Fine-grained complexity of rainbow coloring and its variants

Akanksha Agrawal

Research output: Contribution to journalArticlepeer-review

1 Scopus citations

Abstract

For a graph G and cR:E(G)→[k], a path P between u,v∈V(G) is a rainbow path if for distinct e,e∈E(P), we have cR(e)≠cR(e). RAINBOW k-COLORING takes a graph G and the objective is to check if there is cR:E(G)→[k] such that for all u,v∈V(G) there is a rainbow path between u and v. Two variants of the above problem are SUBSET RAINBOW k-COLORING and STEINER RAINBOW k-COLORING, where we are additionally given a subset S⊆V(G)×V(G) and S⊆V(G), respectively. Moreover, the objective is to check if there is cR:E(G)→[k], such that there is a rainbow path for each (u,v)∈S and u,v∈S, respectively. Under ETH, we obtain that for each k≥3: 1. RAINBOW k-COLORING has no 2o(|E(G)|)nO(1)-time algorithm. 2. STEINER RAINBOW k-COLORING has no 2o(|S|2)nO(1)-time algorithm. We also obtain that SUBSET RAINBOW k-COLORING and STEINER RAINBOW k-COLORING admit 2O(|S|)nO(1)- and 2O(|S|2)nO(1)-time algorithms, respectively.

Original languageEnglish
Pages (from-to)140-158
Number of pages19
JournalJournal of Computer and System Sciences
Volume124
DOIs
StatePublished - 1 Mar 2022
Externally publishedYes

Keywords

  • ETH
  • Fine-grained complexity
  • Lower bound
  • Rainbow coloring

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computer Networks and Communications
  • Computational Theory and Mathematics
  • Applied Mathematics

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