1,868 research outputs found

    Independent transversals in locally sparse graphs

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    Let G be a graph with maximum degree \Delta whose vertex set is partitioned into parts V(G) = V_1 \cup ... \cup V_r. A transversal is a subset of V(G) containing exactly one vertex from each part V_i. If it is also an independent set, then we call it an independent transversal. The local degree of G is the maximum number of neighbors of a vertex v in a part V_i, taken over all choices of V_i and v \not \in V_i. We prove that for every fixed \epsilon > 0, if all part sizes |V_i| >= (1+\epsilon)\Delta and the local degree of G is o(\Delta), then G has an independent transversal for sufficiently large \Delta. This extends several previous results and settles (in a stronger form) a conjecture of Aharoni and Holzman. We then generalize this result to transversals that induce no cliques of size s. (Note that independent transversals correspond to s=2.) In that context, we prove that parts of size |V_i| >= (1+\epsilon)[\Delta/(s-1)] and local degree o(\Delta) guarantee the existence of such a transversal, and we provide a construction that shows this is asymptotically tight.Comment: 16 page

    Model Counting of Query Expressions: Limitations of Propositional Methods

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    Query evaluation in tuple-independent probabilistic databases is the problem of computing the probability of an answer to a query given independent probabilities of the individual tuples in a database instance. There are two main approaches to this problem: (1) in `grounded inference' one first obtains the lineage for the query and database instance as a Boolean formula, then performs weighted model counting on the lineage (i.e., computes the probability of the lineage given probabilities of its independent Boolean variables); (2) in methods known as `lifted inference' or `extensional query evaluation', one exploits the high-level structure of the query as a first-order formula. Although it is widely believed that lifted inference is strictly more powerful than grounded inference on the lineage alone, no formal separation has previously been shown for query evaluation. In this paper we show such a formal separation for the first time. We exhibit a class of queries for which model counting can be done in polynomial time using extensional query evaluation, whereas the algorithms used in state-of-the-art exact model counters on their lineages provably require exponential time. Our lower bounds on the running times of these exact model counters follow from new exponential size lower bounds on the kinds of d-DNNF representations of the lineages that these model counters (either explicitly or implicitly) produce. Though some of these queries have been studied before, no non-trivial lower bounds on the sizes of these representations for these queries were previously known.Comment: To appear in International Conference on Database Theory (ICDT) 201

    Topological transversals to a family of convex sets

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    Let F\mathcal F be a family of compact convex sets in Rd\mathbb R^d. We say that F\mathcal F has a \emph{topological ρ\rho-transversal of index (m,k)(m,k)} (ρ<m\rho<m, 0<kdm0<k\leq d-m) if there are, homologically, as many transversal mm-planes to F\mathcal F as mm-planes containing a fixed ρ\rho-plane in Rm+k\mathbb R^{m+k}. Clearly, if F\mathcal F has a ρ\rho-transversal plane, then F\mathcal F has a topological ρ\rho-transversal of index (m,k),(m,k), for ρ<m\rho<m and kdmk\leq d-m. The converse is not true in general. We prove that for a family F\mathcal F of ρ+k+1\rho+k+1 compact convex sets in Rd\mathbb R^d a topological ρ\rho-transversal of index (m,k)(m,k) implies an ordinary ρ\rho-transversal. We use this result, together with the multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann category of the Grassmannian, and different versions of the colorful Helly theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences

    An asymptotic bound for the strong chromatic number

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    The strong chromatic number χs(G)\chi_{\text{s}}(G) of a graph GG on nn vertices is the least number rr with the following property: after adding rn/rnr \lceil n/r \rceil - n isolated vertices to GG and taking the union with any collection of spanning disjoint copies of KrK_r in the same vertex set, the resulting graph has a proper vertex-colouring with rr colours. We show that for every c>0c > 0 and every graph GG on nn vertices with Δ(G)cn\Delta(G) \ge cn, χs(G)(2+o(1))Δ(G)\chi_{\text{s}}(G) \leq (2 + o(1)) \Delta(G), which is asymptotically best possible.Comment: Minor correction, accepted for publication in Combin. Probab. Compu

    Achieving New Upper Bounds for the Hypergraph Duality Problem through Logic

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    The hypergraph duality problem DUAL is defined as follows: given two simple hypergraphs G\mathcal{G} and H\mathcal{H}, decide whether H\mathcal{H} consists precisely of all minimal transversals of G\mathcal{G} (in which case we say that G\mathcal{G} is the dual of H\mathcal{H}). This problem is equivalent to deciding whether two given non-redundant monotone DNFs are dual. It is known that non-DUAL, the complementary problem to DUAL, is in GC(log2n,PTIME)\mathrm{GC}(\log^2 n,\mathrm{PTIME}), where GC(f(n),C)\mathrm{GC}(f(n),\mathcal{C}) denotes the complexity class of all problems that after a nondeterministic guess of O(f(n))O(f(n)) bits can be decided (checked) within complexity class C\mathcal{C}. It was conjectured that non-DUAL is in GC(log2n,LOGSPACE)\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE}). In this paper we prove this conjecture and actually place the non-DUAL problem into the complexity class GC(log2n,TC0)\mathrm{GC}(\log^2 n,\mathrm{TC}^0) which is a subclass of GC(log2n,LOGSPACE)\mathrm{GC}(\log^2 n,\mathrm{LOGSPACE}). We here refer to the logtime-uniform version of TC0\mathrm{TC}^0, which corresponds to FO(COUNT)\mathrm{FO(COUNT)}, i.e., first order logic augmented by counting quantifiers. We achieve the latter bound in two steps. First, based on existing problem decomposition methods, we develop a new nondeterministic algorithm for non-DUAL that requires to guess O(log2n)O(\log^2 n) bits. We then proceed by a logical analysis of this algorithm, allowing us to formulate its deterministic part in FO(COUNT)\mathrm{FO(COUNT)}. From this result, by the well known inclusion TC0LOGSPACE\mathrm{TC}^0\subseteq\mathrm{LOGSPACE}, it follows that DUAL belongs also to DSPACE[log2n]\mathrm{DSPACE}[\log^2 n]. Finally, by exploiting the principles on which the proposed nondeterministic algorithm is based, we devise a deterministic algorithm that, given two hypergraphs G\mathcal{G} and H\mathcal{H}, computes in quadratic logspace a transversal of G\mathcal{G} missing in H\mathcal{H}.Comment: Restructured the presentation in order to be the extended version of a paper that will shortly appear in SIAM Journal on Computin
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