70 Search Results for "Kavitha, Telikepalli"

Volume

LIPIcs, Volume 18

IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2012)

FSTTCS 2012, December 15-17, 2012, Hyderabad, India

Editors: Deepak D'Souza, Jaikumar Radhakrishnan, and Kavitha Telikepalli

Document
DOI: 10.4230/LIPIcs.FSTTCS.2023.43
Perfect Matchings and Popularity in the Many-To-Many Setting

Authors: Telikepalli Kavitha and Kazuhisa Makino

Published in: LIPIcs, Volume 284, 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)

Abstract
We consider a matching problem in a bipartite graph G where every vertex has a capacity and a strict preference list ranking its neighbors. We assume that G admits a perfect matching, i.e., one that fully matches all vertices. It is only perfect matchings that are feasible here and we seek one that is popular within the set of perfect matchings - it is known that such a matching exists in G and can be efficiently computed. Now we are in the weighted setting, i.e., there is a cost function on the edge set, and we seek a min-cost popular perfect matching in G. We show that such a matching can be computed in polynomial time. Our main technical result shows that every popular perfect matching in a hospitals/residents instance G can be realized as a popular perfect matching in the marriage instance obtained by cloning vertices. Interestingly, it is known that such a mapping does not hold for popular matchings in a hospitals/residents instance.
Cite as

Telikepalli Kavitha and Kazuhisa Makino. Perfect Matchings and Popularity in the Many-To-Many Setting. In 43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 284, pp. 43:1-43:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kavitha_et_al:LIPIcs.FSTTCS.2023.43,
  author =	{Kavitha, Telikepalli and Makino, Kazuhisa},
  title =	{{Perfect Matchings and Popularity in the Many-To-Many Setting}},
  booktitle =	{43rd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2023)},
  pages =	{43:1--43:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-304-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{284},
  editor =	{Bouyer, Patricia and Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2023.43},
  URN =		{urn:nbn:de:0030-drops-194167},
  doi =		{10.4230/LIPIcs.FSTTCS.2023.43},
  annote =	{Keywords: Bipartite graphs, Matchings under preferences, Capacities, Dual certificates}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2023.57
A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees

Authors: Shuai Shao and Stanislav Živný

Published in: LIPIcs, Volume 283, 34th International Symposium on Algorithms and Computation (ISAAC 2023)

Abstract
General factors are a generalization of matchings. Given a graph G with a set π(v) of feasible degrees, called a degree constraint, for each vertex v of G, the general factor problem is to find a (spanning) subgraph F of G such that deg_F(v) ∈ π(v) for every v of G. When all degree constraints are symmetric Δ-matroids, the problem is solvable in polynomial time. The weighted general factor problem is to find a general factor of the maximum total weight in an edge-weighted graph. Strongly polynomial-time algorithms are only known for weighted general factor problems that are reducible to the weighted matching problem by gadget constructions. In this paper, we present a strongly polynomial-time algorithm for a type of weighted general factor problems with real-valued edge weights that is provably not reducible to the weighted matching problem by gadget constructions. As an application, we obtain a strongly polynomial-time algorithm for the terminal backup problem by reducing it to the weighted general factor problem.
Cite as

Shuai Shao and Stanislav Živný. A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 57:1-57:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{shao_et_al:LIPIcs.ISAAC.2023.57,
  author =	{Shao, Shuai and \v{Z}ivn\'{y}, Stanislav},
  title =	{{A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees}},
  booktitle =	{34th International Symposium on Algorithms and Computation (ISAAC 2023)},
  pages =	{57:1--57:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-289-1},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{283},
  editor =	{Iwata, Satoru and Kakimura, Naonori},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2023.57},
  URN =		{urn:nbn:de:0030-drops-193597},
  doi =		{10.4230/LIPIcs.ISAAC.2023.57},
  annote =	{Keywords: matchings, factors, edge constraint satisfaction problems, terminal backup problem, delta matroids}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2022.54
Popular Edges with Critical Nodes

Authors: Kushagra Chatterjee and Prajakta Nimbhorkar

Published in: LIPIcs, Volume 248, 33rd International Symposium on Algorithms and Computation (ISAAC 2022)

