Algorithms and hardness results for the (k,ℓ)-cover problem
Amirali Madani, Anil Maheshwari, Babak Miraftab, Bodhayan Roy · Journal of Computer and System Sciences · 2025
A connected graph has a ( k , ℓ ) -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 , ℓ ) -cover problem. Given a connected graph G , the ( k , ℓ ) -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 , ℓ ) -cover. For every constant k ≥ 3 , we show that the ( k , 1 ) -cover problem is NP -complete for general graphs. Moreover, we show that for every constant k ≥ 3 , the ( k , 1 ) -cover problem admits no polynomial-time constant-factor approximation algorithm unless P = 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 NP -hard even for spiders. However, we show that for every k ≥ 4 , the ( 3 , k − 2 ) -cover and the ( k , 1 ) -cover problems are constant-factor approximable when the input graph is a tree.