Faster exponential algorithms for cut problems via geometric data structures

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 - ($\alpha,\beta$)-Domination: given intervals $\alpha,\beta \subseteq [0,n]$, 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 $\alpha$ and for every vertex $v \notin S$, the number of neighbors of $v$ in $S$ is from $\beta$. 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 $\alpha$ and $\beta$; for the second problem we are not aware of such results.