Parameterized Local Search for Max $c$-Cut

TL;DR

It is shown that using parameterized local search, the results of this heuristic can be further improved on a set of standard benchmark instances and the practical performance of this algorithm in a hill-climbing approach as a post-processing for state-of-the-art heuristics for Max c-Cut.

Abstract

In the NP-hard Max $c$-Cut problem, one is given an undirected edge-weighted graph $G$ and aims to color the vertices of $G$ with $c$ colors such that the total weight of edges with distinctly colored endpoints is maximal. The case with $c=2$ is the famous Max Cut problem. To deal with the NP-hardness of this problem, we study parameterized local search algorithms. More precisely, we study LS Max $c$-Cut where we are also given a vertex coloring and an integer $k$ and the task is to find a better coloring that changes the color of at most $k$ vertices, if such a coloring exists; otherwise, the given coloring is $k$-optimal. We show that, for all $c\ge 2$, LS Max $c$-Cut presumably cannot be solved in $f(k)\cdot n^{\mathcal{O}(1)}$ time even on bipartite graphs. We then present an algorithm for LS Max $c$-Cut with running time $\mathcal{O}((3eΔ)^k\cdot c\cdot k^3\cdotΔ\cdot n)$, where $Δ$ is the maximum degree of the input graph. Finally, we evaluate the practical performance of this algorithm in a hill-climbing approach as a post-processing for a state-of-the-art heuristic for Max $c$-Cut. We show that using parameterized local search, the results of this state-of-the-art heuristic can be further improved on a set of standard benchmark instances.

Figures (2)

• Figure 1: The connections between the different vertex sets in $G'$. Two vertex sets $X$ and $Y$ are adjacent in the figure if $E(X,Y) \neq \emptyset$. Each vertex $v$ in a vertex set with a rectangular node is $k'$-blocked from the opposite part of the partition. The vertex set $V_\Gamma$ is not shown.
• Figure 2: Two solution for the instance of LS Max Cut constructed in the proof of \ref{['permissiveMaxCut']}. In both solutions, the parts of the respective partitions are indicated by the color of the vertices. The left partition shows the initial solution and the right partition shows an improving solution, if one exists. The flip between these partitions is an independent set of size $k$ in $G$ together with the vertex $v^*$.

Theorems & Definitions (34)

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