Efficient Constant-Factor Approximate Enumeration of Minimal Subsets for Monotone Properties with Weight Constraints
Yasuaki Kobayashi, Kazuhiro Kurita, Kunihiro Wasa
TL;DR
The paper introduces the notion of approximate enumeration for minimal subsets satisfying a monotone property under a weight constraint, framed via a supergraph strategy that uses seed solutions and input-restricted subproblems to achieve constant-factor, output-sensitive enumeration. It presents two general frameworks that yield polynomial-delay (and sometimes incremental-polynomial-time) algorithms for a broad class of problems, including Vertex Cover, bounded-degree vertex deletions, $d$-Hitting Set, Star Forest Edge Deletion, and Dominating Set in bounded-degree graphs, with approximation factors that depend on the framework and subroutines. Beyond these frameworks, the authors provide polynomial-delay constant-factor approximate enumerations for Minimal Edge Dominating Sets and Minimal Steiner Subgraphs, addressing non-CK properties with problem-specific neighborhood constructions. The results offer a practical means to enumerate multiple small, near-optimal solutions rather than a single optimal one, with potential benefits for applications where exact objectives are ambiguous or hard to model. Overall, the work broadens the toolkit for output-sensitive enumeration under weight constraints across a range of monotone properties and graph problems, integrating MSO/Courcelle techniques, IRP subproblems, and tailored neighborhood strategies.
Abstract
A property $Π$ on a finite set $U$ is \emph{monotone} if for every $X \subseteq U$ satisfying $Π$, every superset $Y \subseteq U$ of $X$ also satisfies $Π$. Many combinatorial properties can be seen as monotone properties. The problem of finding a minimum subset of $U$ satisfying $Π$ is a central problem in combinatorial optimization. Although many approximate/exact algorithms have been developed to solve this kind of problem on numerous properties, a solution obtained by these algorithms is often unsuitable for real-world applications due to the difficulty of building accurate mathematical models on real-world problems. A promising approach to overcome this difficulty is to \emph{enumerate} multiple small solutions rather than to \emph{find} a single small solution. To this end, given a weight function $w: U \to \mathbb N$ and an integer $k$, we devise algorithms that \emph{approximately} enumerate all minimal subsets of $U$ with weight at most $k$ satisfying $Π$ for various monotone properties $Π$, where "approximate enumeration" means that algorithms output all minimal subsets satisfying $Π$ whose weight at most $k$ and may output some minimal subsets satisfying $Π$ whose weight exceeds $k$ but is at most $ck$ for some constant $c \ge 1$. These algorithms allow us to efficiently enumerate minimal vertex covers, minimal dominating sets in bounded degree graphs, minimal feedback vertex sets, minimal hitting sets in bounded rank hypergraphs, etc., of weight at most $k$ with constant approximation factors.