Abusing the Tutte Matrix: An Algebraic Instance Compression for the K-set-cycle Problem

TL;DR

This paper introduces an algebraic determinant-based algorithm for the K-Cycle problem, achieving fixed-parameter tractability and polynomial compression, which challenges existing assumptions about problem kernelization.

Contribution

It presents a novel algebraic approach to the K-Cycle problem that enables polynomial compression, contrasting with the typical non-existence of such reductions for similar problems.

Findings

Achieves an $O^*(2^{|K|})$ fixed-parameter tractable algorithm.

Provides a polynomial compression of the K-Cycle problem into an algebraic problem.

Shows that polynomial compression does not imply polynomial kernelization for this problem.

Abstract

We give an algebraic, determinant-based algorithm for the K-Cycle problem, i.e., the problem of finding a cycle through a set of specified elements. Our approach gives a simple FPT algorithm for the problem, matching the $O^*(2^{|K|})$ running time of the algorithm of Bj\"orklund et al. (SODA, 2012). Furthermore, our approach is open for treatment by classical algebraic tools (e.g., Gaussian elimination), and we show that it leads to a polynomial compression of the problem, i.e., a polynomial-time reduction of the $K$-Cycle problem into an algebraic problem with coding size $O(|K|^3)$. This is surprising, as several related problems (e.g., k-Cycle and the Disjoint Paths problem) are known not to admit such a reduction unless the polynomial hierarchy collapses. Furthermore, despite the result, we are not aware of any witness for the K-Cycle problem of size polynomial in $|K|+\log n$, which seems (for now) to separate the notions of polynomial compression and polynomial kernelization (as a polynomial kernelization for a problem in NP necessarily implies a small witness).