Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sketching approximability of all finite CSPs

Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Santhoshini Velusamy

TL;DR

This work shows that either a linear sketching algorithm solves the problem in polylogarithmic space, or the problem is not solvable by any sketching algorithm in $o(\sqrt{n})$ space, and gives non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems.

Abstract

A constraint satisfaction problem (CSP), $\textsf{Max-CSP}(\mathcal{F})$, is specified by a finite set of constraints $\mathcal{F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from $\mathcal{F}$ to subsequences of the $n$ variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the $(γ,β)$-approximation version of the problem for parameters $0 \leq β< γ\leq 1$, the goal is to distinguish instances where at least $γ$ fraction of the constraints can be satisfied from instances where at most $β$ fraction of the constraints can be satisfied. In this work we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family $\mathcal{F}$ and every $β< γ$, we show that either a linear sketching algorithm solves the problem in polylogarithmic space, or the problem is not solvable by any sketching algorithm in $o(\sqrt{n})$ space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of $q=k=2$, where we get a dichotomy, and the case when the satisfying assignments of the constraints of $\mathcal{F}$ support a distribution on $[q]^k$ with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variables $q=2$, binary constraints $k=2$, singleton families $|\mathcal{F}|=1$ and only considered the setting where constraints are placed on literals rather than variables.

Sketching approximability of all finite CSPs

TL;DR

This work shows that either a linear sketching algorithm solves the problem in polylogarithmic space, or the problem is not solvable by any sketching algorithm in space, and gives non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems.

Abstract

A constraint satisfaction problem (CSP), , is specified by a finite set of constraints for positive integers and . An instance of the problem on variables is given by applications of constraints from to subsequences of the variables, and the goal is to find an assignment to the variables that satisfies the maximum number of constraints. In the -approximation version of the problem for parameters , the goal is to distinguish instances where at least fraction of the constraints can be satisfied from instances where at most fraction of the constraints can be satisfied. In this work we consider the approximability of this problem in the context of sketching algorithms and give a dichotomy result. Specifically, for every family and every , we show that either a linear sketching algorithm solves the problem in polylogarithmic space, or the problem is not solvable by any sketching algorithm in space. In particular, we give non-trivial approximation algorithms using polylogarithmic space for infinitely many constraint satisfaction problems. We also extend previously known lower bounds for general streaming algorithms to a wide variety of problems, and in particular the case of , where we get a dichotomy, and the case when the satisfying assignments of the constraints of support a distribution on with uniform marginals. Prior to this work, other than sporadic examples, the only systematic classes of CSPs that were analyzed considered the setting of Boolean variables , binary constraints , singleton families and only considered the setting where constraints are placed on literals rather than variables.

Paper Structure

This paper contains 81 sections, 55 theorems, 101 equations, 3 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. CSPs
  3. Streaming algorithms
  4. Past work
  5. Results
  6. This version:
  7. Contrast with dichotomies in the polynomial time setting
  8. Overview of our analysis
  9. Intuition.
  10. Bias-based algorithms.
  11. Lower bounds.
  12. Subsequent results
  13. Explicit Families of Easy and Hard Problems.
  14. $o(n)$-space algorithms.
  15. Random-ordering streaming setting
  16. ...and 66 more sections

Key Result

theorem 1.1

For every $q,k\in\mathbb{N}$, $0 \leq \beta < \gamma \leq 1$ and $\mathcal{F} \subseteq \{f:[q]^k \to \{0,1\}\}$, one of the following two conditions holds: Either $(\gamma,\beta)$-$\textsf{Max-CSP}(\mathcal{F})$ can be solved with $O(\log^3 n)$ space by linear sketches, or for every $\varepsilon >

Figures (3)

  • Figure 1: A plot of $H^\cap$.
  • Figure 2: The roadmap of our lower bounds. The top two rows describe the results of this section, while the remaining rows describe notions and results from \ref{['sec:kpart', 'sec:polar', 'sec:spl']}.
  • Figure 3: An example of shared randomness used in \ref{['lem:SD boolean to non-boolean']}. Here $n=12$, $k=2$, $q=3$, and $\alpha=1/3$. The value of $\mathbf{x}_R\in[q]^n$ is listed in a table. Consider $(u_1,v_1)=(1,3)$ and $(u_2,v_2)=(3,2)$. The variables in sets $T_1,T_2$ are marked grey. The variables correspond to the set $U$ are circled with red lines and the variables correspond to sets $S_1,S_2$ are circled with yellow dashed lines.

Theorems & Definitions (139)

  • theorem 1.1: Succinct version
  • theorem 1.2: Informal
  • theorem 1.3
  • theorem 1.4: Succinct version
  • definition 2.1: $(\gamma,\beta)$-$\textsf{Max-CSP}(\mathcal{F})$
  • definition 2.2: Streaming algorithm
  • definition 2.3: Sketching algorithms
  • remark 2.4
  • proposition 2.5
  • proof
  • ...and 129 more