Summary
This paper investigates how hard it is to compute approximations of certain graph properties related to independent sets, which are groups of nodes that don't touch each other. The authors show that for some special kinds of graphs (bipartite regular graphs), there are efficient algorithms to approximate these counts, but for more general regular graphs, such approximations are much harder and likely impossible to do quickly. Their work provides a mathematical confirmation that these two problems behave differently in terms of computational difficulty. They also introduce a new way to efficiently represent large independent sets, helping the algorithmic part.
What this means in practice
- •For algorithm designers: Design approximation algorithms for counting independent sets specifically in bipartite regular graphs where efficient solutions exist.
- •For complexity theorists: Use the proven separations to better understand the limits of deterministic approximation algorithms for generalized graph problems.
A theory result. No direct application yet.
Abstract
We give complexity-theoretic separations between $\#\mathrm{BIS}$ and $\#\mathrm{IS}$. Given a graph $G$, and a fugacity $λ>0$, the hard-core model is given by $Z_G(λ)=\sum_{I\in\mathcal I(G)}λ^{|I|}$, where $\mathcal I(G)$ denotes the independent sets of $G$. For $F\ge1$, an $F$-factor approximation to $Z_G(λ)$ is an estimate $\widehat Z$ satisfying $F^{-1}Z_G(λ)\le\widehat Z\le FZ_G(λ)$. Let $\mathcal G_Δ$ denote the class of $Δ$-regular graphs and $\mathcal G_{Δ,Δ}$ the class of $Δ$-regular bipartite graphs. We give two separation results. (1) There exist an integer $Δ\ge3$, a sufficiently large fugacity $λ$, and a constant $c>0$ so that the following hold: (a) For every $δ\in(0,1)$, deterministic $(1+δ)$-factor approximation on $\mathcal G_{Δ,Δ}$ is possible in time $\exp(cn)\operatorname{poly}(n,1/δ)$. (b) Under the Gap-Exponential Time Hypothesis, there exists no deterministic algorithm running in time $\exp(2cn)\operatorname{poly}(n)$ on $\mathcal G_Δ$ that gives an $\frac{11}{10}$-factor approximation. (2) For every fixed $Δ\ge3$ and $\varepsilon\in(0,1)$, for all sufficiently large fugacities $λ$, the following hold: (a) There exists an algorithm which outputs a deterministic $(1+\varepsilon)^n$-factor approximation to $Z_G(λ)$ on $\mathcal G_{Δ,Δ}$ in polynomial time. (b) Under the assumption $\mathrm P\ne\mathrm{NP}$, there exists no deterministic polynomial-time algorithm on $\mathcal G_Δ$ that gives a $(1+\varepsilon)^n$-factor approximation. Both our algorithmic results follow from a new spanning-tree encoding of independent sets that enables an efficient representation of large independent sets.