Abstract
In the popular edge problem, the input is a bipartite graph G = (A ∪ B,E) where A and B denote a set of men and a set of women respectively, and each vertex in A∪ B has a strict preference ordering over its neighbours. A matching M in G is said to be popular if there is no other matching M' such that the number of vertices that prefer M' to M is more than the number of vertices that prefer M to M'. The goal is to determine, whether a given edge e belongs to some popular matching in G. A polynomial-time algorithm for this problem appears in [Cseh and Kavitha, 2018]. We consider the popular edge problem when some men or women are prioritized or critical. A matching that matches all the critical nodes is termed as a feasible matching. It follows from [Telikepalli Kavitha, 2014; Kavitha, 2021; Nasre et al., 2021; Meghana Nasre and Prajakta Nimbhorkar, 2017] that, when G admits a feasible matching, there always exists a matching that is popular among all feasible matchings. We give a polynomial-time algorithm for the popular edge problem in the presence of critical men or women. We also show that an analogous result does not hold in the many-to-one setting, which is known as the Hospital-Residents Problem in literature, even when there are no critical nodes.
Cite as

Kushagra Chatterjee and Prajakta Nimbhorkar. Popular Edges with Critical Nodes. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chatterjee_et_al:LIPIcs.ISAAC.2022.54,
  author =	{Chatterjee, Kushagra and Nimbhorkar, Prajakta},
  title =	{{Popular Edges with Critical Nodes}},
  booktitle =	{33rd International Symposium on Algorithms and Computation (ISAAC 2022)},
  pages =	{54:1--54:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-258-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{248},
  editor =	{Bae, Sang Won and Park, Heejin},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.54},
  URN =		{urn:nbn:de:0030-drops-173399},
  doi =		{10.4230/LIPIcs.ISAAC.2022.54},
  annote =	{Keywords: Matching, Stable Matching, Popular feasible Matching}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2022.22
Stable Matchings with One-Sided Ties and Approximate Popularity

Authors: Telikepalli Kavitha

Published in: LIPIcs, Volume 250, 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)

Abstract
We consider a matching problem in a bipartite graph G = (A ∪ B, E) where vertices in A rank their neighbors in a strict order of preference while vertices in B are allowed to have weak rankings, i.e., ties are allowed in their preferences. Stable matchings always exist in G and are easy to find, however popular matchings need not exist and it is NP-complete to decide if one exists. This motivates the "approximately popular" matching problem. A well-known measure of approximate popularity is low unpopularity factor. We show that when each tie in G has length at most k, there always exists a stable matching whose unpopularity factor is at most k. Our proof is algorithmic and we compute such a stable matching in polynomial time. Our result can be considered to be a generalization of Gärdenfors' result (1975) which showed that when rankings are strict, every stable matching is popular. There are several applications where the size of the matching is its most important attribute. What one seeks here is a maximum matching M such that there is no maximum matching more popular than M. When rankings are weak, it is NP-hard to decide if G admits such a matching. When ties are one-sided and of length at most k, we show a polynomial time algorithm to find a maximum matching whose unpopularity factor within the set of maximum matchings is at most 2k.
Cite as

Telikepalli Kavitha. Stable Matchings with One-Sided Ties and Approximate Popularity. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kavitha:LIPIcs.FSTTCS.2022.22,
  author =	{Kavitha, Telikepalli},
  title =	{{Stable Matchings with One-Sided Ties and Approximate Popularity}},
  booktitle =	{42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
  pages =	{22:1--22:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-261-7},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{250},
  editor =	{Dawar, Anuj and Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.22},
  URN =		{urn:nbn:de:0030-drops-174144},
  doi =		{10.4230/LIPIcs.FSTTCS.2022.22},
  annote =	{Keywords: Bipartite graphs, Maximum matchings, Unpopularity factor}
}
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.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
DOI: 10.4230/LIPIcs.FSTTCS.2021.25
Matchings, Critical Nodes, and Popular Solutions

Authors: Telikepalli Kavitha

Published in: LIPIcs, Volume 213, 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)

