255 research outputs found

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    The Potts model and the independence polynomial:Uniqueness of the Gibbs measure and distributions of complex zeros

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    Part 1 of this dissertation studies the antiferromagnetic Potts model, which originates in statistical physics. In particular the transition from multiple Gibbs measures to a unique Gibbs measure for the antiferromagnetic Potts model on the infinite regular tree is studied. This is called a uniqueness phase transition. A folklore conjecture about the parameter at which the uniqueness phase transition occurs is partly confirmed. The proof uses a geometric condition, which comes from analysing an associated dynamical system.Part 2 of this dissertation concerns zeros of the independence polynomial. The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the independence polynomial is related to phase transitions in terms of the analycity of the free energy and plays an important role in the design of efficient algorithms to approximately compute evaluations of the independence polynomial. Chapter 5 directly relates the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. This is done by moreover relating the set of zeros of the independence polynomial to chaotic behaviour of a naturally associated family of rational functions; the occupation ratios. Chapter 6 studies boundedness of zeros of the independence polynomial of tori for sequences of tori converging to the integer lattice. It is shown that zeros are bounded for sequences of balanced tori, but unbounded for sequences of highly unbalanced tori

    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Asymptotics for Palette Sparsification

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    It is shown that the following holds for each ε>0\varepsilon>0. For GG an nn-vertex graph of maximum degree DD and "lists" LvL_v (vV(G)v \in V(G)) chosen independently and uniformly from the ((1+ε)lnn(1+\varepsilon)\ln n)-subsets of {1,...,D+1}\{1, ..., D+1\}, G admits a proper coloring σ with σvLvv G \text{ admits a proper coloring } \sigma \text{ with } \sigma_v \in L_v \forall v with probability tending to 1 as DD \to \infty. This is an asymptotically optimal version of a recent "palette sparsification" theorem of Assadi, Chen, and Khanna.Comment: 29 page

    On the structure of graphs with forbidden induced substructures

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    One of the central goals in extremal combinatorics is to understand how the global structure of a combinatorial object, e.g. a graph, hypergraph or set system, is affected by local constraints. In this thesis we are concerned with structural properties of graphs and hypergraphs which locally do not look like some type of forbidden induced pattern. Patterns can be single subgraphs, families of subgraphs, or in the multicolour version colourings or families of colourings of subgraphs. Erdős and Szekeres\u27s quantitative version of Ramsey\u27s theorem asserts that in every 22-edge-colouring of the complete graph on nn vertices there is a monochromatic clique on at least 12logn\frac{1}{2}\log n vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed colourings on subgraphs ensures much larger monochromatic cliques. The conjecture is open in general, though a few partial results are known. The first part of this thesis will be concerned with different variants of this conjecture: A bipartite variant, a multicolour variant, and an order-size variant for hypergraphs. In the second part of this thesis we focus more on order-size pairs; an order-size pair (n,e)(n,e) is the family consisting of all graphs of order nn and size ee, i.e. on nn vertices with ee edges. We consider order-size pairs in different settings: The graph setting, the bipartite setting and the hypergraph setting. In all these settings we investigate the existence of absolutely avoidable pairs, i.e. fixed pairs that are avoided by all order-size pairs with sufficiently large order, and also forcing densities of order-size pairs (m,f)(m,f), i.e. for nn approaching infinity, the limit superior of the fraction of all possible sizes ee, such that the order-size pair (n,e)(n,e) does not avoid the pair (m,f)(m,f)

    Integrality and cutting planes in semidefinite programming approaches for combinatorial optimization

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    Many real-life decision problems are discrete in nature. To solve such problems as mathematical optimization problems, integrality constraints are commonly incorporated in the model to reflect the choice of finitely many alternatives. At the same time, it is known that semidefinite programming is very suitable for obtaining strong relaxations of combinatorial optimization problems. In this dissertation, we study the interplay between semidefinite programming and integrality, where a special focus is put on the use of cutting-plane methods. Although the notions of integrality and cutting planes are well-studied in linear programming, integer semidefinite programs (ISDPs) are considered only recently. We show that manycombinatorial optimization problems can be modeled as ISDPs. Several theoretical concepts, such as the Chvátal-Gomory closure, total dual integrality and integer Lagrangian duality, are studied for the case of integer semidefinite programming. On the practical side, we introduce an improved branch-and-cut approach for ISDPs and a cutting-plane augmented Lagrangian method for solving semidefinite programs with a large number of cutting planes. Throughout the thesis, we apply our results to a wide range of combinatorial optimization problems, among which the quadratic cycle cover problem, the quadratic traveling salesman problem and the graph partition problem. Our approaches lead to novel, strong and efficient solution strategies for these problems, with the potential to be extended to other problem classes

    The existence of subspace designs

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    We prove the existence of subspace designs with any given parameters, provided that the dimension of the underlying space is sufficiently large in terms of the other parameters of the design and satisfies the obvious necessary divisibility conditions. This settles an open problem from the 1970s. Moreover, we also obtain an approximate formula for the number of such designs.Comment: 61 page

    Fast Decoding of Explicit Almost Optimal ?-Balanced q-Ary Codes And Fast Approximation of Expanding k-CSPs

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    A general approach to transversal versions of Dirac-type theorems

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    Given a collection of hypergraphs =(1,...,) with the same vertex set, an -edge graph ⊂∪∈[] is atransversal if there is a bijection ∶()→[] such that ∈(()) for each ∈(). How large does the minimum degree of each need to be so that necessarily contains a copy of that is a transversal? Each in the collection could be the same hypergraph,hence the minimum degree of each needs to be large enough to ensure that ⊆. Since its general introduction by Joos and Kim (Bull. Lond. Math. Soc. 52 (2020)498–504), a growing body of work has shown that inmany cases this lower bound is tight. In this paper, wegive a unified approach to this problem by providinga widely applicable sufficient condition for this lowerbound to be asymptotically tight. This is general enoughto recover many previous results in the area and obtainnovel transversal variants of several classical Dirac-typeresults for (powers of) Hamilton cycles. For example, wederive that any collection of graphs on an -vertex set, each with minimum degree at least (∕( + 1) +(1)), contains a transversal copy of the th power of a Hamilton cycle. This can be viewed as a rainbow versionof the Pósa–Seymour conjecture
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