Parameterized Algorithms and Kernels for Rainbow Matching

Sushmita Gupta, Sanjukta Roy, Saket Saurabh, Meirav Zehavi

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

In this paper, we study the NP-complete colorful variant of the classical Matching problem, namely, the Rainbow Matching problem. Given an edge-colored graph G and a positive integer k, this problem asks whether there exists a matching of size at least k such that all the edges in the matching have distinct colors. We first develop a deterministic algorithm that solves Rainbow Matching on paths in time O⋆((1+52)k) and polynomial space. This algorithm is based on a curious combination of the method of bounded search trees and a “divide-and-conquer-like” approach, where the branching process is guided by the maintenance of an auxiliary bipartite graph where one side captures “divided-and-conquered” pieces of the path. Our second result is a randomized algorithm that solves Rainbow Matching on general graphs in time O (2 k ) and polynomial-space. Here, we show how a result by Björklund et al. (J Comput Syst Sci 87:119–139, 2017) can be invoked as a black box, wrapped by a probability-based analysis tailored to our problem. We also complement our two main results by designing kernels for Rainbow Matching on general and bounded-degree graphs.

Original languageEnglish
Pages (from-to)1684-1698
Number of pages15
JournalAlgorithmica
Volume81
Issue number4
DOIs
StatePublished - 1 Apr 2019

Keywords

  • 3-Dimensional matching
  • 3-Set packing
  • Bounded search trees
  • Divide-and-conquer
  • Parameterized algorithm
  • Rainbow matching

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