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

Amirali Madani; Anil Maheshwari; Babak Miraftab; Bodhayan Roy

doi:10.1016/j.jcss.2025.103727

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arXiv · 2502.02572

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

Amirali Madani·Anil Maheshwari·Babak Miraftab·Bodhayan Roy

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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.

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Amirali Madani, Anil Maheshwari, Babak Miraftab, Bodhayan Roy. 2025-02-04. Algorithms and Hardness Results for the $(k,\ell)$-Cover Problem. https://doi.org/10.1016/j.jcss.2025.103727

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