2025/02/04 by Madani, Amirali, Maheshwari, Anil, Miraftab, Babak +1
#Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2502.02572
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 \mathbbNP-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=\mathbbNP. 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 \mathbbNP-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.