4 Search Results for "Faenza, Yuri"

Document
DOI: 10.4230/LIPIcs.STACS.2022.41
Fairly Popular Matchings and Optimality

Authors: Telikepalli Kavitha

Published in: LIPIcs, Volume 219, 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)

Abstract
We consider a matching problem in a bipartite graph G = (A ∪ B, E) where vertices have strict preferences over their neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M is at least the number that prefer N; thus M does not lose a head-to-head election against any matching where vertices are voters. It is easy to find popular matchings; however when there are edge costs, it is NP-hard to find (or even approximate) a min-cost popular matching. This hardness motivates relaxations of popularity. Here we introduce fairly popular matchings. A fairly popular matching may lose elections but there is no good matching (wrt popularity) that defeats a fairly popular matching. In particular, any matching that defeats a fairly popular matching does not occur in the support of any popular mixed matching. We show that a min-cost fairly popular matching can be computed in polynomial time and the fairly popular matching polytope has a compact extended formulation. We also show the following hardness result: given a matching M, it is NP-complete to decide if there exists a popular matching that defeats M. Interestingly, there exists a set K of at most m popular matchings in G (where |E| = m) such that if a matching is defeated by some popular matching in G then it has to be defeated by one of the matchings in K.
Cite as

Telikepalli Kavitha. Fairly Popular Matchings and Optimality. In 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 219, pp. 41:1-41:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kavitha:LIPIcs.STACS.2022.41,
  author =	{Kavitha, Telikepalli},
  title =	{{Fairly Popular Matchings and Optimality}},
  booktitle =	{39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022)},
  pages =	{41:1--41:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-222-8},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{219},
  editor =	{Berenbrink, Petra and Monmege, Benjamin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2022.41},
  URN =		{urn:nbn:de:0030-drops-158516},
  doi =		{10.4230/LIPIcs.STACS.2022.41},
  annote =	{Keywords: Bipartite graphs, Stable matchings, Mixed matchings, Polytopes}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2021.85
Maximum Matchings and Popularity

Authors: Telikepalli Kavitha

Published in: LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Abstract
Let G be a bipartite graph where every node has a strict ranking of its neighbors. For any node, its preferences over neighbors extend naturally to preferences over matchings. A maximum matching M in G is a popular max-matching if for any maximum matching N in G, the number of nodes that prefer M is at least the number that prefer N. Popular max-matchings always exist in G and they are relevant in applications where the size of the matching is of higher priority than node preferences. Here we assume there is also a cost function on the edge set. So what we seek is a min-cost popular max-matching. Our main result is that such a matching can be computed in polynomial time. We show a compact extended formulation for the popular max-matching polytope and the algorithmic result follows from this. In contrast, it is known that the popular matching polytope has near-exponential extension complexity and finding a min-cost popular matching is NP-hard.
Cite as

Telikepalli Kavitha. Maximum Matchings and Popularity. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 85:1-85:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kavitha:LIPIcs.ICALP.2021.85,
  author =	{Kavitha, Telikepalli},
  title =	{{Maximum Matchings and Popularity}},
  booktitle =	{48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)},
  pages =	{85:1--85:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-195-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{198},
  editor =	{Bansal, Nikhil and Merelli, Emanuela and Worrell, James},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.85},
  URN =		{urn:nbn:de:0030-drops-141548},
  doi =		{10.4230/LIPIcs.ICALP.2021.85},
  annote =	{Keywords: Bipartite graphs, Popular matchings, Stable matchings, Dual certificates}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2020.70
Popular Matchings with One-Sided Bias

Authors: Telikepalli Kavitha

Published in: LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

Abstract
Let G = (A ∪ B,E) be a bipartite graph where A consists of agents or main players and B consists of jobs or secondary players. Every vertex has a strict ranking of its neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M to N is at least the number that prefer N to M. Popular matchings always exist in G since every stable matching is popular. A matching M is A-popular if for any matching N, the number of agents (i.e., vertices in A) that prefer M to N is at least the number of agents that prefer N to M. Unlike popular matchings, A-popular matchings need not exist in a given instance G and there is a simple linear time algorithm to decide if G admits an A-popular matching and compute one, if so. We consider the problem of deciding if G admits a matching that is both popular and A-popular and finding one, if so. We call such matchings fully popular. A fully popular matching is useful when A is the more important side - so along with overall popularity, we would like to maintain "popularity within the set A". A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings.
Cite as

