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

TL;DR

This work investigates the algorithmic version of the $(k,\ell)$-cover problem, where each edge must lie in at least $\ell$ copies of $K_k$ after adding non-edges. It proves NP-completeness for general graphs for all fixed $k\ge 3$ and shows no constant-factor approximation unless $\mathbb{P}=\mathbb{NP}$, via SET-COVER based reductions; however, it identifies a polynomial-time algorithm for the $(3,1)$-cover on chordal graphs and, separately, NP-hardness on trees including spiders. For trees, it provides constant-factor approximations: an $8/3$-approximation for $k\ge 5$ (and the $(3,k-2)$-cover) in $O(nk)$ time, and a 2-approximation for $k=4$ in $O(n\log n)$ time. The results illuminate the landscape of tractable and intractable instances and establish a foundation for further approximation or exact algorithms in structured graph classes.

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.

Figures (9)

• Figure 1: An example of the reduction graph $\mathbb{G}$ for $\mathcal{F}=\{S_1, S_2,S_3\}$, $\mathcal{X}=\{x_1,x_2,x_3\}$, $S_1=\{x_1,x_2\}$, $S_2=\{x_2,x_3\}$, and $S_3=\{x_3\}$. Every black edge in this graph is contained in a triangle consisting of its endpoints plus one other auxiliary vertex omitted from this figure for simplicity (see Step 5 of the construction). Therefore, only the red edges of this graph are not contained in any triangles.
• Figure 2: An example of the reduction graph $\mathbb{G}$ for $k=4$, $\mathcal{F}=\{S_1, S_2,S_3\}$, $\mathcal{X}=\{x_1,x_2,x_3\}$, $S_1=\{x_1,x_2\}$, $S_2=\{x_2,x_3\}$, and $S_3=\{x_3\}$. Every black edge in this graph is contained in a $k$-clique consisting of its endpoints plus $k-2$ other vertices omitted from this figure for simplicity (see Step 5 and Step 6 of the construction). Therefore, only the red edges of this graph are not contained in any $k$-cliques.
• Figure 3: An example of a chordal graph and its trees (depicted in red). The outer and boundary vertices are depicted in black and blue, respectively.
• Figure 4: An example of a chordal graph $G$ with trees $T_1$ and $T_2$, and two completion sets $E'$ (a) and $E"$ (b). Solid edges belong to $G$, and the blue dashed edges are the edges of the completion sets.
• Figure 5: An example of \ref{['iterell']} for $\ell=3$: Solid red edges are the edges of $F_{\ell-1}$. Green edges belong to $T_j$ and $T_i$ but not to $F_{\ell-1}$. We remove the red dashed edges and replace them with the blue dashed ones. In this example, $\mathbb{C}=\{e_1, e_2, e_3\}$ at the end of iteration $\ell$. Moreover, we add $e'_1$, $e'_2$, and $e'_3$ in the first three iterations after removing $e_1$, $e_2$, and $e_3$, respectively. Observe how $F_{\ell-1} \cup (u,v)$ remains a forest with exactly two components.

Theorems & Definitions (64)

• Definition 1
• Definition 2
• Definition 3
• Lemma 1
• proof
• Lemma 2
• proof
• Claim 1
• proof
• Theorem 1