Complex contraction on trees without proof of correlation decay

TL;DR

This paper introduces a novel complex contraction method in the complex plane for counting problems on trees, enabling efficient approximation schemes without relying on correlation decay proofs.

Contribution

It presents a unified approach for identifying contraction regions in the complex plane for various counting problems, broadening the applicability of FPTAS via Barvinok's algorithm.

Findings

Established complex contraction regions for weighted set cover problems

Unified the approach for multiple counting problems like edge covers and bipartite independent sets

Achieved FPTAS without correlation decay proofs

Abstract

We prove complex contraction for zero-free regions of counting weighted set cover problem in which an element can appear in an unbounded number of sets, thus obtaining fully polynomial-time approximation schemes(FPTAS) via Barvinok's algorithmic paradigm\cite{barvinok2016combinatorics}. Relying on the computation tree expansion, our approach does not need proof of correlation decay in the real axis. We directly look in the complex plane for a region that contracts into its interior as the tree recursion procedure goes from leaves to the root. For the class of problems under the framework of weighted set covers, we are able to give a general approach for describing the contraction regions and draw a unified algorithmic conclusion. Several previous results, including counting (weighted-)edge covers, counting bipartite independent sets and counting monotone CNFs can be completely or partially covered by our main theorem. In contrast to the correlation decay method which also depends on tree expansions and needs different potential functions for different problems, our approach is more generic in the sense that our contraction region for different problems shares a common shape in the complex plane.