[Paper Review] Space complexity of list H-colouring: a dichotomy

László Egri, Pavol Hell+2 more|arXiv (Cornell University)|Aug 1, 2013

Advanced Graph Theory Research3 citations

TL;DR

This paper establishes a logspace vs. NL-hardness dichotomy for the list homomorphism problem LHOM(H) on digraph templates H, proving that LHOM(H) is solvable in logspace if and only if H contains no 'circular N' structure; otherwise, it is hard for NL. The result confirms a conjecture by Larose and Tesson and provides a polynomial-time algorithm to detect circular N in H.

ABSTRACT

The Dichotomy Conjecture for constraint satisfaction problems (CSPs) states that every CSP is in P or is NP-complete (Feder-Vardi, 1993). It has been verified for conservative problems (also known as list homomorphism problems) by A. Bulatov (2003). We augment this result by showing that for digraph templates H, every conservative CSP, denoted LHOM(H), is solvable in logspace or is hard for NL. More precisely, we introduce a digraph structure we call a circular N, and prove the following dichotomy: if H contains no circular N then LHOM(H) admits a logspace algorithm, and otherwise LHOM(H) is hard for NL. Our algorithm operates by reducing the lists in a complex manner based on a novel decomposition of an auxiliary digraph, combined with repeated applications of Reingold's algorithm for undirected reachability (2005). We also prove an algebraic version of this dichotomy: the digraphs without a circular N are precisely those that admit a finite chain of polymorphisms satisfying the Hagemann-Mitschke identities. This confirms a conjecture of Larose and Tesson (2007) for LHOM(H). Moreover, we show that the presence of a circular N can be decided in time polynomial in the size of H.

Motivation & Objective

• To resolve the refined complexity conjecture of Larose and Tesson for conservative CSPs, specifically LHOM(H) on digraph templates.
• To characterize which list homomorphism problems are solvable in logspace and which are hard for NL.
• To identify a structural forbidden subgraph—'circular N'—that separates the two complexity classes.
• To provide a polynomial-time algorithm for detecting the presence of a circular N in a given digraph H.
• To confirm the algebraic conjecture that digraphs without circular N admit a finite chain of Hagemann-Mitschke polymorphisms.

Proposed method

• Introduce the concept of a 'circular N'—a specific digraph structure defined by three walks with avoidance and edge direction constraints.
• Prove that LHOM(H) is in logspace if and only if H contains no circular N, using a novel decomposition of an auxiliary digraph.
• Apply Reingold’s logspace algorithm for undirected reachability as a core subroutine in the proposed logspace algorithm.
• Construct a colored triple digraph H++ to encode potential circular N structures, using edge colors (green, blue, brown) to track forbidden edge patterns.
• Use reachability analysis in H++ to detect the existence of a circular N via path constraints (no brown edge before blue) between specific vertex triples.
• Establish a polynomial-time algorithm to detect circular N by checking reachability in H++ with color-constrained paths.

Experimental results

Research questions

• RQ1 Under what structural conditions on a digraph H is the list homomorphism problem LHOM(H) solvable in logspace?
• RQ2 Is there a forbidden substructure that characterizes the NL-hardness of LHOM(H) for conservative CSPs?
• RQ3 Can the presence of a circular N in H be detected in polynomial time?
• RQ4 Does the absence of a circular N in H correspond to the existence of a finite chain of Hagemann-Mitschke polymorphisms?
• RQ5 Can the complexity classification of LHOM(H) be refined beyond P/NP-completeness to include logspace and NL-hardness?

Key findings

• LHOM(H) is in logspace if and only if H contains no circular N structure.
• If H contains a circular N, then LHOM(H) is hard for NL, establishing a complete dichotomy between logspace and NL-hardness.
• The presence of a circular N in H can be detected in time polynomial in |V(H)| using the colored triple digraph H++.
• The absence of a circular N in H is equivalent to the existence of a finite chain of Hagemann-Mitschke polymorphisms, confirming an algebraic conjecture by Larose and Tesson.
• When H contains no circular N, LHOM(H) is definable in first-order logic, and if H also lacks independent edges and bicycles, it is in L.
• The characterization via circular N unifies and refines prior results on forbidden substructures like DAT and bicycles in the context of conservative CSPs.