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Inapproximability of the independent set polynomial in the complex plane

Ivona Bezakova, Andreas Galanis, Leslie Ann Goldberg, Daniel Stefankovic

TL;DR

The paper resolves the complexity of approximately evaluating the independent set partition function $Z_G(\lambda)$ for graphs of bounded degree when the activity $\lambda$ is complex. It introduces the cardioid region $\Lambda_\Delta$ as the natural boundary for approximability and proves that, for $\lambda$ outside $\Lambda_\Delta$ (and not a nonnegative real), approximating $Z_G(\lambda)$ is $\#\mathsf{P}$-hard, with stronger $\#\mathsf{P}$-hardness for nonreal values and a hardness result for deciding positivity on negative reals. The core method blends complex dynamics (iteration of multivariate rational maps) with gadget-based reductions: repelling fixpoints of $f(z)=\frac{1}{1+\lambda z^{\Delta-1}}$ are exploited via contracting maps that cover neighborhoods to implement arbitrary complex activities with exponentially small error, converting approximate counting into exact counting. The results extend the real-parameter hardness picture and provide a rigorous threshold between tractable and intractable regimes, while new developments show zero-free regions can yield future FPTAS in parts of the region. This work advances our understanding of the boundary between NP-hard and #P-hard regimes for complex-valued combinatorial partition functions and demonstrates a powerful toolkit for proving intractability via complex-analytic dynamics and gadget constructions.

Abstract

We study the complexity of approximating the independent set polynomial $Z_G(λ)$ of a graph $G$ with maximum degree $Δ$ when the activity $λ$ is a complex number. This problem is already well understood when $λ$ is real using connections to the $Δ$-regular tree $T$. The key concept in that case is the "occupation ratio" of the tree $T$. This ratio is the contribution to $Z_T(λ)$ from independent sets containing the root of the tree, divided by $Z_T(λ)$ itself. If $λ$ is such that the occupation ratio converges to a limit, as the height of $T$ grows, then there is an FPTAS for approximating $Z_G(λ)$ on a graph $G$ with maximum degree $Δ$. Otherwise, the approximation problem is NP-hard. Unsurprisingly, the case where $λ$ is complex is more challenging. Peters and Regts identified the complex values of $λ$ for which the occupation ratio of the $Δ$-regular tree converges. These values carve a cardioid-shaped region $Λ_Δ$ in the complex plane. Motivated by the picture in the real case, they asked whether $Λ_Δ$ marks the true approximability threshold for general complex values $λ$. Our main result shows that for every $λ$ outside of $Λ_Δ$, the problem of approximating $Z_G(λ)$ on graphs $G$ with maximum degree at most $Δ$ is indeed NP-hard. In fact, when $λ$ is outside of $Λ_Δ$ and is not a positive real number, we give the stronger result that approximating $Z_G(λ)$ is actually #P-hard. If $λ$ is a negative real number outside of $Λ_Δ$, we show that it is #P-hard to even decide whether $Z_G(λ)>0$, resolving in the affirmative a conjecture of Harvey, Srivastava and Vondrak. Our proof techniques are based around tools from complex analysis -- specifically the study of iterative multivariate rational maps.

Inapproximability of the independent set polynomial in the complex plane

TL;DR

The paper resolves the complexity of approximately evaluating the independent set partition function for graphs of bounded degree when the activity is complex. It introduces the cardioid region as the natural boundary for approximability and proves that, for outside (and not a nonnegative real), approximating is -hard, with stronger -hardness for nonreal values and a hardness result for deciding positivity on negative reals. The core method blends complex dynamics (iteration of multivariate rational maps) with gadget-based reductions: repelling fixpoints of are exploited via contracting maps that cover neighborhoods to implement arbitrary complex activities with exponentially small error, converting approximate counting into exact counting. The results extend the real-parameter hardness picture and provide a rigorous threshold between tractable and intractable regimes, while new developments show zero-free regions can yield future FPTAS in parts of the region. This work advances our understanding of the boundary between NP-hard and #P-hard regimes for complex-valued combinatorial partition functions and demonstrates a powerful toolkit for proving intractability via complex-analytic dynamics and gadget constructions.

Abstract

We study the complexity of approximating the independent set polynomial of a graph with maximum degree when the activity is a complex number. This problem is already well understood when is real using connections to the -regular tree . The key concept in that case is the "occupation ratio" of the tree . This ratio is the contribution to from independent sets containing the root of the tree, divided by itself. If is such that the occupation ratio converges to a limit, as the height of grows, then there is an FPTAS for approximating on a graph with maximum degree . Otherwise, the approximation problem is NP-hard. Unsurprisingly, the case where is complex is more challenging. Peters and Regts identified the complex values of for which the occupation ratio of the -regular tree converges. These values carve a cardioid-shaped region in the complex plane. Motivated by the picture in the real case, they asked whether marks the true approximability threshold for general complex values . Our main result shows that for every outside of , the problem of approximating on graphs with maximum degree at most is indeed NP-hard. In fact, when is outside of and is not a positive real number, we give the stronger result that approximating is actually #P-hard. If is a negative real number outside of , we show that it is #P-hard to even decide whether , resolving in the affirmative a conjecture of Harvey, Srivastava and Vondrak. Our proof techniques are based around tools from complex analysis -- specifically the study of iterative multivariate rational maps.

Paper Structure

This paper contains 31 sections, 41 theorems, 222 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proof approach
  3. New Developments
  4. Proof Outline
  5. Iteration of complex rational maps
  6. A characterisation of the cardioid
  7. Applying the theory
  8. Chasing repelling fixpoints
  9. Exponential precision via contracting maps that cover
  10. Constructing the maps
  11. Fitting the pieces together and proof for the real case
  12. Dependencies between Lemmas
  13. Preliminaries
  14. Implementing activities
  15. Finding roots of polynomials
  16. ...and 16 more sections

Key Result

Theorem 1

Let $\Delta\geq 3$ and $\lambda\in \mathbb{C}_{\mathbb{Q}}$ be such that $\lambda\not\in (\Lambda_\Delta \cup \mathbb{R}_{\geq 0})$. Then, $\#\mathsf{BipHardCoreNorm}(\lambda,\Delta,1.01)$ is $\#\mathsf{P}$-hard.

Figures (3)

  • Figure 1: The cardioid-shaped region $\Lambda_\Delta$ in the complex plane. We show that for all $\lambda\in \mathbb{C}\backslash (\Lambda_\Delta\cup \mathbb{R}_{\geq 0})$, approximating $Z_G(\lambda)$ is $\#\mathsf{P}$-hard. Previously, it was known that the problem is $\mathsf{NP}$-hard on the real line in the intervals $\lambda<-\lambda^*$ and $\lambda>\lambda_c$. Note, we have that the thresholds $-\lambda^*,\lambda_c$ belong to $\Lambda_\Delta$, by taking $z=\pm 1/(\Delta-1)$ in \ref{['fig:cardioid']}.
  • Figure 2: The binary equality gadgets $B_i$ and $B'_i$ used in the proof of Lemma \ref{['lem:equality']}.
  • Figure 3: The gadget $B_i"$ with activity vector $\boldsymbol{\lambda}"$ used in the proof of Lemma \ref{['lem:bipartitegadget']}.

Theorems & Definitions (79)

  • Theorem 1
  • Remark
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Definition 5
  • Proposition 6
  • Lemma 7: e.g., Milnor
  • Theorem 8: see, e.g., Milnor
  • Lemma 9
  • ...and 69 more