Algorithms and Hardness Results for the $(k,\ell)$-Cover Problem

TL;DR

This paper investigates the computational complexity and approximation possibilities of the $(k,\, ext{ell})$-cover problem in graphs, revealing NP-hardness, inapproximability, and polynomial-time solutions for special graph classes.

Contribution

It establishes NP-completeness and inapproximability results for the $(k,1)$-cover problem, and provides polynomial algorithms for chordal graphs and approximation algorithms for trees.

Findings

$(k,1)$-cover is NP-complete for general graphs when $k\,\geq 3$.

No polynomial-time constant-factor approximation exists for $(k,1)$-cover unless P=NP.

Polynomial-time solution for $(3,1)$-cover on chordal graphs.

Abstract

A connected graph has a $(k,\ell)$-cover if each of its edges is contained in at least $\ell$ cliques of order $k$. Motivated by recent advances in extremal combinatorics and the literature on edge modification problems, we study the algorithmic version of the $(k,\ell)$-cover problem. Given a connected graph $G$, the $(k, \ell)$-cover problem is to identify the smallest subset of non-edges of $G$ such that their addition to $G$ results in a graph with a $(k, \ell)$-cover. For every constant $k\geq3$, we show that the $(k,1)$-cover problem is $\mathbb{NP}$-complete for general graphs. Moreover, we show that for every constant $k\geq 3$, the $(k,1)$-cover problem admits no polynomial-time constant-factor approximation algorithm unless $\mathbb{P}=\mathbb{NP}$. However, we show that the $(3,1)$-cover problem can be solved in polynomial time when the input graph is chordal. For the class of trees and general values of $k$, we show that the $(k,1)$-cover problem is $\mathbb{NP}$-hard even for spiders. However, we show that for every $k\geq4$, the $(3,k-2)$-cover and the $(k,1)$-cover problems are constant-factor approximable when the input graph is a tree.