2,048 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

    Local geometry of NAE-SAT solutions in the condensation regime

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    The local behavior of typical solutions of random constraint satisfaction problems (CSP) describes many important phenomena including clustering thresholds, decay of correlations, and the behavior of message passing algorithms. When the constraint density is low, studying the planted model is a powerful technique for determining this local behavior which in many examples has a simple Markovian structure. Work of Coja-Oghlan, Kapetanopoulos, Muller (2020) showed that for a wide class of models, this description applies up to the so-called condensation threshold. Understanding the local behavior after the condensation threshold is more complex due to long-range correlations. In this work, we revisit the random regular NAE-SAT model in the condensation regime and determine the local weak limit which describes a random solution around a typical variable. This limit exhibits a complicated non-Markovian structure arising from the space of solutions being dominated by a small number of large clusters, a result rigorously verified by Nam, Sly, Sohn (2021). This is the first characterization of the local weak limit in the condensation regime for any sparse random CSPs in the so-called one-step replica symmetry breaking (1RSB) class. Our result is non-asymptotic, and characterizes the tight fluctuation O(n1/2)O(n^{-1/2}) around the limit. Our proof is based on coupling the local neighborhoods of an infinite spin system, which encodes the structure of the clusters, to a broadcast model on trees whose channel is given by the 1RSB belief-propagation fixed point. We believe that our proof technique has broad applicability to random CSPs in the 1RSB class.Comment: 43 pages, 2 figure

    Homogeneous colourings of graphs

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    A proper vertex kk-colouring of a graph GG is called ll-homogeneous if the number of colours in the neigbourhood of each vertex of GG equals ll. We explore basic properties (the existence and the number of used colours) of homogeneous colourings of graphs in general as well as of some specific graph families, in particular planar graphs

    A restricted L(2, 1)-labelling problem on interval graphs

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    In a graph G = (V, E), L(2, 1)-labelling is considered by a function ` whose domain is V and codomain is set of non-negative integers with a condition that the vertices which are adjacent assign labels whose difference is at least two and the vertices whose distance is two, assign distinct labels. The difference between maximum and minimum labels among all possible labels is denoted by λ2,1(G). This paper contains a variant of L(2, 1)-labelling problem. In L(2, 1)-labelling problem, all the vertices are L(2, 1)-labeled by least number of labels. In this paper, maximum allowable label K is given. The problem is: L(2, 1)-label the vertices of G by using the labels {0, 1, 2, . . . , K} such that maximum number of vertices get label. If K labels are adequate for labelling all the vertices of the graph then all vertices get label, otherwise some vertices remains unlabeled. An algorithm is designed to solve this problem. The algorithm is also illustrated by examples. Also, an algorithm is designed to test whether an interval graph is no hole label or not for the purpose of L(2, 1)-labelling.Publisher's Versio

    Structural Parameterizations for Two Bounded Degree Problems Revisited

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    Product structure of graph classes with strongly sublinear separators

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    We investigate the product structure of hereditary graph classes admitting strongly sublinear separators. We characterise such classes as subgraphs of the strong product of a star and a complete graph of strongly sublinear size. In a more precise result, we show that if any hereditary graph class G\mathcal{G} admits O(n1ϵ)O(n^{1-\epsilon}) separators, then for any fixed δ(0,ϵ)\delta\in(0,\epsilon) every nn-vertex graph in G\mathcal{G} is a subgraph of the strong product of a graph HH with bounded tree-depth and a complete graph of size O(n1ϵ+δ)O(n^{1-\epsilon+\delta}). This result holds with δ=0\delta=0 if we allow HH to have tree-depth O(loglogn)O(\log\log n). Moreover, using extensions of classical isoperimetric inequalties for grids graphs, we show the dependence on δ\delta in our results and the above td(H)O(loglogn)\text{td}(H)\in O(\log\log n) bound are both best possible. We prove that nn-vertex graphs of bounded treewidth are subgraphs of the product of a graph with tree-depth tt and a complete graph of size O(n1/t)O(n^{1/t}), which is best possible. Finally, we investigate the conjecture that for any hereditary graph class G\mathcal{G} that admits O(n1ϵ)O(n^{1-\epsilon}) separators, every nn-vertex graph in G\mathcal{G} is a subgraph of the strong product of a graph HH with bounded tree-width and a complete graph of size O(n1ϵ)O(n^{1-\epsilon}). We prove this for various classes G\mathcal{G} of interest.Comment: v2: added bad news subsection; v3: removed section "Polynomial Expansion Classes" which had an error, added section "Lower Bounds", and added a new author; v4: minor revisions and corrections

    Improved NP-Hardness of Approximation for Orthogonality Dimension and Minrank

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    The orthogonality dimension of a graph G over ? is the smallest integer k for which one can assign a nonzero k-dimensional real vector to each vertex of G, such that every two adjacent vertices receive orthogonal vectors. We prove that for every sufficiently large integer k, it is NP-hard to decide whether the orthogonality dimension of a given graph over ? is at most k or at least 2^{(1-o(1))?k/2}. We further prove such hardness results for the orthogonality dimension over finite fields as well as for the closely related minrank parameter, which is motivated by the index coding problem in information theory. This in particular implies that it is NP-hard to approximate these graph quantities to within any constant factor. Previously, the hardness of approximation was known to hold either assuming certain variants of the Unique Games Conjecture or for approximation factors smaller than 3/2. The proofs involve the concept of line digraphs and bounds on their orthogonality dimension and on the minrank of their complement

    Make a graph singly connected by edge orientations

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    A directed graph DD is singly connected if for every ordered pair of vertices (s,t)(s,t), there is at most one path from ss to tt in DD. Graph orientation problems ask, given an undirected graph GG, to find an orientation of the edges such that the resultant directed graph DD has a certain property. In this work, we study the graph orientation problem where the desired property is that DD is singly connected. Our main result concerns graphs of a fixed girth gg and coloring number cc. For every g,c3g,c\geq 3, the problem restricted to instances of girth gg and coloring number cc, is either NP-complete or in P. As further algorithmic results, we show that the problem is NP-hard on planar graphs and polynomial time solvable distance-hereditary graphs
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