen +1 more
TL;DR: In this paper, the authors give a characterization of the quadratic minimum spanning tree problem for complete bipartite graphs and cactuses, which can be verified in O(|E|) time.
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Neural Graph Matching Network: Learning Lawler’s Quadratic Assignment Problem with Extension to Hypergraph and Multiple-graph Matching
Abstract: Graph matching involves combinatorial optimization based on edge-to-edge affinity matrix, which can be generally formulated as Lawlers Quadratic Assignment Problem (QAP). This paper presents a QAP network directly learning with the affinity matrix (equivalently the association graph) whereby the matching problem is translated into a vertex classification task. The association graph is learned by an embedding network for vertex classification, followed by Sinkhorn normalization and a cross-entropy loss for end-to-end learning. We further improve the embedding model on association graph by introducing Sinkhorn based matching-aware constraint, as well as dummy nodes to deal with unequal sizes of graphs. To our best knowledge, this is the first network to directly learn with the general Lawlers QAP. In contrast, recent deep matching methods focus on the learning of node/edge features in two graphs respectively. We also show how to extend our network to hypergraph matching, and matching of multiple graphs. Experimental results on both synthetic graphs and real-world images show its effectiveness. For pure QAP tasks on synthetic data and QAPLIB benchmark, our method can perform competitively and even surpass state-of-the-art graph matching and QAP solvers with notable less time cost. We provide a project homepage at http://thinklab.sjtu.edu.cn/project/NGM/index.html
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Linearizable special cases of the QAP
TL;DR: In this paper, the authors consider special cases of the quadratic assignment problem (QAP) that are linearizable in the sense of Bookhold and provide combinatorial characterizations of the linearizable instances of the weighted feedback arc set QAP and of the traveling salesman QAP.
References
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Book
Network Flows: Theory, Algorithms, and Applications
TL;DR: In-depth, self-contained treatments of shortest path, maximum flow, and minimum cost flow problems, including descriptions of polynomial-time algorithms for these core models are presented.
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Assignment Problems and the Location of Economic Activities
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Assignment Problems and the Location of Economic Activities
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P-Complete Approximation Problems
Sartaj Sahni,Teofilo F. Gonzalez +1 more
TL;DR: For P- complete problems such as traveling salesperson, cycle covers, 0-1 integer programming, multicommodity network flows, quadratic assignment, etc., it is shown that the approximation problem is also P-complete.
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The Quadratic Assignment Problem
TL;DR: In this article, the equivalence of the Koopmans-beckmann problem to a linear assignment problem with certain additional constraints is demonstrated, and a method for calculating a lower bound on the cost function is presented, and this forms the basis for an algorithm to determine optimal solutions.