DeepAI

# Efficient Constant-Factor Approximate Enumeration of Minimal Subsets for Monotone Properties with Cardinality Constraints

A property Π on a finite set U is monotone if for every X ⊆ U satisfying Π, every superset Y ⊆ U of X also satisfies Π. Many combinatorial properties can be seen as monotone properties, and 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 problem on numerous properties, a solution obtained by these algorithms is often unsuitable for real-world applications due to the difficulty of building mathematical models on real-world problems. A promising approach to overcome this difficulty is to enumerate multiple small solutions rather than to find a single small solution. To this end, given an integer k, we devise algorithms that approximately enumerate all minimal subsets of U with cardinality at most k satisfying Π for various monotone properties Π, where "approximate enumeration" means that algorithms may output some minimal subsets satisfying Π whose cardinality exceeds k and is at most ck for some constant c ≥ 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., with constant approximation factors.

• 29 publications
• 16 publications
• 18 publications
01/21/2022

### An Approximation Algorithm for K-best Enumeration of Minimal Connected Edge Dominating Sets with Cardinality Constraints

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### Non-monotone target sets for threshold values restricted to 0, 1, and the vertex degree

We consider a non-monotone activation process (X_t)_t∈{ 0,1,2,…} on a gr...
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### Bounded-Degree Cut is Fixed-Parameter Tractable

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### Infinitely exchangeable random graphs generated from a Poisson point process on monotone sets and applications to cluster analysis for networks

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01/05/2016

### The minimal hitting set generation problem: algorithms and computation

Finding inclusion-minimal "hitting sets" for a given collection of sets ...
06/10/2016

### Tunable Online MUS/MSS Enumeration

In various areas of computer science, the problem of dealing with a set ...
12/13/2014

### Combinatorial Structure of the Deterministic Seriation Method with Multiple Subset Solutions

Seriation methods order a set of descriptions given some criterion (e.g....