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Dichotomy for orderings?

Gábor Kun, Jaroslav Nešetřil

TL;DR

The paper proves that the class of F^<-ordering problems has full NP power: any NP language can be encoded by forbidding a finite set of ordered subgraphs, implying no general dichotomy for these problems. It introduces a Temporal Sparse Incomparability Lemma (SIL) for orderings, using the Lovász Local Lemma, with a deterministic variant for bounded-degree instances, to transfer hardness to high-girth structures. A dichotomy is established for biconnected patterns, showing these ordering problems are either NP-complete or tractable, guided by CSP reductions and the Bodirsky–Kára temporal CSP dichotomy. For a single ordered biconnected graph not complete, the corresponding ordering problem is NP-complete, confirming Duffus–Ginn–Rödl; broader results connect temporal CSP languages, pp interpretations, and polymorphisms to tractability and hardness in this ordered-context framework.

Abstract

The class $NP$ can be defined by the means of Monadic Second-Order logic going back to Fagin and Feder-Vardi, and also by forbidden expanded substructures (cf. lifts and shadows of Kun and Nešetřil). Consequently, for such problems there is no dichotomy, unlike for $CSP$'s. We prove that ordering problems for graphs defined by finitely many forbidden ordered subgraphs still capture the class $NP$. In particular, we refute a conjecture of Hell, Mohar and Rafiey that dichotomy holds for this class. On the positive side, we confirm the conjecture of Duffus, Ginn and Rödl that ordering problems defined by one single biconnected ordered graph are $NP$-complete but for the ordered complete graph. An interesting feature appeared and was noticed several times. For finite sets of biconnected patterns (which may be colored structures or ordered structures) complexity dichotomy holds. A principal tool for obtaining this result is known as the Sparse Incomparability Lemma, a classical result in the theory of homomorphisms of graphs and structures. We prove it here in the setting of ordered graphs as a Temporal Sparse Incomparability Lemma for orderings. Interestingly, our proof involves the Lovász Local Lemma.

Dichotomy for orderings?

TL;DR

The paper proves that the class of F^<-ordering problems has full NP power: any NP language can be encoded by forbidding a finite set of ordered subgraphs, implying no general dichotomy for these problems. It introduces a Temporal Sparse Incomparability Lemma (SIL) for orderings, using the Lovász Local Lemma, with a deterministic variant for bounded-degree instances, to transfer hardness to high-girth structures. A dichotomy is established for biconnected patterns, showing these ordering problems are either NP-complete or tractable, guided by CSP reductions and the Bodirsky–Kára temporal CSP dichotomy. For a single ordered biconnected graph not complete, the corresponding ordering problem is NP-complete, confirming Duffus–Ginn–Rödl; broader results connect temporal CSP languages, pp interpretations, and polymorphisms to tractability and hardness in this ordered-context framework.

Abstract

The class can be defined by the means of Monadic Second-Order logic going back to Fagin and Feder-Vardi, and also by forbidden expanded substructures (cf. lifts and shadows of Kun and Nešetřil). Consequently, for such problems there is no dichotomy, unlike for 's. We prove that ordering problems for graphs defined by finitely many forbidden ordered subgraphs still capture the class . In particular, we refute a conjecture of Hell, Mohar and Rafiey that dichotomy holds for this class. On the positive side, we confirm the conjecture of Duffus, Ginn and Rödl that ordering problems defined by one single biconnected ordered graph are -complete but for the ordered complete graph. An interesting feature appeared and was noticed several times. For finite sets of biconnected patterns (which may be colored structures or ordered structures) complexity dichotomy holds. A principal tool for obtaining this result is known as the Sparse Incomparability Lemma, a classical result in the theory of homomorphisms of graphs and structures. We prove it here in the setting of ordered graphs as a Temporal Sparse Incomparability Lemma for orderings. Interestingly, our proof involves the Lovász Local Lemma.

Paper Structure

This paper contains 10 sections, 25 theorems, 3 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. Notation and basic notions
  3. Full power
  4. The biconnected phenomenon by two examples
  5. The sparse incomparability lemma for orderings
  6. A deterministic algorithm for the Temporal Sparse Incomparability Lemma
  7. Dichotomy for biconnected patterns
  8. The complexity of temporal $CSP$ languages
  9. pp formulas, interpretations and polymorphisms
  10. The complexity of temporal $CSP$ languages corresponding to a single ordered graph

Key Result

Theorem 1

For every language $L$ in the class $NP$ there exists a finite set $\mathcal{F}^<$ such that the $\mathcal{F}^<$-ordering problem and $L$ are polynomially equivalent.

Figures (3)

  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (70)

  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Theorem 6
  • proof
  • Claim 7
  • proof
  • Claim 8
  • proof
  • ...and 60 more