Disjoint covering of bipartite graphs with $s$-clubs

TL;DR

We address partitioning and disjoint covering of bipartite graphs by $s$-clubs, focusing on two problems: PC($k,s$) (partition into at most $k$ disjoint $s$-clubs) and MAX-DCC($t,s$) (maximize vertices covered by disjoint $(t,s)$-clubs). The main techniques are reductions from $k$-List Coloring and Max-2B3DM, complemented by a dedicated gadget construction that enforces color constraints and club diameters. The paper proves NP-hardness for fixed $s \ge 6$ and $k \ge 5$, and APX-hardness for MAX-DCC$(t,3)$ with $t \ge 8$ and MAX-DCC$(t,2)$ with $t \ge 5$ (on bipartite graphs), while providing a polynomial-time algorithm for MAX-DCC$(2,2)$. These results delineate hardness boundaries for flexible clustering models in bipartite networks and guide future research toward unresolved cases and tractable instances.

Abstract

For a positive integer $s$, an $s$-club in a graph $G$ is a set of vertices inducing a subgraph with diameter at most $s$. As generalizations of cliques, $s$-clubs offer a flexible model for real-world networks. This paper addresses the problems of partitioning and disjoint covering of vertices with $s$-clubs on bipartite graphs. First we prove that for any fixed $s \geq 6$ and fixed $k \geq 5$, determining whether the vertices of $G$ can be partitioned into at most $k$ disjoint $s$-clubs is NP-complete even for bipartite graphs. Note that our NP-completeness result is stronger than the one in Abbas and Stewart (1999), as we assume that both $s$ and $k$ are constants and not part of the input. Additionally, we study the Maximum Disjoint $(t,s)$-Club Covering problem (MAX-DCC($t,s$)), which aims to find a collection of vertex-disjoint $(t,s)$-clubs (i.e. $s$-clubs with at least $t$ vertices) that covers the maximum number of vertices in $G$. We prove that it is NP-hard to achieve an approximation factor of $\frac{95}{94} $ for MAX-DCC($t,3$) for any fixed $t\geq 8$ and for MAX-DCC($t,2$) for any fixed $t\geq 5$ even for bipartite graphs. Previously, results were known only for MAX-DCC($3,2$). Finally, we provide a polynomial-time algorithm for MAX-DCC($2,2$).

Figures (3)

• Figure 1: (a) A bipartite graph $G$ with the list coloring $L: V(G) \to [3]$ and (b) the corresponding graph $G'$ for $s=6$ and $k=5$. The gray vertices represent the auxiliary vertices added in the construction.
• Figure 2: The graph $G_{M,8,3}$ obtained when the instance of Max-2B3DM problem is the set $M=\{(x_1, y_1, z_1), \,\,(x_2, y_1, z_1)\,\,(x_1, y_2, z_2)\}$.
• Figure 3: The graph $G_{M,5,2}$ obtained when the instance of Max-2B3DM problem is the set $M=\{(x_1, y_1, z_1), \,\,(x_2, y_1, z_1)\,\,(x_1, y_2, z_2)\}$.

Theorems & Definitions (22)

• Lemma 1
• proof
• Lemma 2
• proof
• Claim 1
• proof
• Theorem 1
• proof
• Claim 2
• proof