Algorithms and hardness results for the (k,ℓ)-cover problem

Abstract

A connected graph has a $(k,\ell )$-cover if each of its edges is contained in at least ℓ 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 $k\ge 3$, we show that the $(k,1)$-cover problem is $\mathrm{NP}$-complete for general graphs. Moreover, we show that for every constant $k\ge 3$, the $(k,1)$-cover problem admits no polynomial-time constant-factor approximation algorithm unless $P=\mathrm{NP}$. 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 $\mathrm{NP}$-hard even for spiders. However, we show that for every $k\ge 4$, the $(3,k-2)$-cover and the $(k,1)$-cover problems are constant-factor approximable when the input graph is a tree.