Abstract
We consider a matching problem in a marriage instance G. Every node has a strict preference order ranking its neighbors. There is a set C of prioritized or critical nodes and we are interested in only those matchings that match as many critical nodes as possible. Such matchings are useful in several applications and we call them critical matchings. A stable matching need not be critical. We consider a well-studied relaxation of stability called popularity. Our goal is to find a popular critical matching, i.e., a weak Condorcet winner within the set of critical matchings where nodes are voters. We show that popular critical matchings always exist in G and min-size/max-size such matchings can be efficiently computed.
Cite as

Telikepalli Kavitha. Matchings, Critical Nodes, and Popular Solutions. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 25:1-25:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{kavitha:LIPIcs.FSTTCS.2021.25,
  author =	{Kavitha, Telikepalli},
  title =	{{Matchings, Critical Nodes, and Popular Solutions}},
  booktitle =	{41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021)},
  pages =	{25:1--25:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-215-0},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{213},
  editor =	{Boja\'{n}czyk, Miko{\l}aj and Chekuri, Chandra},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2021.25},
  URN =		{urn:nbn:de:0030-drops-155360},
  doi =		{10.4230/LIPIcs.FSTTCS.2021.25},
  annote =	{Keywords: Bipartite graphs, Stable matchings, LP-duality}
}
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.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
DOI: 10.4230/LIPIcs.FSTTCS.2020.25
Min-Cost Popular Matchings

Authors: Telikepalli Kavitha

Published in: LIPIcs, Volume 182, 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)

Abstract
Let G = (A ∪ B, E) be a bipartite graph on n vertices where every vertex ranks its neighbors in a strict order of preference. A matching M in G is popular if there is no matching N such that vertices that prefer N to M outnumber those that prefer M to N. Popular matchings always exist in G since every stable matching is popular. Thus it is easy to find a popular matching in G - however it is NP-hard to compute a min-cost popular matching in G when there is a cost function on the edge set; moreover it is NP-hard to approximate this to any multiplicative factor. An O^*(2ⁿ) algorithm to compute a min-cost popular matching in G follows from known results. Here we show: - an algorithm with running time O^*(2^{n/4}) ≈ O^*(1.19ⁿ) to compute a min-cost popular matching; - assume all edge costs are non-negative - then given ε > 0, a randomized algorithm with running time poly(n,1/(ε)) to compute a matching M such that cost(M) is at most twice the optimal cost and with high probability, the fraction of all matchings more popular than M is at most 1/2+ε.
Cite as

Telikepalli Kavitha. Min-Cost Popular Matchings. In 40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 182, pp. 25:1-25:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{kavitha:LIPIcs.FSTTCS.2020.25,
  author =	{Kavitha, Telikepalli},
  title =	{{Min-Cost Popular Matchings}},
  booktitle =	{40th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2020)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-174-0},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{182},
  editor =	{Saxena, Nitin and Simon, Sunil},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2020.25},
  URN =		{urn:nbn:de:0030-drops-132668},
  doi =		{10.4230/LIPIcs.FSTTCS.2020.25},
  annote =	{Keywords: Bipartite graphs, 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.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.2019.20
Popular Roommates in Simply Exponential Time

Authors: Telikepalli Kavitha

Published in: LIPIcs, Volume 150, 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)

Abstract
We consider the popular matching problem in a graph G = (V,E) on n vertices with strict preferences. A matching M is popular if there is no matching N in G such that vertices that prefer N to M outnumber those that prefer M to N. It is known that it is NP-hard to decide if G has a popular matching or not. There is no faster algorithm known for this problem than the brute force algorithm that could take n! time. Here we show a simply exponential time algorithm for this problem, i.e., one that runs in O^*(k^n) time, where k is a constant. We use the recent breakthrough result on the maximum number of stable matchings possible in such instances to analyze our algorithm for the popular matching problem. We identify a natural (also, hard) subclass of popular matchings called truly popular matchings and show an O^*(2^n) time algorithm for the truly popular matching problem.
Cite as