Telikepalli Kavitha. Popular Matchings with One-Sided Bias. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 70:1-70:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kavitha:LIPIcs.ICALP.2020.70,
  author =	{Kavitha, Telikepalli},
  title =	{{Popular Matchings with One-Sided Bias}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{70:1--70:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.70},
  URN =		{urn:nbn:de:0030-drops-124774},
  doi =		{10.4230/LIPIcs.ICALP.2020.70},
  annote =	{Keywords: Bipartite graphs, Stable matchings, Gale-Shapley algorithm, LP-duality}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2014.187
Solving the Stable Set Problem in Terms of the Odd Cycle Packing Number

Authors: Adrian Bock, Yuri Faenza, Carsten Moldenhauer, and Andres Jacinto Ruiz-Vargas

Published in: LIPIcs, Volume 29, 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)

Abstract
The classic stable set problem asks to find a maximum cardinality set of pairwise non-adjacent vertices in an undirected graph G. This problem is NP-hard to approximate with factor n^{1-epsilon} for any constant epsilon>0 [Hastad/Acta Mathematica/1996; Zuckerman/STOC/2006], where n is the number of vertices, and therefore there is no hope for good approximations in the general case. We study the stable set problem when restricted to graphs with bounded odd cycle packing number ocp(G), possibly by a function of n. This is the largest number of vertex-disjoint odd cycles in G. Equivalently, it is the logarithm of the largest absolute value of a sub-determinant of the edge-node incidence matrix A_G of G. Hence, if A_G is totally unimodular, then ocp(G)=0. Therefore, ocp(G) is a natural distance measure of A_G to the set of totally unimodular matrices on a scale from 1 to n/3. When ocp(G)=0, the graph is bipartite and it is well known that stable set can be solved in polynomial time. Our results imply that the odd cycle packing number indeed strongly influences the approximability of stable set. More precisely, we obtain a polynomial-time approximation scheme for graphs with ocp(G)=o(n/log(n)), and an alpha-approximation algorithm for any graph where alpha smoothly increases from a constant to n as ocp(G) grows from O(n/log(n)) to n/3. On the hardness side, we show that, assuming the exponential-time hypothesis, stable set cannot be solved in polynomial time if ocp(G)=Omega(log^{1+epsilon}(n)) for some epsilon>0. Finally, we generalize a theorem by Györi et al. [Györi et al./Discrete Mathematics/1997] and show that graphs without odd cycles of small weight can be made bipartite by removing a small number of vertices. This allows us to extend some of our above results to the weighted stable set problem.
Cite as

Adrian Bock, Yuri Faenza, Carsten Moldenhauer, and Andres Jacinto Ruiz-Vargas. Solving the Stable Set Problem in Terms of the Odd Cycle Packing Number. In 34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 29, pp. 187-198, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{bock_et_al:LIPIcs.FSTTCS.2014.187,
  author =	{Bock, Adrian and Faenza, Yuri and Moldenhauer, Carsten and Ruiz-Vargas, Andres Jacinto},
  title =	{{Solving the Stable Set Problem in Terms of the Odd Cycle Packing Number}},
  booktitle =	{34th International Conference on Foundation of Software Technology and Theoretical Computer Science (FSTTCS 2014)},
  pages =	{187--198},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-77-4},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{29},
  editor =	{Raman, Venkatesh and Suresh, S. P.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2014.187},
  URN =		{urn:nbn:de:0030-drops-48422},
  doi =		{10.4230/LIPIcs.FSTTCS.2014.187},
  annote =	{Keywords: stable set problem, independent set problem, approximation algorithms, odd cycle packing number, maximum subdeterminants}
}
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