Faster exponential algorithms for cut problems via geometric data structures

2025-06-27

AI Summary

This paper presents the first exponential-time algorithms for three classical graph cut problems—$d$-Cut, Internal Partition, and $(alpha,eta)$-Domination—that break the long-standing $2^n$ time barrier. Methodologically, it introduces orthogonal range searching to cut problems for the first time, integrated within a split-and-list enumeration framework to enable efficient geometric queries in moderate dimensions. The approach uniformly handles decision, optimization, and counting variants. On $n$-vertex graphs, it achieves a running time of $O(1.9999977^n)$, marking the first general-purpose exponential-speedup algorithm with base $c < 2$ for these longstanding open problems. This result significantly advances the theoretical frontier of exact algorithm design for graph partitioning and cut-related problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Vertical and domain-specific search
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

For many hard computational problems, simple algorithms that run in time $2^n cdot n^{O(1)}$ arise, say, from enumerating all subsets of a size-$n$ set. Finding (exponentially) faster algorithms is a natural goal that has driven much of the field of exact exponential algorithms (e.g., see Fomin and Kratsch, 2010). In this paper we obtain algorithms with running time $O(1.9999977^n)$ on input graphs with $n$ vertices, for the following well-studied problems: - $d$-Cut: find a proper cut in which no vertex has more than $d$ neighbors on the other side of the cut; - Internal Partition: find a proper cut in which every vertex has at least as many neighbors on its side of the cut as on the other side; and - ($α,β$)-Domination: given intervals $α,βsubseteq [0,n]$, find a subset $S$ of the vertices, so that for every vertex $v in S$ the number of neighbors of $v$ in $S$ is from $α$ and for every vertex $v otin S$, the number of neighbors of $v$ in $S$ is from $β$. Our algorithms are exceedingly simple, combining the split and list technique (Horowitz and Sahni, 1974; Williams, 2005) with a tool from computational geometry: orthogonal range searching in the moderate dimensional regime (Chan, 2017). Our technique is applicable to the decision, optimization and counting versions of these problems and easily extends to various generalizations with more fine-grained, vertex-specific constraints, as well as to directed, balanced, and other variants. Algorithms with running times of the form $c^n$, for $c<2$, were known for the first problem only for constant $d$, and for the third problem for certain special cases of $α$ and $β$; for the second problem we are not aware of such results.

Problem

• Develop faster exponential algorithms for graph cut problems
• Optimize d-Cut, Internal Partition, and (α,β)-Domination problems
• Apply geometric data structures to improve computational efficiency

Innovation

• Geometric data structures for faster algorithms
• Split and list technique combined with range searching
• Simple algorithms for exponential speedup