Approximating Maximum Edge 2-Coloring by Normalizing Graphs

TL;DR

The paper studies maximum edge 2-coloring (ME2C), seeking to maximize the number of colors in an edge coloring where each vertex sees at most two colors. It introduces a normalization preprocessing framework that preserves the optimal ME2C value and augments the classic matching-based algorithm, enabling stronger guarantees. By analyzing normalized graphs via character graphs and a upper-bound lemma, the authors derive improved approximation ratios: $1.5$-approximation for claw-free and subcubic graphs, and a refined $1.625$-approximation for graphs that contain a perfect matching. These results advance the state of ME2C approximations, with implications for anti-Ramsey problems and network design where channel coloring constraints arise.

Abstract

In a simple, undirected graph G, an edge 2-coloring is a coloring of the edges such that no vertex is incident to edges with more than 2 distinct colors. The problem maximum edge 2-coloring (ME2C) is to find an edge 2-coloring in a graph G with the goal to maximize the number of colors. For a relevant graph class, ME2C models anti-Ramsey numbers and it was considered in network applications. For the problem a 2-approximation algorithm is known, and if the input graph has a perfect matching, the same algorithm has been shown to have a performance guarantee of 5/3. It is known that ME2C is APX-hard and that it is UG-hard to obtain an approximation ratio better than 1.5. We show that if the input graph has a perfect matching, there is a polynomial time 1.625-approximation and if the graph is claw-free or if the maximum degree of the input graph is at most three (i.e., the graph is subcubic), there is a polynomial time 1.5-approximation algorithm for ME2C

Figures (6)

• Figure 1: Modifications 1 and 2, and a simple cactus (Modification 3) as a subgraph of $G$.
• Figure 2: Subfigures a) and b): Modification 3 for a simple triangular cactus consisting of just one triangle. Subfigures a) and c) show a variant of Modification 3 that keeps the perfect matching (used in \ref{['sec:pm']}).
• Figure 3: End vertices are marked by hollow circles, inner vertices are marked by filled circles. A half-filled vertex can be either an inner- or an end vertex.
• Figure 4: Worst-case instance containing a perfect matching from AdamaszekPopa2016. Without applying modifications, \ref{['alg:basic']} may chose a perfect matching such that removing the matching leaves a connected graph -- resulting in a $5/3 \approx 1.667$ approximation. An improved approximation ratio of $1.625$ is possible due to Modification 3. In particular, Modification 3 replaces all simple cacti by independent edges, which results in a modified graph $G'$ consisting only of independent edges, which means that \ref{['alg:basic']} actually computes an optimal solution.
• Figure 5: Modification 4: Bridge removal, when $\deg(u)=3$ and $\deg(v)=1$.

Theorems & Definitions (58)

• theorem 1
• theorem 2
• theorem 3
• Lemma 1
• Definition 1
• Definition 2
• Lemma 2
• proof
• Lemma 3
• proof