2,048 research outputs found
Algorithms and complexity for approximately counting hypergraph colourings and related problems
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
LIPIcs, Volume 251, ITCS 2023, Complete Volum
Local geometry of NAE-SAT solutions in the condensation regime
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
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
A proper vertex -colouring of a graph is called -homogeneous if the number of colours in the neigbourhood of each vertex of equals . 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
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
Product structure of graph classes with strongly sublinear separators
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
admits separators, then for any fixed
every -vertex graph in is a subgraph
of the strong product of a graph with bounded tree-depth and a complete
graph of size . This result holds with if
we allow to have tree-depth . Moreover, using extensions of
classical isoperimetric inequalties for grids graphs, we show the dependence on
in our results and the above bound are
both best possible. We prove that -vertex graphs of bounded treewidth are
subgraphs of the product of a graph with tree-depth and a complete graph of
size , which is best possible. Finally, we investigate the
conjecture that for any hereditary graph class that admits
separators, every -vertex graph in is a
subgraph of the strong product of a graph with bounded tree-width and a
complete graph of size . We prove this for various classes
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
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
A directed graph is singly connected if for every ordered pair of
vertices , there is at most one path from to in . Graph
orientation problems ask, given an undirected graph , to find an orientation
of the edges such that the resultant directed graph has a certain property.
In this work, we study the graph orientation problem where the desired property
is that is singly connected. Our main result concerns graphs of a fixed
girth and coloring number . For every , the problem
restricted to instances of girth and coloring number , 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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