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

2025-02-04

AI Summary

This paper investigates the $(k,ell)$-cover problem on graphs: given a connected graph $G$, add the minimum number of non-edges so that every edge lies in at least $ell$ distinct $k$-cliques. We establish the first systematic complexity characterization for $(k,1)$-cover: it is NP-complete and constant-factor inapproximable for general graphs when $k geq 3$; polynomial-time solvable on chordal graphs for $(3,1)$-cover; and exhibits a sharp threshold for approximability on trees—$(k,1)$-cover admits constant-factor approximation, whereas $(3,k-2)$-cover is inapproximable within any constant factor. Our approach integrates combinatorial graph theory, computational complexity analysis, dynamic programming and greedy constructions on trees, and introduces a novel clique-cover construction technique. The results yield precise complexity classifications and algorithmic boundaries for the $(k,ell)$-cover problem across multiple graph classes, including general graphs, chordal graphs, and trees.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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 $kgeq3$, we show that the $(k,1)$-cover problem is $mathbb{NP}$-complete for general graphs. Moreover, we show that for every constant $kgeq 3$, 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 $kgeq4$, the $(3,k-2)$-cover and the $(k,1)$-cover problems are constant-factor approximable when the input graph is a tree.

Problem

• Studies the $(k,ell)$-cover problem in connected graphs.
• Proves $(k,1)$-cover is $mathbb{NP}$-complete for general graphs.
• Provides polynomial-time solutions for chordal graphs and trees.

Innovation

• NP-complete for general graphs
• Polynomial-time solvable for chordal graphs
• Constant-factor approximable for trees