312 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

    Structural optimization in steel structures, algorithms and applications

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    L'abstract è presente nell'allegato / the abstract is in the attachmen

    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

    Efficient Distributed Decomposition and Routing Algorithms in Minor-Free Networks and Their Applications

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    In the LOCAL model, low-diameter decomposition is a useful tool in designing algorithms, as it allows us to shift from the general graph setting to the low-diameter graph setting, where brute-force information gathering can be done efficiently. Recently, Chang and Su [PODC 2022] showed that any high-conductance network excluding a fixed minor contains a high-degree vertex, so the entire graph topology can be gathered to one vertex efficiently in the CONGEST model using expander routing. Therefore, in networks excluding a fixed minor, many problems that can be solved efficiently in LOCAL via low-diameter decomposition can also be solved efficiently in CONGEST via expander decomposition. In this work, we show improved decomposition and routing algorithms for networks excluding a fixed minor in the CONGEST model. Our algorithms cost poly(logn,1/ϵ)\text{poly}(\log n, 1/\epsilon) rounds deterministically. For bounded-degree graphs, our algorithms finish in O(ϵ1logn)+ϵO(1)O(\epsilon^{-1}\log n) + \epsilon^{-O(1)} rounds. Our algorithms have a wide range of applications, including the following results in CONGEST. 1. A (1ϵ)(1-\epsilon)-approximate maximum independent set in a network excluding a fixed minor can be computed deterministically in O(ϵ1logn)+ϵO(1)O(\epsilon^{-1}\log^\ast n) + \epsilon^{-O(1)} rounds, nearly matching the Ω(ϵ1logn)\Omega(\epsilon^{-1}\log^\ast n) lower bound of Lenzen and Wattenhofer [DISC 2008]. 2. Property testing of any additive minor-closed property can be done deterministically in O(logn)O(\log n) rounds if ϵ\epsilon is a constant or O(ϵ1logn)+ϵO(1)O(\epsilon^{-1}\log n) + \epsilon^{-O(1)} rounds if the maximum degree Δ\Delta is a constant, nearly matching the Ω(ϵ1logn)\Omega(\epsilon^{-1}\log n) lower bound of Levi, Medina, and Ron [PODC 2018].Comment: To appear in PODC 202

    A Fine-Grained Classification of the Complexity of Evaluating the Tutte Polynomial on Integer Points Parameterized by Treewidth and Cutwidth

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    We give a fine-grained classification of evaluating the Tutte polynomial T(G;x,y) on all integer points on graphs with small treewidth and cutwidth. Specifically, we show for any point (x,y) ∈ ℤ² that either - T(G; x, y) can be computed in polynomial time, - T(G; x, y) can be computed in 2^O(tw) n^O(1) time, but not in 2^o(ctw) n^O(1) time assuming the Exponential Time Hypothesis (ETH), - T(G; x, y) can be computed in 2^O(tw log tw) n^O(1) time, but not in 2^o(ctw log ctw) n^O(1) time assuming the ETH, where we assume tree decompositions of treewidth tw and cutwidth decompositions of cutwidth ctw are given as input along with the input graph on n vertices and point (x,y). To obtain these results, we refine the existing reductions that were instrumental for the seminal dichotomy by Jaeger, Welsh and Vertigan [Math. Proc. Cambridge Philos. Soc'90]. One of our technical contributions is a new rank bound of a matrix that indicates whether the union of two forests is a forest itself, which we use to show that the number of forests of a graph can be counted in 2^O(tw) n^O(1) time

    Geometry and Topology in Memory and Navigation

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    Okinawa Institute of Science and Technology Graduate UniversityDoctor of PhilosophyGeometry and topology offer rich mathematical worlds and perspectives with which to study and improve our understanding of cognitive function. Here I present the following examples: (1) a functional role for inhibitory diversity in associative memories with graph- ical relationships; (2) improved memory capacity in an associative memory model with setwise connectivity, with implications for glial and dendritic function; (3) safe and effi- cient group navigation among conspecifics using purely local geometric information; and (4) enhancing geometric and topological methods to probe the relations between neural activity and behaviour. In each work, tools and insights from geometry and topology are used in essential ways to gain improved insights or performance. This thesis contributes to our knowledge of the potential computational affordances of biological mechanisms (such as inhibition and setwise connectivity), while also demonstrating new geometric and topological methods and perspectives with which to deepen our understanding of cognitive tasks and their neural representations.doctoral thesi

    Superpolynomial Lower Bounds for Learning Monotone Classes

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    Determinantal Sieving

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    We introduce determinantal sieving, a new, remarkably powerful tool in the toolbox of algebraic FPT algorithms. Given a polynomial P(X)P(X) on a set of variables X={x1,,xn}X=\{x_1,\ldots,x_n\} and a linear matroid M=(X,I)M=(X,\mathcal{I}) of rank kk, both over a field F\mathbb{F} of characteristic 2, in 2k2^k evaluations we can sieve for those terms in the monomial expansion of PP which are multilinear and whose support is a basis for MM. Alternatively, using 2k2^k evaluations of PP we can sieve for those monomials whose odd support spans MM. Applying this framework, we improve on a range of algebraic FPT algorithms, such as: 1. Solving qq-Matroid Intersection in time O(2(q2)k)O^*(2^{(q-2)k}) and qq-Matroid Parity in time O(2qk)O^*(2^{qk}), improving on O(4qk)O^*(4^{qk}) (Brand and Pratt, ICALP 2021) 2. TT-Cycle, Colourful (s,t)(s,t)-Path, Colourful (S,T)(S,T)-Linkage in undirected graphs, and the more general Rank kk (S,T)(S,T)-Linkage problem, all in O(2k)O^*(2^k) time, improving on O(2k+S)O^*(2^{k+|S|}) respectively O(2S+O(k2log(k+F)))O^*(2^{|S|+O(k^2 \log(k+|\mathbb{F}|))}) (Fomin et al., SODA 2023) 3. Many instances of the Diverse X paradigm, finding a collection of rr solutions to a problem with a minimum mutual distance of dd in time O(2r(r1)d/2)O^*(2^{r(r-1)d/2}), improving solutions for kk-Distinct Branchings from time 2O(klogk)2^{O(k \log k)} to O(2k)O^*(2^k) (Bang-Jensen et al., ESA 2021), and for Diverse Perfect Matchings from O(22O(rd))O^*(2^{2^{O(rd)}}) to O(2r2d/2)O^*(2^{r^2d/2}) (Fomin et al., STACS 2021) All matroids are assumed to be represented over a field of characteristic 2. Over general fields, we achieve similar results at the cost of using exponential space by working over the exterior algebra. For a class of arithmetic circuits we call strongly monotone, this is even achieved without any loss of running time. However, the odd support sieving result appears to be specific to working over characteristic 2
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