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A Dichotomy for Maximum PCSPs on Graphs

Tamio-Vesa Nakajima, Stanislav Živný

TL;DR

A complete classification of this problem under Khot's Unique Games Conjecture is given, including an efficient approximation algorithm for the following problem: Given a (multi)graph X, which contains a bipartite subgraph with $\rho$ edges, what is the largest triangle-free subgraph of X that can be found efficiently?

Abstract

Fix two non-empty loopless graphs $G$ and $H$ such that $G$ maps homomorphically to $H$. The Maximum Promise Constraint Satisfaction Problem parameterised by $G$ and $H$ is the following computational problem, denoted by MaxPCSP($G$, $H$): Given an input (multi)graph $X$ that admits a map to $G$ preserving a $ρ$-fraction of the edges, find a map from $X$ to $H$ that preserves a $ρ$-fraction of the edges. As our main result, we give a complete classification of this problem under Khot's Unique Games Conjecture: The only tractable cases are when $G$ is bipartite and $H$ contains a triangle. Along the way, we establish several results, including an efficient approximation algorithm for the following problem: Given a (multi)graph $X$ which contains a bipartite subgraph with $ρ$ edges, what is the largest triangle-free subgraph of $X$ that can be found efficiently? We present an SDP-based algorithm that finds one with at least $0.8823 ρ$ edges, thus improving on the subgraph with $0.878 ρ$ edges obtained by the classic Max-Cut algorithm of Goemans and Williamson.

A Dichotomy for Maximum PCSPs on Graphs

TL;DR

A complete classification of this problem under Khot's Unique Games Conjecture is given, including an efficient approximation algorithm for the following problem: Given a (multi)graph X, which contains a bipartite subgraph with edges, what is the largest triangle-free subgraph of X that can be found efficiently?

Abstract

Fix two non-empty loopless graphs and such that maps homomorphically to . The Maximum Promise Constraint Satisfaction Problem parameterised by and is the following computational problem, denoted by MaxPCSP(, ): Given an input (multi)graph that admits a map to preserving a -fraction of the edges, find a map from to that preserves a -fraction of the edges. As our main result, we give a complete classification of this problem under Khot's Unique Games Conjecture: The only tractable cases are when is bipartite and contains a triangle. Along the way, we establish several results, including an efficient approximation algorithm for the following problem: Given a (multi)graph which contains a bipartite subgraph with edges, what is the largest triangle-free subgraph of that can be found efficiently? We present an SDP-based algorithm that finds one with at least edges, thus improving on the subgraph with edges obtained by the classic Max-Cut algorithm of Goemans and Williamson.
Paper Structure (28 sections, 33 theorems, 65 equations, 3 figures)

This paper contains 28 sections, 33 theorems, 65 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Paper organisation
  4. Preliminaries
  5. Graphs and (partial) homomorphisms.
  6. Maximum PCSPs.
  7. SDP
  8. Results
  9. Approximation of maxPCSP(K2,G3)
  10. Approximation of maxPCSP(K2,K3)
  11. Technical bounds.
  12. Randomised algorithm.
  13. Derandomised algorithm.
  14. Hardness of maxPCSP(K2,G3)
  15. Reduction.
  16. ...and 13 more sections

Key Result

Theorem 1

Let $G$ and $H$ be two fixed graphs such that there is a homomorphism from $G$ to $H$. If $G$ is bipartite and $H$ contains a triangle then $\mathop{\mathrm{\textup{MaxPCSP}}}\nolimits(G,H)$ is 1-approximable. Otherwise, 1-approximation of $\mathop{\mathrm{\textup{MaxPCSP}}}\nolimits(G,H)$ is -hard

Figures (3)

  • Figure 1: Function giving rise to $\alpha_{GW}, \tau_{GW}$.
  • Figure 2: Bounds from \ref{['lem:bounds']}
  • Figure 3: Plot of expressions from \ref{['lem:ineq']}.

Theorems & Definitions (65)

  • Theorem 1
  • Lemma 2
  • Theorem 2
  • Conjecture 3: BG21:sicomp
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • proof : Proof of \ref{['thm:UGCdichotomy']}
  • Theorem 6
  • Lemma 7
  • ...and 55 more