236 research outputs found

    An approximate version of Sidorenko's conjecture

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    A beautiful conjecture of Erd\H{o}s-Simonovits and Sidorenko states that if H is a bipartite graph, then the random graph with edge density p has in expectation asymptotically the minimum number of copies of H over all graphs of the same order and edge density. This conjecture also has an equivalent analytic form and has connections to a broad range of topics, such as matrix theory, Markov chains, graph limits, and quasirandomness. Here we prove the conjecture if H has a vertex complete to the other part, and deduce an approximate version of the conjecture for all H. Furthermore, for a large class of bipartite graphs, we prove a stronger stability result which answers a question of Chung, Graham, and Wilson on quasirandomness for these graphs.Comment: 12 page

    Separating Automatic Relations

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    We study the separability problem for automatic relations (i.e., relations on finite words definable by synchronous automata) in terms of recognizable relations (i.e., finite unions of products of regular languages). This problem takes as input two automatic relations R and R\u27, and asks if there exists a recognizable relation S that contains R and does not intersect R\u27. We show this problem to be undecidable when the number of products allowed in the recognizable relation is fixed. In particular, checking if there exists a recognizable relation S with at most k products of regular languages that separates R from R\u27 is undecidable, for each fixed k ? 2. Our proofs reveal tight connections, of independent interest, between the separability problem and the finite coloring problem for automatic graphs, where colors are regular languages

    Separating Automatic Relations

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    We study the separability problem for automatic relations (i.e., relations on finite words definable by synchronous automata) in terms of recognizable relations (i.e., finite unions of products of regular languages). This problem takes as input two automatic relations RR and R′R', and asks if there exists a recognizable relation SS that contains RR and does not intersect R′R'. We show this problem to be undecidable when the number of products allowed in the recognizable relation is fixed. In particular, checking if there exists a recognizable relation SS with at most kk products of regular languages that separates RR from R′R' is undecidable, for each fixed k≥2k \geq 2. Our proofs reveal tight connections, of independent interest, between the separability problem and the finite coloring problem for automatic graphs, where colors are regular languages.Comment: Long version of a paper accepted at MFCS 202

    Foundations for decision problems in separation logic with general inductive predicates

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    Abstract. We establish foundational results on the computational com-plexity of deciding entailment in Separation Logic with general induc-tive predicates whose underlying base language allows for pure formulas, pointers and existentially quantified variables. We show that entailment is in general undecidable, and ExpTime-hard in a fragment recently shown to be decidable by Iosif et al. Moreover, entailment in the base language is ΠP2-complete, the upper bound even holds in the presence of list predicates. We additionally show that entailment in essentially any fragment of Separation Logic allowing for general inductive predicates is intractable even when strong syntactic restrictions are imposed.

    Universality of graph homomorphism games and the quantum coloring problem

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    We show that quantum graph parameters for finite, simple, undirected graphs encode winning strategies for all possible synchronous non-local games. Given a synchronous game G=(I,O,λ)\mathcal{G}=(I,O,\lambda) with ∣I∣=n|I|=n and ∣O∣=k|O|=k, we demonstrate what we call a weak ∗*-equivalence between G\mathcal{G} and a 33-coloring game on a graph with at most 3+n+9n(k−2)+6∣λ−1({0})∣3+n+9n(k-2)+6|\lambda^{-1}(\{0\})| vertices, strengthening and simplifying work implied by Z. Ji (arXiv:1310.3794) for winning quantum strategies for synchronous non-local games. As an application, we obtain a quantum version of L. Lov\'{a}sz's reduction (Proc. 4th SE Conf. on Comb., Graph Theory & Computing, 1973) of the kk-coloring problem for a graph GG with nn vertices and mm edges to the 33-coloring problem for a graph with 3+n+9n(k−2)+6mk3+n+9n(k-2)+6mk vertices. We also show that, for ``graph of the game" X(G)X(\mathcal{G}) associated to G\mathcal{G} from A. Atserias et al (J. Comb. Theory Series B, Vol. 136, 2019), the independence number game Hom(K∣I∣,X(G)‾)\text{Hom}(K_{|I|},\overline{X(\mathcal{G})}) is hereditarily ∗*-equivalent to G\mathcal{G}, so that the possibility of winning strategies is the same in both games for all models, except the game algebra. Thus, the quantum versions of the chromatic number, independence number and clique number encode winning strategies for all synchronous games in all quantum models.Comment: 28 pages; 2 figure
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