Computational Thresholds for Balanced and Fixed-Slice Independent Sets in Bipartite Graphs

2026-08-03

AI Summary

This work investigates the worst-case computational complexity of approximately counting and sampling fixed-slice and balanced independent sets in bipartite graphs. Focusing on the fixed-slice model with prescribed left- and right-side sizes and the balanced hard-core model conditioned on equal partition sizes, the study combines complexity reductions, phase transition analysis, and methods from statistical physics to establish new hardness results. It proves for the first time that when the overall density α lies in (1/Δ, 1/2) and the distribution is more balanced than typical Δ-regular bipartite graphs, no fully polynomial randomized approximation scheme (FPRAS) or efficient sampler exists. For the balanced hard-core model, the work completely characterizes the tractable and intractable regimes separated by a critical fugacity λ_c(Δ), revealing that its computational phase transition threshold coincides with that of the general bounded-degree hard-core model.

Abstract

Motivated by recent work of Kocurek, Oveis Gharan, and Tjowasi, which gives an efficient sampling algorithm for the hard-core model on random regular bipartite graphs by decomposing into fixed-size slices, we study the worst-case tractability of approximate counting and sampling of fixed-size slices for bipartite independent set problems. Let $G=(L\sqcup R,E)$ be a bipartite graph with $|L|=|R|=n$ and maximum degree $Δ$. The fixed-slice problem asks to sample uniformly from independent sets satisfying $|I\cap L|=α_L n$ and $|I\cap R|=α_R n$. We show that if the overall density $α$ lies in the interval $(\frac{1}Δ, \tfrac{1}{2})$, and the densities on the two sides are more balanced than the typical phase densities of a random $Δ$-regular bipartite graph, then there is no FPRAS or efficient sampling scheme unless $\mathbf{NP}=\mathbf{RP}$. We then study a related fugacity model in which the densities are not fixed, but the independent set is required to be balanced between the two sides of the bipartition. For $λ>0$, the balanced hard-core model is the ordinary hard-core model with fugacity $λ$, conditioned on the event $|I\cap L|=|I\cap R|$. We prove that this model has the same computational threshold as the hard-core model on general bounded-degree graphs. That is, for every fixed $Δ\ge 3$, if $λ<λ_c(Δ)$, then the balanced partition function admits an FPTAS and the balanced hard-core distribution admits an efficient sampling scheme. Conversely, if $λ>λ_c(Δ)$, then no FPRAS or efficient sampler exists on this graph class unless $\mathbf{NP}=\mathbf{RP}$.

Problem

• bipartite graphs
• independent sets
• approximate counting
• sampling
• computational thresholds

Innovation

• computational threshold
• bipartite independent set
• fixed-slice sampling
• balanced hard-core model
• approximate counting