Telikepalli Kavitha. Popular Roommates in Simply Exponential Time. In 39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 150, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kavitha:LIPIcs.FSTTCS.2019.20,
  author =	{Kavitha, Telikepalli},
  title =	{{Popular Roommates in Simply Exponential Time}},
  booktitle =	{39th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2019)},
  pages =	{20:1--20:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-131-3},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{150},
  editor =	{Chattopadhyay, Arkadev and Gastin, Paul},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2019.20},
  URN =		{urn:nbn:de:0030-drops-115820},
  doi =		{10.4230/LIPIcs.FSTTCS.2019.20},
  annote =	{Keywords: Roommates instance, Popular matching, Stable matching, Dual certificate}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.MFCS.2019.4
Popular Matchings: Good, Bad, and Mixed (Invited Talk)

Authors: Telikepalli Kavitha

Published in: LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Abstract
We consider the landscape of popular matchings in a bipartite graph G where every vertex has strict preferences over its neighbors. This is a very well-studied model in two-sided matching markets. A matching M is popular if it does not lose a head-to-head election against any matching, where each vertex casts a vote for the matching where it gets a better assignment. Roughly speaking, a popular matching is one such that there is no matching where more vertices are happier. The notion of popularity is more relaxed than stability: a classical notion studied for the last several decades. Popular matchings always exist in G since stable matchings always exist in a bipartite graph and every stable matching is popular. Algorithmically speaking, the landscape of popular matching seems to have only a few bright spots. Every stable matching is a min-size popular matching and there are also simple linear time algorithms for computing a max-size popular matching and for the popular edge problem. All these algorithms reduce the popular matching problem to an appropriate question in stable matchings and solve the corresponding stable matching problem. We now know NP-hardness results for many popular matching problems. These include the min-cost/max-weight popular matching problem and the problem of deciding if G admits a popular matching that is neither a min-size nor a max-size popular matching. For non-bipartite graphs, it is NP-hard to even decide if a popular matching exists or not. A mixed matching is a probability distribution or a lottery over matchings. A popular mixed matching is one that never loses a head-to-head election against any mixed matching. As an allocation mechanism, a popular mixed matching has several nice properties. Moreover, finding a max-weight or min-cost popular mixed matching in G is easy (by solving a linear program). Interestingly, there is always an optimal popular mixed matching Pi with a simple structure: Pi = {(M_0,1/2),(M_1,1/2)} where M_0 and M_1 are matchings in G. Popular mixed matchings always exist in non-bipartite graphs as well and can be computed in polynomial time.
Cite as

Telikepalli Kavitha. Popular Matchings: Good, Bad, and Mixed (Invited Talk). In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, p. 4:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kavitha:LIPIcs.MFCS.2019.4,
  author =	{Kavitha, Telikepalli},
  title =	{{Popular Matchings: Good, Bad, and Mixed}},
  booktitle =	{44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)},
  pages =	{4:1--4:1},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-117-7},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{138},
  editor =	{Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.4},
  URN =		{urn:nbn:de:0030-drops-109489},
  doi =		{10.4230/LIPIcs.MFCS.2019.4},
  annote =	{Keywords: Matchings under preferences, Algorithms, NP-Hardness}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2018.42
An Algorithm for the Maximum Weight Strongly Stable Matching Problem

Authors: Adam Kunysz

Published in: LIPIcs, Volume 123, 29th International Symposium on Algorithms and Computation (ISAAC 2018)

Abstract
An instance of the maximum weight strongly stable matching problem with incomplete lists and ties is an undirected bipartite graph G = (A cup B, E), with an adjacency list being a linearly ordered list of ties, which are vertices equally good for a given vertex. We are also given a weight function w on the set E. An edge (x, y) in E setminus M is a blocking edge for M if by getting matched to each other neither of the vertices x and y would become worse off and at least one of them would become better off. A matching is strongly stable if there is no blocking edge with respect to it. The goal is to compute a strongly stable matching of maximum weight with respect to w. We give a polyhedral characterisation of the problem and prove that the strongly stable matching polytope is integral. This result implies that the maximum weight strongly stable matching problem can be solved in polynomial time. Thereby answering an open question by Gusfield and Irving [Dan Gusfield and Robert W. Irving, 1989]. The main result of this paper is an efficient O(nm log{(Wn)}) time algorithm for computing a maximum weight strongly stable matching, where we denote n = |V|, m = |E| and W is a maximum weight of an edge in G. For small edge weights we show that the problem can be solved in O(nm) time. Note that the fastest known algorithm for the unweighted version of the problem has O(nm) runtime [Telikepalli Kavitha et al., 2007]. Our algorithm is based on the rotation structure which was constructed for strongly stable matchings in [Adam Kunysz et al., 2016].
Cite as

