Solving Vertex Integrity Faster than $2^n$

2026-09-27

AI Summary

This study addresses the exponential-time bottleneck in exactly computing graph vertex integrity by investigating its algorithmic upper bounds and complexity lower bounds. Methodologically, it innovatively combines balanced partition techniques based on optimal non-redundant separators with bounded-size dynamic programming strategies, and establishes a reduction from Vertex Cover on subcubic graphs to derive the lower bound. The primary contributions include the first deterministic exact algorithm that reduces both time and space complexities to $O(1.9602^n)$, breaking the naive $O^*(2^n)$ barrier. Furthermore, under the Exponential Time Hypothesis (ETH), it proves a complexity lower bound of $2^{o(n)}n^{O(1)}$, thereby establishing a tight complexity characterization for the exact computation of vertex integrity.

Abstract

The vertex integrity of a graph $G$ is the minimum of $|S|+\max_{C\in\operatorname{cc}(G-S)}|V(C)|$ over all vertex sets $S\subseteq V(G)$, where the maximum is zero if $G-S$ is empty. We study its exact exponential complexity in terms of $n=|V(G)|$. First, we give a reduction from Vertex Cover on subcubic graphs that increases the number of vertices by only a constant factor. Consequently, unless the Exponential Time Hypothesis fails, Vertex Integrity admits no $2^{o(n)}n^{O(1)}$-time algorithm. We also give a deterministic exact algorithm running in $O(1.9602^n)$ time and space, improving on the direct $O^*(2^n)$ algorithm. Its key ingredients are a balanced partition of the components left by an optimal irredundant separator and a subset dynamic program restricted to sets of at most $\lceil 2n/5\rceil$ vertices. An optimal separator can be recovered within the same bounds.

Problem

• Vertex Integrity
• Exact Exponential Complexity
• Exponential Time Hypothesis

Innovation

• Vertex Integrity
• Exact Exponential Algorithm
• Exponential Time Hypothesis
• Subset Dynamic Programming
• Balanced Partition