The Complexity of Promise Constraint Satisfaction Problem Seen from the Other Side

TL;DR

The paper analyzes the parameterized complexity of left-hand side restricted CSPs and their promise counterparts (PCSPs). It formalizes left-hand side restrictions via $PHom$ and investigates when such problems are fixed-parameter tractable, notably showing tractability when the left-hand class has bounded tree width modulo homomorphic equivalence and hardness when this condition fails. The main technical contribution is a sufficient condition for $W[1]$-hardness in the left-hand side restricted PCSP, adapting Grohe's grid-based construction to the promise setting using grid-like mappings and $k\times k$ grids. These results help delineate the parameterized complexity landscape for left-hand side restrictions and provide tools for proving hardness for approximation variants like Gap-Clique.

Abstract

We introduce the framework of the left-hand side restricted promise constraint satisfaction problem, which includes problems like approximating clique number of a graph. We study the parameterized complexity of problems in this class and provide some initial results. The main technical contribution is a sufficient condition for W[1]-hardness which, in particular, covers left-hand side restricted bounded arity CSPs.

Figures (2)

• Figure 1: An example of a minor of a graph.
• Figure 2: A tree-decomposition of a $(3\times 3$)-grid.

Theorems & Definitions (12)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5
• Theorem 6
• Definition 7
• Definition 8
• Proposition 9
• Corollary 10