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A Quadratic Vertex Kernel and a Subexponential Algorithm for Subset-FAST

Satyabrata Jana, Lawqueen Kanesh, Madhumita Kundu, Daniel Lokshtanov, Saket Saurabh

TL;DR

The paper tackles Subset-FAST in tournaments, introducing the first polynomial kernel of size $|V(T')|=O(k^2)$ and a subexponential FPT algorithm. It achieves this by proving an equivalence to Subset-FAST-Rev, bounding the terminal set to $O(k)$, and exploiting an $S$-topological ordering to bound neighborhood-classes, which together yield a compact kernel and enable efficient dynamic-programming on reduced instances. Central ideas include the reversal-equivalence for minimal solutions, the no-$S$-triangle characterization of $S$-acyclicity, and the $S$-topological ordering that tightens the instance structure. Collectively, these results advance the tractability of Subset-FAST, offering practical reductions and a subexponential-time route for instances where $k$ is the parameter.

Abstract

In the Subset Feedback Arc Set in Tournaments, Subset-FAST problem we are given as input a tournament $T$ with a vertex set $V(T)$ and an arc set $A(T)$, along with a terminal set $S \subseteq V(T)$, and an integer $ k$. The objective is to determine whether there exists a set $ F \subseteq A(T) $ with $|F| \leq k$ such that the resulting graph $T-F $ contains no cycle that includes any vertex of $S$. When $S=V(T)$ this is the classic Feedback Arc Set in Tournaments (FAST) problem. We obtain the first polynomial kernel for this problem parameterized by the solution size. More precisely, we obtain an algorithm that, given an input instance $(T, S, k)$, produces an equivalent instance $(T',S',k')$ with $k'\leq k$ and $V(T')=O(k^2)$. It was known that FAST admits a simple quadratic vertex kernel and a non-trivial linear vertex kernel. However, no such kernel was previously known for Subset-FAST. Our kernel employs variants of the most well-known reduction rules for FAST and introduces two new reduction rules to identify irrelevant vertices. As a result of our kernelization, we also obtain the first sub-exponential time FPT algorithm for Subset-FAST.

A Quadratic Vertex Kernel and a Subexponential Algorithm for Subset-FAST

TL;DR

The paper tackles Subset-FAST in tournaments, introducing the first polynomial kernel of size and a subexponential FPT algorithm. It achieves this by proving an equivalence to Subset-FAST-Rev, bounding the terminal set to , and exploiting an -topological ordering to bound neighborhood-classes, which together yield a compact kernel and enable efficient dynamic-programming on reduced instances. Central ideas include the reversal-equivalence for minimal solutions, the no--triangle characterization of -acyclicity, and the -topological ordering that tightens the instance structure. Collectively, these results advance the tractability of Subset-FAST, offering practical reductions and a subexponential-time route for instances where is the parameter.

Abstract

In the Subset Feedback Arc Set in Tournaments, Subset-FAST problem we are given as input a tournament with a vertex set and an arc set , along with a terminal set , and an integer . The objective is to determine whether there exists a set with such that the resulting graph contains no cycle that includes any vertex of . When this is the classic Feedback Arc Set in Tournaments (FAST) problem. We obtain the first polynomial kernel for this problem parameterized by the solution size. More precisely, we obtain an algorithm that, given an input instance , produces an equivalent instance with and . It was known that FAST admits a simple quadratic vertex kernel and a non-trivial linear vertex kernel. However, no such kernel was previously known for Subset-FAST. Our kernel employs variants of the most well-known reduction rules for FAST and introduces two new reduction rules to identify irrelevant vertices. As a result of our kernelization, we also obtain the first sub-exponential time FPT algorithm for Subset-FAST.

Paper Structure

This paper contains 6 sections, 4 theorems, 2 figures.

Table of Contents

  1. Introduction
  2. Our Results and Methods
  3. Preliminaries
  4. Kernelization.
  5. Characterization of Subset-FAST as Arcs Reversal Problem
  6. $\boldsymbol{S}$-Topological Ordering

Key Result

Theorem 1

Subset-FAST admits a kernel with ${\mathcal{O}}\xspace(k^2)$ vertices.

Figures (2)

  • Figure 1: A summary of the steps of our kernelization algorithm.
  • Figure 2: Illustration of \ref{['lem:minimal_equivalence']}: equivalence between Subset-FAST and Subset-FAST-Rev. Red vertices indicates $S$-vertices.

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 5
  • Definition 6: ${\mathsf{S} \text{-}\textsf{topological ordering}}\xspace$