An extension of matching theory
Journal of Combinatorial Theory Series B
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Journal of Combinatorial Theory Series B
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Journal of Combinatorial Theory Series B
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ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming: Part II
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ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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We define a problem called Simplex Matching and show that it is solvable in polynomial time. While Simplex Matching is interesting in its own right as a nontrivial extension of nonbipartite min-cost matching, its main value lies in many (seemingly very different) problems that can be solved using our algorithm. For example, suppose that we are given a graph with terminal nodes, nonterminal nodes, and edge costs. Then, the Terminal Backup problem, which consists of finding the cheapest forest connecting every terminal to at least one other terminal, is reducible to Simplex Matching. Simplex Matching is also useful for various tasks that involve forming groups of at least two members, such as project assignment and variants of facility location. In an instance of Simplex Matching, we are given a hypergraph $H$ with edge costs and edge size at most 3. We show how to find the min-cost perfect matching of $H$ efficiently if the edge costs obey a simple and realistic inequality that we call the Simplex Condition. The algorithm we provide is relatively simple to understand and implement but difficult to prove correct. In the process of this proof we show some powerful new results about covering cubic graphs with simple combinatorial objects.