2 Search Results for "Abolhassani, Melika"

Document
DOI: 10.4230/LIPIcs.ESA.2025.52
Beating Competitive Ratio 4 for Graphic Matroid Secretary

Authors: Kiarash Banihashem, MohammadTaghi Hajiaghayi, Dariusz R. Kowalski, Piotr Krysta, Danny Mittal, and Jan Olkowski

Published in: LIPIcs, Volume 351, 33rd Annual European Symposium on Algorithms (ESA 2025)

Abstract
One of the classic problems in online decision-making is the secretary problem, where the goal is to hire the best secretary out of n rankable applicants or, in a natural extension, to maximize the probability of selecting the largest number from a sequence arriving in random order. Many works have considered generalizations of this problem where one can accept multiple values subject to a combinatorial constraint. The seminal work of Babaioff, Immorlica, Kempe, and Kleinberg (SODA'07, JACM'18) proposed the matroid secretary conjecture, suggesting that there exists an O(1)-competitive algorithm for the matroid constraint, and many works since have attempted to obtain algorithms for both general matroids and specific classes of matroids. The ultimate goal of these results is to obtain an e-competitive algorithm, and the strong matroid secretary conjecture states that this is possible for general matroids. One of the most important classes of matroids is the graphic matroid, where a set of edges in a graph is deemed independent if it contains no cycle. Given the rich combinatorial structure of graphs, obtaining algorithms for these matroids is often seen as a good first step towards solving the problem for general matroids. For matroid secretary, Babaioff et al. (SODA'07, JACM'18) first studied graphic matroid case and obtained a 16-competitive algorithm. Subsequent works have improved the competitive ratio, most recently to 4 by Soto, Turkieltaub, and Verdugo (SODA'18). In this paper, we break the 4-competitive barrier for the problem, obtaining a new algorithm with a competitive ratio of 3.95. For the special case of simple graphs (i.e., graphs that do not contain parallel edges) we further improve this to 3.77. Intuitively, solving the problem for simple graphs is easier as they do not contain cycles of length two. A natural question that arises is whether we can obtain a ratio arbitrarily close to e by assuming the graph has a large enough girth. We answer this question affirmatively, proving that one can obtain a competitive ratio arbitrarily close to e even for constant values of girth, providing further evidence for the strong matroid secretary conjecture. We further show that this bound is tight: for any constant g, one cannot obtain a competitive ratio better than e even if we assume that the input graph has girth at least g. To our knowledge, such a bound was not previously known even for simple graphs.
Cite as

Kiarash Banihashem, MohammadTaghi Hajiaghayi, Dariusz R. Kowalski, Piotr Krysta, Danny Mittal, and Jan Olkowski. Beating Competitive Ratio 4 for Graphic Matroid Secretary. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 52:1-52:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{banihashem_et_al:LIPIcs.ESA.2025.52,
  author =	{Banihashem, Kiarash and Hajiaghayi, MohammadTaghi and Kowalski, Dariusz R. and Krysta, Piotr and Mittal, Danny and Olkowski, Jan},
  title =	{{Beating Competitive Ratio 4 for Graphic Matroid Secretary}},
  booktitle =	{33rd Annual European Symposium on Algorithms (ESA 2025)},
  pages =	{52:1--52:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-395-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{351},
  editor =	{Benoit, Anne and Kaplan, Haim and Wild, Sebastian and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2025.52},
  URN =		{urn:nbn:de:0030-drops-245205},
  doi =		{10.4230/LIPIcs.ESA.2025.52},
  annote =	{Keywords: online algorithms, graphic matroids, secretary problem}
}
Document
DOI: 10.4230/LIPIcs.ESA.2016.3
Beating Ratio 0.5 for Weighted Oblivious Matching Problems

Authors: Melika Abolhassani, T.-H. Hubert Chan, Fei Chen, Hossein Esfandiari, MohammadTaghi Hajiaghayi, Mahini Hamid, and Xiaowei Wu

Published in: LIPIcs, Volume 57, 24th Annual European Symposium on Algorithms (ESA 2016)

Abstract
We prove the first non-trivial performance ratios strictly above 0.5 for weighted versions of the oblivious matching problem. Even for the unweighted version, since Aronson, Dyer, Frieze, and Suen first proved a non-trivial ratio above 0.5 in the mid-1990s, during the next twenty years several attempts have been made to improve this ratio, until Chan, Chen, Wu and Zhao successfully achieved a significant ratio of 0.523 very recently (SODA 2014). To the best of our knowledge, our work is the first in the literature that considers the node-weighted and edge-weighted versions of the problem in arbitrary graphs (as opposed to bipartite graphs). (1) For arbitrary node weights, we prove that a weighted version of the Ranking algorithm has ratio strictly above 0.5. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors at the end of algorithm, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the node-weighted and the unweighted oblivious matching problems. As a result, we prove that the ratio for the node-weighted case is at least 0.501512. Interestingly via the structural property, we can also improve slightly the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014). (2) For a bounded number of distinct edge weights, we show that ratio strictly above 0.5 can be achieved by partitioning edges carefully according to the weights, and running the (unweighted) Ranking algorithm on each part. Our analysis is based on a new primal-dual framework known as \emph{matching coverage}, in which dual feasibility is bypassed. Instead, only dual constraints corresponding to edges in an optimal matching are satisfied. Using this framework we also design and analyze an algorithm for the edge-weighted online bipartite matching problem with free disposal. We prove that for the case of bounded online degrees, the ratio is strictly above 0.5.
Cite as

Melika Abolhassani, T.-H. Hubert Chan, Fei Chen, Hossein Esfandiari, MohammadTaghi Hajiaghayi, Mahini Hamid, and Xiaowei Wu. Beating Ratio 0.5 for Weighted Oblivious Matching Problems. In 24th Annual European Symposium on Algorithms (ESA 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 57, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{abolhassani_et_al:LIPIcs.ESA.2016.3,
  author =	{Abolhassani, Melika and Chan, T.-H. Hubert and Chen, Fei and Esfandiari, Hossein and Hajiaghayi, MohammadTaghi and Hamid, Mahini and Wu, Xiaowei},
  title =	{{Beating Ratio 0.5 for Weighted Oblivious Matching Problems}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{3:1--3:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Sankowski, Piotr and Zaroliagis, Christos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2016.3},
  URN =		{urn:nbn:de:0030-drops-63443},
  doi =		{10.4230/LIPIcs.ESA.2016.3},
  annote =	{Keywords: Weighted matching, oblivious algorithms, Ranking, linear programming}
}
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