AI 中文总结
该研究针对二分图的固定切片独立集问题，证明了其近似计数与采样的计算阈值，明确了平衡硬核模型与一般有界度图硬核模型具有相同的阈值。
AI 中文摘要
受Kocurek、Oveis Gharan和Tjowasi近期工作的启发，该工作通过分解为固定大小的切片，给出了随机正则二分图上硬核模型的高效采样算法，我们研究二分图独立集问题的固定切片近似计数与采样的最坏情况可处理性。设G=(L∪R,E)为二分图，|L|=|R|=n，最大度为Δ。固定切片问题要求从满足|I∩L|=α_L n且|I∩R|=α_R n的独立集中均匀采样。我们证明，若整体密度α位于区间(1/Δ,1/2)，且两侧密度比随机Δ-正则二分图的典型相密度更平衡，则除非NP=RP，否则不存在FPRAS或高效采样方案。随后我们研究了一种相关的 fugacity 模型，其中密度不固定，但要求独立集在二分图两侧平衡。对于λ>0，平衡硬核模型是具有 fugacity λ的普通硬核模型，且满足|I∩L|=|I∩R|。我们证明该模型与一般有界度图上的硬核模型具有相同的计算阈值，即对于每个固定Δ≥3，若λ<λ_c(Δ)，则平衡配分函数存在FPTAS，且平衡硬核分布存在高效采样方案；反之，若λ>λ_c(Δ)，则除非NP=RP，否则该图类上不存在FPRAS或高效采样器。
英文摘要
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 $\mathbf{NP}=\mathbf{RP}$.
