Exact Algorithms for MaxCut on Split Graphs

TL;DR

The paper addresses finding maximum cuts on split graphs, introducing two polynomial-space exponential-time subroutines that yield an $O^{*}(1.42^{n})$-time exact algorithm. It also presents a subexponential-time decision algorithm running in $2^{O(\,\sqrt{k}\,)}\,\text{poly}$ by exploiting a bound on the clique size, with ETH-based optimality considerations. The contributions establish tight complexity bounds for MaxCut on split graphs and suggest avenues for extending these results to related graph classes, while highlighting limitations due to known reductions. This work advances exact algorithm techniques for graphs with large homogeneous parts and informs future research on leveraging structured graph decompositions for hard combinatorial problems.

Abstract

This paper presents an $O^{*}(1.42^{n})$ time algorithm for the Maximum Cut problem on split graphs, along with a subexponential time algorithm for its decision variant.

Figures (4)

• Figure 1: A graph $G$.
• Figure 2: A split graph $G'$.
• Figure 3: MaxCut$(G, I)$
• Figure 4: MaxCut$(G, C)$

Theorems & Definitions (10)

• Proposition 1
• proof
• Lemma 1
• proof
• Lemma 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof