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Faster parameterized algorithm for 3-Hitting Set

Dekel Tsur

TL;DR

An $O*(2.0409^k)$-time algorithm for 3-Hitting Set, where O is the number of vertices in the hypergraph and k is the number of integers in the hypergraph.

Abstract

In the 3-Hitting Set problem, the input is a hypergraph $G$ such that the size of every hyperedge of $G$ is at most 3, and an integers $k$, and the goal is to decide whether there is a set $S$ of at most $k$ vertices such that every hyperedge of $G$ contains at least one vertex from $S$. In this paper we give an $O^*(2.0409^k)$-time algorithm for 3-Hitting Set.

Faster parameterized algorithm for 3-Hitting Set

TL;DR

An -time algorithm for 3-Hitting Set, where O is the number of vertices in the hypergraph and k is the number of integers in the hypergraph.

Abstract

In the 3-Hitting Set problem, the input is a hypergraph such that the size of every hyperedge of is at most 3, and an integers , and the goal is to decide whether there is a set of at most vertices such that every hyperedge of contains at least one vertex from . In this paper we give an -time algorithm for 3-Hitting Set.
Paper Structure (10 sections, 22 theorems, 25 equations, 2 figures, 2 tables)

This paper contains 10 sections, 22 theorems, 25 equations, 2 figures, 2 tables.

Table of Contents

  1. Keywords
  2. Introduction
  3. Preliminaries
  4. The algorithm
  5. Analysis
  6. Analysis of Rules (\ref{['rule:cycle']})--(\ref{['rule:deg2-33']})
  7. Analysis of Rule (\ref{['rule:final']})
  8. $d' = 1$
  9. $d' = 2$
  10. Analysis of the algorithm

Key Result

Lemma 1

Let $(G,k)$ be an instance and let $(G',k')$ be an instance that is obtained by applying a reduction rule or a branching rule on $(G,k)$. If $G$ does not contain an empty hyperedge, then $G'$ does not contain an empty hyperedge.

Figures (2)

  • Figure 1: The cases of Lemma \ref{['lem:rule-deg2-2x-2']}. Ellipses with solid lines are hyperedges that must exist, and ellipses with dashed lines are 2-hyperedges that may exist.
  • Figure 2: The cases of Lemma \ref{['lem:rule-deg2-2x-1']}.

Theorems & Definitions (58)

  • Lemma 1
  • proof
  • Claim 2
  • proof
  • Claim 3
  • proof
  • Claim 4
  • proof
  • Claim 5
  • proof
  • ...and 48 more