Adam Kunysz. An Algorithm for the Maximum Weight Strongly Stable Matching Problem. In 29th International Symposium on Algorithms and Computation (ISAAC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 123, pp. 42:1-42:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kunysz:LIPIcs.ISAAC.2018.42,
  author =	{Kunysz, Adam},
  title =	{{An Algorithm for the Maximum Weight Strongly Stable Matching Problem}},
  booktitle =	{29th International Symposium on Algorithms and Computation (ISAAC 2018)},
  pages =	{42:1--42:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-094-1},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{123},
  editor =	{Hsu, Wen-Lian and Lee, Der-Tsai and Liao, Chung-Shou},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2018.42},
  URN =		{urn:nbn:de:0030-drops-99902},
  doi =		{10.4230/LIPIcs.ISAAC.2018.42},
  annote =	{Keywords: Stable marriage, Strongly stable matching, Weighted matching, Rotation}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2018.17
Popular Matchings in Complete Graphs

Authors: Ágnes Cseh and Telikepalli Kavitha

Published in: LIPIcs, Volume 122, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)

Abstract
Our input is a complete graph G = (V,E) on n vertices where each vertex has a strict ranking of all other vertices in G. The goal is to construct a matching in G that is "globally stable" or popular. A matching M is popular if M does not lose a head-to-head election against any matching M': here each vertex casts a vote for the matching in {M,M'} where it gets a better assignment. Popular matchings need not exist in the given instance G and the popular matching problem is to decide whether one exists or not. The popular matching problem in G is easy to solve for odd n. Surprisingly, the problem becomes NP-hard for even n, as we show here.
Cite as

Ágnes Cseh and Telikepalli Kavitha. Popular Matchings in Complete Graphs. In 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 122, pp. 17:1-17:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{cseh_et_al:LIPIcs.FSTTCS.2018.17,
  author =	{Cseh, \'{A}gnes and Kavitha, Telikepalli},
  title =	{{Popular Matchings in Complete Graphs}},
  booktitle =	{38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018)},
  pages =	{17:1--17:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-093-4},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{122},
  editor =	{Ganguly, Sumit and Pandya, Paritosh},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2018.17},
  URN =		{urn:nbn:de:0030-drops-99164},
  doi =		{10.4230/LIPIcs.FSTTCS.2018.17},
  annote =	{Keywords: popular matching, complete graph, complexity, linear programming}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2017.19
Popular Matchings with Multiple Partners

Authors: Florian Brandl and Telikepalli Kavitha

Published in: LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

Abstract
Our input is a bipartite graph G=(A\cup B,E) where each vertex in A\cup B has a preference list strictly ranking its neighbors. The vertices in A and in B are called students and courses, respectively. Each student a seeks to be matched to cap(a)\geq 1 many courses while each course b seeks cap(b)\geq 1 many students to be matched to it. The Gale-Shapley algorithm computes a pairwise-stable matching (one with no blocking edge) in G in linear time. We consider the problem of computing a popular matching in G - a matching M is popular if M cannot lose an election to any matching where vertices cast votes for one matching versus another. Our main contribution is to show that a max-size popular matching in G can be computed by the 2-level Gale-Shapley algorithm in linear time. This is an extension of the classical Gale-Shapley algorithm and we prove its correctness via linear programming.
Cite as

Florian Brandl and Telikepalli Kavitha. Popular Matchings with Multiple Partners. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 19:1-19:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{brandl_et_al:LIPIcs.FSTTCS.2017.19,
  author =	{Brandl, Florian and Kavitha, Telikepalli},
  title =	{{Popular Matchings with Multiple Partners}},
  booktitle =	{37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)},
  pages =	{19:1--19:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-055-2},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{93},
  editor =	{Lokam, Satya and Ramanujam, R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.19},
  URN =		{urn:nbn:de:0030-drops-83765},
  doi =		{10.4230/LIPIcs.FSTTCS.2017.19},
  annote =	{Keywords: Bipartite graphs, Linear programming duality, Gale-Shapley algorithm}
}
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