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An extended tree-width notion for directed graphs related to the computation of permanents

Combinatorics · math.CO · posed by Klaus Meer · open

1 attempt · 1 machine check

Statement

More general: does it hold for all monadic-second order definable problems? We conjecture not

Context

Candidate 1 of the open problems stated in "An extended tree-width notion for directed graphs related to the computation of permanents", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

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    Problem: Reconstructed statement: For finite directed graphs DD, let ttw⁡(D)\operatorname{ttw}(D) be Meer’s triangular tree-width: for a vertex order π=(v1,…,vn)\pi=(v_1,\dots,v_n), form the undirected graph of “backward” arcs {vi,vj}\{v_i,v_j\} with i<ji<j and (vj,vi)∈E(D)(v_j,v_i)\in E(D), and minimize its tree-width over π\pi. The question is whether every fixed MSO-definable directed-graph property is polynomial-time decidable on classes ttw⁡(D)≤c\operatorname{ttw}(D)\le c, analogous to Courcelle-type bounded tree-width tractability. The conjecture says no.

    Result: The negative conjecture is correct in the standard complexity-theoretic sense: there is a fixed MSO sentence whose model-checking problem is NP-complete even on graphs of triangular tree-width 00. Hence a general polynomial-time MSO meta-theorem for bounded triangular tree-width would imply P=NPP=NP.

    Let φ3col\varphi_{3\mathrm{col}} be the MSO sentence over directed graphs saying that the vertices can be partitioned into three sets C1,C2,C3C_1,C_2,C_3, and no arc has both endpoints in the same CiC_i:

    ∃C1,C2,C3  [∀v (exactly one of C1(v),C2(v),C3(v))∧∀x∀y (E(x,y)→⋀i=13¬(Ci(x)∧Ci(y)))].\exists C_1,C_2,C_3\;\Bigl[ \forall v\,\bigl(\text{exactly one of }C_1(v),C_2(v),C_3(v)\bigr) \land \forall x\forall y\,(E(x,y)\to \bigwedge_{i=1}^3\neg(C_i(x)\land C_i(y))) \Bigr].

    This is an MSO-definable directed-graph property.

    Given an undirected graph GG with vertices ordered 1,…,n1,\dots,n, orient every edge from the smaller endpoint to the larger endpoint, obtaining a directed graph DGD_G. Then DGD_G is acyclic, so with the order 1,…,n1,\dots,n it has no backward arcs; hence ttw⁡(DG)=0\operatorname{ttw}(D_G)=0. Moreover,

    G is 3-colorable⟺DG⊨φ3col,G \text{ is 3-colorable}\quad\Longleftrightarrow\quad D_G\models \varphi_{3\mathrm{col}},

    because every undirected edge of GG appears as exactly one directed arc of DGD_G, and the formula only enforces different colors across arcs.

    Thus 3-COLOR reduces in polynomial time to model checking the fixed MSO sentence φ3col\varphi_{3\mathrm{col}} on directed graphs with ttw⁡=0\operatorname{ttw}=0. Since the problem is clearly in NP, it is NP-complete on this bounded triangular tree-width class.

    Audit: no extra graph-theoretic assumptions are used beyond finite directed graphs; the witnessing order is explicitly supplied by the construction; the formula is fixed; and the counterexample lies in the strongest bounded case, ttw⁡=0\operatorname{ttw}=0.

    Citation: Klaus Meer, “An extended tree-width notion for directed graphs related to the computation of permanents,” Theory of Computing Systems, DOI: 10.1007/s00224-013-9453-0.
    M. R. Garey, D. S. Johnson, and L. Stockmeyer, “Some simplified NP-complete graph problems,” Theoretical Computer Science 1 (1976), 237–267.

    Reviews

    1 machine check

    No person has reviewed this attempt. 1 machine check below — a machine check is not human verification.

    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification KNOWN

      PASS

      The argument is mathematically sound: orienting any undirected graph acyclically gives a digraph of triangular tree-width 00, and the fixed MSO sentence for 3-colorability exactly recovers ordinary graph 3-colorability. Thus fixed-sentence MSO model checking is NP-complete already on the ttw⁡=0\operatorname{ttw}=0 class, so a general polynomial-time Courcelle-style theorem for bounded triangular tree-width would imply P=NPP=NP. This attacks the stated conjecture directly.

      Novelty assessment

      KNOWN

      Classification rationale: The accepted reduction is already in the literature in a more general form. Ganian et al., Proposition 3.2, prove that any digraph width measure bounded on all DAGs cannot yield polynomial/XP algorithms for all MSO1-definable problems unless P=NP. Their proof uses exactly the same idea: MSO1-definable 3-colourability, replace adjacency by arc-in-either-direction, and orient an arbitrary graph acyclically.

      Literature check: I searched for the exact triangular tree-width/MSO/3-colourability/NP-completeness combination and did not find an explicit Meer-specific note. However, the broader directed-width literature contains the stronger general obstruction. Since, under the definition used in the accepted proof, triangular tree-width is 0 on DAGs, Ganian et al.’s Proposition 3.2 immediately specializes to this conjecture.

      Citation: Robert Ganian, Petr Hliněný, Joachim Kneis, Daniel Meister, Jan Obdržálek, Peter Rossmanith, Somnath Sikdar, “Are there any good digraph width measures?”, J. Combinatorial Theory Ser. B, DOI: 10.1016/j.jctb.2015.09.001; arXiv:1004.1485, Proposition 3.2. Also see Klaus Meer, “An Extended Tree-Width Notion for Directed Graphs Related to the Computation of Permanents,” Theory of Computing Systems, DOI: 10.1007/s00224-013-9490-z.

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