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

2025-03-10

AI Summary

This paper studies the Subset Feedback Arc Set in Tournaments (Subset-FAST) problem: given a tournament $T$, a terminal set $S subseteq V(T)$, and an integer $k$, determine whether at most $k$ arcs can be deleted to eliminate all cycles containing at least one vertex from $S$. When $S = V(T)$, this reduces to the classical Feedback Arc Set in Tournaments (FAST) problem. We present the first polynomial kernel of size $O(k^2)$ for Subset-FAST, resolving an open question on kernelizability. Our kernelization employs two novel reduction rules that integrate classical FAST techniques with strategies for identifying irrelevant vertices. Based on this kernel, we design the first subexponential-time fixed-parameter tractable (FPT) algorithm, running in time $2^{O(sqrt{k} log k)}$. The primary contributions are (i) the first polynomial kernel for Subset-FAST, and (ii) a significantly accelerated exact algorithmic framework enabled by this kernel.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Develops a polynomial kernel for Subset-FAST problem.
• Introduces new reduction rules for identifying irrelevant vertices.
• Provides a sub-exponential time FPT algorithm for Subset-FAST.

Innovation

• Developed polynomial kernel for Subset-FAST problem
• Introduced new reduction rules for kernelization
• Achieved sub-exponential time FPT algorithm