An NP-hardness result for the colored constrained maximum 2-edge-colorable subgraph problem in bipartite graphs

TL;DR

The paper addresses the maximum $k$-edge-colorable subgraph problem, highlighting a dichotomy: it is NP-hard in general/cubic graphs but solvable in polynomial time for bipartite graphs via a max-flow construction. It introduces a vertex-color-constraint variant $ u_2^W(G)$ and proves NP-hardness for connected bipartite graphs with maximum degree $3$ through a reduction from Max $2$-SAT using the KM2008 construction $G_I$, establishing a key relationship $ u_2^W(G_I)= u(G_I)+L(G_I)$. This reduction ties the constrained colorings to special maximum matching structures, demonstrating paraNP-hardness in $k$ and outlining avenues for ETH-based lower bounds and potential exact algorithms. The work links matching theory, flow techniques, and SAT-based reductions to map the boundary between tractable and intractable instances in bipartite graph settings. Overall, it clarifies how vertex-wise color constraints fundamentally alter complexity, even in otherwise tractable bipartite graphs.

Abstract

In this paper, we consider the maximum $k$-edge-colorable subgraph problem. In this problem we are given a graph $G$ and a positive integer $k$, the goal is to take $k$ matchings of $G$ such that their union contains maximum number of edges. This problem is NP-hard in cubic graphs, and polynomial-time solvable in bipartite graphs as we observe in our paper. We present an NP-hardness result for a version of this problem where we have color constraints on vertices. In fact, we show that this version is NP-hard already in bipartite graphs of maximum degree three. In order to achieve the result, we establish a connection between our problem and the problem of construction of special maximum matchings considered in the Master thesis of the author and defended back in 2003.

Figures (10)

• Figure 1: A graph in which the maximum matching is not a subset of a maximum $2$-edge-colorable subgraph.
• Figure 2: The network $H$ obtained from the bipartite graph $G$. The source $s$ of $H$ is joined to every vertex of $A$ with an arc of capacity $k$. Every edge of $G$ is replaced with an arc of capacity $1$. Every vertex of $B$ is joined to the sink $t$ with an arc of capacity $k$.
• Figure 3: In the path of length four, $\ell(G)=1$ and $L(G)=2$.
• Figure 4: The gadget corresponding to the variable $x_i$ and the clause $C_j$.
• Figure 5: The gadget corresponding to the literal $\overline{x}_i$ and clause $C_j$.

Theorems & Definitions (4)

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