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Hardness of 4-Colourings G-Colourable Graphs

Sergey Avvakumov, Marek Filakovský, Jakub Opršal, Gianluca Tasinato, Uli Wagner

TL;DR

This work resolves a central hardness conjecture for promise constraint satisfaction problems by proving NP-hardness of $PCSP(C_\ell,K_4)$ for every odd $\ell\ge3$, which implies hardness for all non-bipartite loopless graphs $G$ containing an odd cycle. The authors fuse an algebraic PCSP framework based on polymorphisms and minion homomorphisms with topological methods, notably equivariant obstruction theory and homomorphism complexes, to translate graph-coloring challenges into equivariant maps and then into affine $\mathbb Z_2$-maps. They introduce a topologically simplified target space $Y$ (an Eilenberg–MacLane space) to classify equivariant maps $T^n\to Y$ and prove a minion isomorphism to a bounded-arity, non-constant minion, enabling NP-hardness via BBKO21. A key technical achievement is bounding the essential arity by $|I|=O(\ell^2)$, obtained through combinatorial averaging over slices and height arguments, which permits the hardness reduction. The results advance the understanding of PCSP hardness for colorability-type problems and illustrate a powerful synthesis of algebra, topology, and combinatorics in computational complexity.

Abstract

We study the complexity of a class of promise graph homomorphism problems. For a fixed graph H, the H-colouring problem is to decide whether a given graph has a homomorphism to H. By a result of Hell and Nešetřil, this problem is NP-hard for any non-bipartite loop-less graph H. Brakensiek and Guruswami [SODA 2018] conjectured the hardness extends to promise graph homomorphism problems as follows: fix a pair of non-bipartite loop-less graphs G, H such that there is a homomorphism from G to H, it is NP-hard to distinguish between graphs that are G-colourable and those that are not H-colourable. We confirm this conjecture in the cases when both G and H are 4-colourable. This is a common generalisation of previous results of Khanna, Linial, and Safra [Comb. 20(3): 393-415 (2000)] and of Krokhin and Opršal [FOCS 2019]. The result is obtained by combining the algebraic approach to promise constraint satisfaction with methods of topological combinatorics and equivariant obstruction theory.

Hardness of 4-Colourings G-Colourable Graphs

TL;DR

This work resolves a central hardness conjecture for promise constraint satisfaction problems by proving NP-hardness of for every odd , which implies hardness for all non-bipartite loopless graphs containing an odd cycle. The authors fuse an algebraic PCSP framework based on polymorphisms and minion homomorphisms with topological methods, notably equivariant obstruction theory and homomorphism complexes, to translate graph-coloring challenges into equivariant maps and then into affine -maps. They introduce a topologically simplified target space (an Eilenberg–MacLane space) to classify equivariant maps and prove a minion isomorphism to a bounded-arity, non-constant minion, enabling NP-hardness via BBKO21. A key technical achievement is bounding the essential arity by , obtained through combinatorial averaging over slices and height arguments, which permits the hardness reduction. The results advance the understanding of PCSP hardness for colorability-type problems and illustrate a powerful synthesis of algebra, topology, and combinatorics in computational complexity.

Abstract

We study the complexity of a class of promise graph homomorphism problems. For a fixed graph H, the H-colouring problem is to decide whether a given graph has a homomorphism to H. By a result of Hell and Nešetřil, this problem is NP-hard for any non-bipartite loop-less graph H. Brakensiek and Guruswami [SODA 2018] conjectured the hardness extends to promise graph homomorphism problems as follows: fix a pair of non-bipartite loop-less graphs G, H such that there is a homomorphism from G to H, it is NP-hard to distinguish between graphs that are G-colourable and those that are not H-colourable. We confirm this conjecture in the cases when both G and H are 4-colourable. This is a common generalisation of previous results of Khanna, Linial, and Safra [Comb. 20(3): 393-415 (2000)] and of Krokhin and Opršal [FOCS 2019]. The result is obtained by combining the algebraic approach to promise constraint satisfaction with methods of topological combinatorics and equivariant obstruction theory.

Paper Structure

This paper contains 29 sections, 32 theorems, 51 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Acknowledgement
  4. Preliminaries
  5. Polymorphisms and a Hardness Criterion
  6. Topology and Homomorphism Complexes
  7. Simplicial sets
  8. Products
  9. Group actions
  10. Relational simplicial sets
  11. Order complexes of posets
  12. Homomorphism complexes
  13. Homotopy
  14. Overview of the Proof
  15. Step 1
  16. ...and 14 more sections

Key Result

theorem 1

Let $G$ be a non-bipartite 4-colourable graph. Then $\mathop{\mathrm{PCSP}}\nolimits(G, K_4)$ is NP-hard.

Figures (5)

  • Figure 1: The simplicial set $\Sigma^2$
  • Figure 2: The simplicial set $\Gamma_{12}\cong \mathop{\mathrm{Hom}}\nolimits(K_2,K_3)$; see also Examples \ref{['ex:GammaL']} and \ref{['ex:odd-cycles']}.
  • Figure 3: The simplicial set $\mathop{\mathrm{Hom}}\nolimits(K_2, K_4)$; see also Example \ref{['ex:box-k4']}.
  • Figure 4: Coordinate cycle and band in $T^2$ and $T^3$.
  • Figure 5: A path starting with the point $\boldsymbol u_0 = (1, 0, 0, 0)$ shown as projection on the first two (left) and last two coordinates (right).

Theorems & Definitions (67)

  • theorem 1
  • definition 1
  • definition 2
  • theorem 2: BBKO21
  • definition 3
  • definition 4
  • lemma 1
  • proof
  • theorem 4
  • lemma 2
  • ...and 57 more