Popular Edges and Dominant Matchings

Cseh, Ágnes; Kavitha, Telikepalli

doi:10.1007/978-3-319-33461-5_12

Cited by 22 publications

(77 citation statements)

References 16 publications

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“…Note that there is a one-to-one correspondence between branchings in G and arborescences in D (simply make r the parent of all roots of the branching). A branching is popular in G if and only if the corresponding arborescence is popular among all arborescences in D. 9 We will therefore prove our results for arborescences in D. The corresponding results for branchings in G follow immediately by projection, i.e., removing node r and its incident edges.…”

Section: Dual Certificatesmentioning

confidence: 66%

Popular Branchings and Their Dual Certificates

Kavitha

Király

Matuschke

et al. 2020

Integer Programming and Combinatorial Optimization

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Let G be a digraph where every node has preferences over its incoming edges. The preferences of a node extend naturally to preferences over branchings, i.e., directed forests; a branching B is popular if B does not lose a head-to-head election (where nodes cast votes) against any branching. Such popular branchings have a natural application in liquid democracy. The popular branching problem is to decide if G admits a popular branching or not. We give a characterization of popular branchings in terms of dual certificates and use this characterization to design an efficient combinatorial algorithm for the popular branching problem. When preferences are weak rankings, we use our characterization to formulate the popular branching polytope in the original space and also show that our algorithm can be modified to compute a branching with least unpopularity margin. When preferences are strict rankings, we show that "approximately popular" branchings always exist. 6 See www.democracy.earth and www.interaktive-demokratie.org, respectively. 7 Typically, such a delegation is represented by an edge (y, x); for the sake of consistency with downward edges in a branching, we use (x, y).

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Section: Dual Certificatesmentioning

confidence: 66%

Popular Branchings and Their Dual Certificates

Kavitha

Király

Matuschke

et al. 2020

Integer Programming and Combinatorial Optimization

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“…The augmentation of G into G * is based on a certain subgraph of G called its "popular subgraph". Call an edge e in G = (A ∪ B, E) popular if there is some popular matching in G that contains e. Let E F ⊆ E be the set of popular edges in G. The set E F can be computed in linear time, see [11]. Call F G = (A ∪ B, E F ) the popular subgraph of G.…”

Section: The Popular Subgraphmentioning

confidence: 99%

“…[11]). A popular matching M in G = (A ∪ B, E) is dominant if and only if M admits a witness α such that α v ∈ {±1} for every popular vertex v.…”

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confidence: 99%

Quasi-popular Matchings, Optimality, and Extended Formulations

Faenza¹,

Kavitha²

2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

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Let G = (A ∪ B, E) be an instance of the stable marriage problem where every vertex ranks its neighbors in a strict order of preference. A matching M in G is popular if M does not lose a headto-head election against any matching. Popular matchings are a well-studied generalization of stable matchings, introduced with the goal of enlarging the set of admissible solutions, while maintaining a certain level of fairness. Every stable matching is a min-size popular matching. Unfortunately, when there are edge costs, it is NP-hard to find a popular matching of minimum cost -even worse, the min-cost popular matching problem is hard to approximate up to any factor. Let opt be the cost of a min-cost popular matching. Our goal is to efficiently compute a matching of cost at most opt by paying the price of mildly relaxing popularity. Our main positive results are two bi-criteria algorithms that find in polynomial time a near-popular or "quasi-popular" matching of cost at most opt. Moreover, one of the algorithms finds a quasi-popular matching of cost at most that of a min-cost popular fractional matching -this cost could be much smaller than opt. Key to the other algorithm are a number of results for certain polytopes related to matchings. In particular, we give a polynomial-size extended formulation for an integral polytope sandwiched between the popular and quasi-popular matching polytopes. We complement these results by showing that it is NP-hard to find a quasi-popular matching of minimum cost, and that both the popular and quasi-popular matching polytopes have near-exponential extension complexity.

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“…Dominant matchings always exist in a bipartite graph and such a matching can be computed in linear time [18]. Every polynomial time algorithm currently known to find a popular matching in a bipartite graph finds either a stable matching [9] or a dominant matching [14,18,7].…”

Section: Definitionmentioning

confidence: 99%

“…For instance, Kavitha [18] showed that a max-size popular matching can be found efficiently by a combination of the Gale-Shapley algorithm and promotion of nodes rejected once by all neighbors. Cseh and Kavitha [7] showed that a pair of nodes is matched together in some popular matching if and only if this pair is matched together either in some stable matching or in some dominant matching. Dominant matchings are a subclass of max-size popular matchings, and these are equivalent (under a simple linear map) to stable matchings in a larger graph.…”

Section: Introductionmentioning

confidence: 99%

Popular Matchings and Limits to Tractability

Faenza

Kavitha

Powers

et al. 2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider popular matching problems in both bipartite and non-bipartite graphs with strict preference lists. It is known that every stable matching is a min-size popular matching. A subclass of max-size popular matchings called dominant matchings has been well-studied in bipartite graphs: they always exist and there is a simple linear time algorithm to find one. We show that stable and dominant matchings are the only two tractable subclasses of popular matchings in bipartite graphs; more precisely, we show that it is NP-complete to decide if G admits a popular matching that is neither stable nor dominant. We also show a number of related hardness results, such as (tight) inapproximability of the maximum weight popular matching problem. In non-bipartite graphs, we show a strong negative result: it is NP-hard to decide whether a popular matching exists or not, and the same result holds if we replace popular with dominant. On the positive side, we show the following results in any graph:we identify a subclass of dominant matchings called strongly dominant matchings and show a linear time algorithm to decide if a strongly dominant matching exists or not; we show an efficient algorithm to compute a popular matching of minimum cost in a graph with edge costs and bounded treewidth.This paper is a merger of results shown in the arXiv papers [8,21] along with one in [20] and new results.

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Popular Edges and Dominant Matchings

Cited by 22 publications

References 16 publications

Popular Branchings and Their Dual Certificates

Popular Branchings and Their Dual Certificates

Quasi-popular Matchings, Optimality, and Extended Formulations

Popular Matchings and Limits to Tractability

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