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Moser-Tardos Algorithm with small number of random bits

Endre Csóka, Łukasz Grabowski, András Máthé, Oleg Pikhurko, Konstantinos Tyros

TL;DR

It is proved that if the authors restrict attention to a class of problems whose dependency graphs have subexponential growth, then the expected total number of random bits used by the algorithm is constant; in particular, it is independent from the number of variables.

Abstract

We study a variant of the parallel Moser-Tardos Algorithm. We prove that if we restrict attention to a class of problems whose dependency graphs have subexponential growth, then the expected total number of random bits used by the algorithm is constant; in particular, it is independent from the number of variables. This is achieved by using the same random bits to resample variables which are far enough in the dependency graph. There are two corollaries. First, we obtain a deterministic algorithm for finding a satisfying assignment, which for any class of problems as in the previous paragraph runs in time O(n), where n is the number of variables. Second, we present a Borel version of the Lovász Local Lemma.

Moser-Tardos Algorithm with small number of random bits

TL;DR

It is proved that if the authors restrict attention to a class of problems whose dependency graphs have subexponential growth, then the expected total number of random bits used by the algorithm is constant; in particular, it is independent from the number of variables.

Abstract

We study a variant of the parallel Moser-Tardos Algorithm. We prove that if we restrict attention to a class of problems whose dependency graphs have subexponential growth, then the expected total number of random bits used by the algorithm is constant; in particular, it is independent from the number of variables. This is achieved by using the same random bits to resample variables which are far enough in the dependency graph. There are two corollaries. First, we obtain a deterministic algorithm for finding a satisfying assignment, which for any class of problems as in the previous paragraph runs in time O(n), where n is the number of variables. Second, we present a Borel version of the Lovász Local Lemma.
Paper Structure (18 sections, 19 theorems, 83 equations, 3 figures)

This paper contains 18 sections, 19 theorems, 83 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main result.
  3. Connections to descriptive combinatorics and distributed algorithms.
  4. Algorithm and the results
  5. Notation and conventions
  6. Description of the algorithm
  7. Statement of the main result
  8. Analysis of the algorithm
  9. Landscapes
  10. Landscapes associated to a run of the MTA
  11. The sequence encoded by a finalised landscape
  12. Counting landscapes
  13. Interlude: Analysis of the classical MTA
  14. Some ideas behind re-using random bits
  15. Landscape restrictions
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $G$ be a digraph and let ${\Delta}$ be the maximal vertex degree in $\operatorname{Rel}(G)$. If for every $x\in V(G)$ we have where $\operatorname{Var}(x)$ denotes the out-neighbourhood of $x$, then there exists $f\in b^{ V(G)}$ which satisfies $\mathbf R$.

Figures (3)

  • Figure 1:
  • Figure 2: The forest from Figure \ref{['fig-lala']}(b), after the operation described in (Step 2)
  • Figure 3:

Theorems & Definitions (52)

  • Theorem 1.1: Lovász Local Lemma MR0491337
  • Remark 1.3
  • Remark 2.1
  • Remark 2.2
  • Remark 2.4
  • Theorem 2.5
  • Remark 2.8
  • Corollary 2.10
  • Remark 2.11
  • proof : Proof of Corollary \ref{['cory-main']}
  • ...and 42 more