The Maximum Clique Problem in Multiple Interval Graphs

TL;DR

This paper investigates the computational complexity of the maximum clique problem across various classes of multiple interval graphs, establishing NP-completeness and APX-completeness results, and introducing new graph classes with complexity analyses.

Contribution

It proves NP-completeness and APX-completeness for maximum clique in several multiple interval graph classes and introduces new circular interval graph classes with complexity results.

Findings

NP-complete for unit 2-interval and 3-track graphs

APX-complete for 2-interval, 3-track, unit 3-interval, and unit 4-track graphs

Provides a polynomial-time t-approximation algorithm for weighted maximum clique

Abstract

Multiple interval graphs are variants of interval graphs where instead of a single interval, each vertex is assigned a set of intervals on the real line. We study the complexity of the MAXIMUM CLIQUE problem in several classes of multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding the size of the maximum clique, is known to be NP-complete for $t$-interval graphs when $t\geq 3$ and polynomial-time solvable when $t=1$. The problem is also known to be NP-complete in $t$-track graphs when $t\geq 4$ and polynomial-time solvable when $t\leq 2$. We show that MAXIMUM CLIQUE is already NP-complete for unit 2-interval graphs and unit 3-track graphs. Further, we show that the problem is APX-complete for 2-interval graphs, 3-track graphs, unit 3-interval graphs and unit 4-track graphs. We also introduce two new classes of graphs called $t$-circular interval graphs and $t$-circular track graphs and study the complexity of the MAXIMUM CLIQUE problem in them. On the positive side, we present a polynomial time $t$-approximation algorithm for WEIGHTED MAXIMUM CLIQUE on $t$-interval graphs, improving earlier work with approximation ratio $4t$.