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Popular Edges with Critical Nodes

Authors Kushagra Chatterjee, Prajakta Nimbhorkar

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  • Mathematics of computing → Graph algorithms
Keywords
  • Matching
  • Stable Matching
  • Popular feasible Matching

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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.

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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) https://doi.org/10.4230/LIPIcs.ISAAC.2022.54

Author Details

Kushagra Chatterjee
  • National University of Singapore, Singapore
Prajakta Nimbhorkar
  • Chennai Mathematical Institute, India

Funding

  • Chatterjee, Kushagra: Supported in part by an MoE AcRF Tier 2 grant (WBS No. A-8000416-00-00) and an ODPRT grant (WBS No. A-0008078-00-00).
  • Nimbhorkar, Prajakta: Supported in part by SERB Award CRG/2019/004757.

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