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Approximating the volume of a truncated relaxation of the independence polytope

Ferenc Bencs, Guus Regts

TL;DR

This work deterministically approximates the volume of a truncated relaxation P_{G,δ} of the independent set polytope for bounded-degree graphs by recasting vol(P_{G,δ}) as a graph-polynomial evaluation p_{G,δ}(1). The authors prove a zero-free disk for p_{G,δ} when δ = O(1/Δ) and develop an efficient Δ^{O(k)}-time method to compute the coefficients λ_{H,k} of ind(H,G) in p_{G,δ}, enabling Barvinok’s interpolation to yield polynomial-time approximation in n/ε. The main contributions are the zero-free region, the combinatorial and algorithmic framework to compute forest-based weights w_T and the induced-subgraph coefficients, and the resulting deterministic algorithm that improves on prior quasi-polynomial time results. The results illuminate a path for deterministic volume approximation via graph-polynomial techniques, while highlighting current limits and open directions to extend beyond the δ = O(1/Δ) regime.

Abstract

Answering a question of Gamarnik and Smedira, we give a polynomial time algorithm that approximately computes the volume of a truncation of a relaxation of the independent set polytope, improving on their quasi-polynomial time algorithm. Our algorithm is obtained by viewing the volume as an evaluation of a graph polynomial and we approximate this evaluation using Barvinok's interpolation method.

Approximating the volume of a truncated relaxation of the independence polytope

TL;DR

This work deterministically approximates the volume of a truncated relaxation P_{G,δ} of the independent set polytope for bounded-degree graphs by recasting vol(P_{G,δ}) as a graph-polynomial evaluation p_{G,δ}(1). The authors prove a zero-free disk for p_{G,δ} when δ = O(1/Δ) and develop an efficient Δ^{O(k)}-time method to compute the coefficients λ_{H,k} of ind(H,G) in p_{G,δ}, enabling Barvinok’s interpolation to yield polynomial-time approximation in n/ε. The main contributions are the zero-free region, the combinatorial and algorithmic framework to compute forest-based weights w_T and the induced-subgraph coefficients, and the resulting deterministic algorithm that improves on prior quasi-polynomial time results. The results illuminate a path for deterministic volume approximation via graph-polynomial techniques, while highlighting current limits and open directions to extend beyond the δ = O(1/Δ) regime.

Abstract

Answering a question of Gamarnik and Smedira, we give a polynomial time algorithm that approximately computes the volume of a truncation of a relaxation of the independent set polytope, improving on their quasi-polynomial time algorithm. Our algorithm is obtained by viewing the volume as an evaluation of a graph polynomial and we approximate this evaluation using Barvinok's interpolation method.
Paper Structure (7 sections, 10 theorems, 52 equations)

This paper contains 7 sections, 10 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. The volume as an evaluation of a graph polynomial
  3. A zero-free disk for $p_{G,\delta}$
  4. Computing the coefficients of $\text{ind}(H,\cdot)$
  5. Computation of $w_T$
  6. Computation of $\lambda_{H,k}$
  7. Concluding remarks, open problems and discussion

Key Result

Theorem 1.1

Let $\Delta>0$. There exists a constant $C>0$ such that for each and $\delta\in [0,C/\Delta]$ there exists an algorithm that on input of an $n$-vertex graph of maximum degree at most $\Delta$ and $\varepsilon\in (0,1)$ computes a number $\xi$ such that in time polynomial in $n/\varepsilon$.

Theorems & Definitions (17)

  • Theorem 1.1
  • Lemma 2.1
  • Remark 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • ...and 7 more