169 research outputs found

    Algorithms and complexity for approximately counting hypergraph colourings and related problems

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    The past decade has witnessed advancements in designing efficient algorithms for approximating the number of solutions to constraint satisfaction problems (CSPs), especially in the local lemma regime. However, the phase transition for the computational tractability is not known. This thesis is dedicated to the prototypical problem of this kind of CSPs, the hypergraph colouring. Parameterised by the number of colours q, the arity of each hyperedge k, and the vertex maximum degree Δ, this problem falls into the regime of Lovász local lemma when Δ ≲ qᵏ. In prior, however, fast approximate counting algorithms exist when Δ ≲ qᵏ/³, and there is no known inapproximability result. In pursuit of this, our contribution is two-folded, stated as follows. • When q, k ≥ 4 are evens and Δ ≥ 5·qᵏ/², approximating the number of hypergraph colourings is NP-hard. • When the input hypergraph is linear and Δ ≲ qᵏ/², a fast approximate counting algorithm does exist

    LIPIcs, Volume 251, ITCS 2023, Complete Volume

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    LIPIcs, Volume 251, ITCS 2023, Complete Volum

    On linear, fractional, and submodular optimization

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    In this thesis, we study four fundamental problems in the theory of optimization. 1. In fractional optimization, we are interested in minimizing a ratio of two functions over some domain. A well-known technique for solving this problem is the Newton– Dinkelbach method. We propose an accelerated version of this classical method and give a new analysis using the Bregman divergence. We show how it leads to improved or simplified results in three application areas. 2. The diameter of a polyhedron is the maximum length of a shortest path between any two vertices. The circuit diameter is a relaxation of this notion, whereby shortest paths are not restricted to edges of the polyhedron. For a polyhedron in standard equality form with constraint matrix A, we prove an upper bound on the circuit diameter that is quadratic in the rank of A and logarithmic in the circuit imbalance measure of A. We also give circuit augmentation algorithms for linear programming with similar iteration complexity. 3. The correlation gap of a set function is the ratio between its multilinear and concave extensions. We present improved lower bounds on the correlation gap of a matroid rank function, parametrized by the rank and girth of the matroid. We also prove that for a weighted matroid rank function, the worst correlation gap is achieved with uniform weights. Such improved lower bounds have direct applications in submodular maximization and mechanism design. 4. The last part of this thesis concerns parity games, a problem intimately related to linear programming. A parity game is an infinite-duration game between two players on a graph. The problem of deciding the winner lies in NP and co-NP, with no known polynomial algorithm to date. Many of the fastest (quasi-polynomial) algorithms have been unified via the concept of a universal tree. We propose a strategy iteration framework which can be applied on any universal tree

    Finding a Highly Connected Steiner Subgraph and its Applications

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    Given a (connected) undirected graph G, a set X ? V(G) and integers k and p, the Steiner Subgraph Extension problem asks whether there exists a set S ? X of at most k vertices such that G[S] is a p-edge-connected subgraph. This problem is a natural generalization of the well-studied Steiner Tree problem (set p = 1 and X to be the terminals). In this paper, we initiate the study of Steiner Subgraph Extension from the perspective of parameterized complexity and give a fixed-parameter algorithm (i.e., FPT algorithm) parameterized by k and p on graphs of bounded degeneracy (removing the assumption of bounded degeneracy results in W-hardness). Besides being an independent advance on the parameterized complexity of network design problems, our result has natural applications. In particular, we use our result to obtain new single-exponential FPT algorithms for several vertex-deletion problems studied in the literature, where the goal is to delete a smallest set of vertices such that: (i) the resulting graph belongs to a specified hereditary graph class, and (ii) the deleted set of vertices induces a p-edge-connected subgraph of the input graph

    An EPTAS for Budgeted Matching and Budgeted Matroid Intersection via Representative Sets

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    LIPIcs, Volume 261, ICALP 2023, Complete Volume

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    LIPIcs, Volume 261, ICALP 2023, Complete Volum

    Deterministic Constrained Multilinear Detection

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