Incremental complexity of a bi-objective hypergraph transversal problem

The hypergraph transversal problem has been intensively studied, from both a theoretical and a practical point of view. In particular , its incremental complexity is known to be quasi-polynomial in general and polynomial for bounded hypergraphs. Recent applications in computational biology however require to solve a generalization of this problem, that we call bi-objective transversal problem. The instance is in this case composed of a pair of hypergraphs (A, B), and the aim is to find minimal sets which hit all the hyperedges of A while intersecting a minimal set of hyperedges of B. In this paper, we formalize this problem, link it to a problem on monotone boolean $\land$ -- $\lor$ formulae of depth 3 and study its incremental complexity

Independence densities of hypergraphs

We consider the number of independent sets in hypergraphs, which allows us to define the independence density of countable hypergraphs. Hypergraph independence densities include a broad family of densities over graphs and relational structures, such as $F$-free densities of graphs for a given graph $F.$ In the case of $k$-uniform hypergraphs, we prove that the independence density is always rational. In the case of finite but unbounded hyperedges, we show that the independence density can be any real number in $[0,1].$ Finally, we extend the notion of independence density via independence polynomials

On the Enumeration of Minimal Dominating Sets and Related Notions

A dominating set $D$ in a graph is a subset of its vertex set such that each vertex is either in $D$ or has a neighbour in $D$. In this paper, we are interested in the enumeration of (inclusion-wise) minimal dominating sets in graphs, called the Dom-Enum problem. It is well known that this problem can be polynomially reduced to the Trans-Enum problem in hypergraphs, i.e., the problem of enumerating all minimal transversals in a hypergraph. Firstly we show that the Trans-Enum problem can be polynomially reduced to the Dom-Enum problem. As a consequence there exists an output-polynomial time algorithm for the Trans-Enum problem if and only if there exists one for the Dom-Enum problem. Secondly, we study the Dom-Enum problem in some graph classes. We give an output-polynomial time algorithm for the Dom-Enum problem in split graphs, and introduce the completion of a graph to obtain an output-polynomial time algorithm for the Dom-Enum problem in $P_6$-free chordal graphs, a proper superclass of split graphs. Finally, we investigate the complexity of the enumeration of (inclusion-wise) minimal connected dominating sets and minimal total dominating sets of graphs. We show that there exists an output-polynomial time algorithm for the Dom-Enum problem (or equivalently Trans-Enum problem) if and only if there exists one for the following enumeration problems: minimal total dominating sets, minimal total dominating sets in split graphs, minimal connected dominating sets in split graphs, minimal dominating sets in co-bipartite graphs.Comment: 15 pages, 3 figures, In revisio

Distributed local approximation algorithms for maximum matching in graphs and hypergraphs

We describe approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in a hypergraph of rank $r$. Our main result is a deterministic algorithm to generate a matching which is an $O(r)$-approximation to the maximum weight matching, running in $\tilde O(r \log \Delta + \log^2 \Delta + \log^* n)$ rounds. (Here, the $\tilde O()$ notations hides $\text{polyloglog } \Delta$ and $\text{polylog } r$ factors). This is based on a number of new derandomization techniques extending methods of Ghaffari, Harris & Kuhn (2017). As a main application, we obtain nearly-optimal algorithms for the long-studied problem of maximum-weight graph matching. Specifically, we get a $(1+\epsilon)$ approximation algorithm using $\tilde O(\log \Delta / \epsilon^3 + \text{polylog}(1/\epsilon, \log \log n))$ randomized time and $\tilde O(\log^2 \Delta / \epsilon^4 + \log^*n / \epsilon)$ deterministic time. The second application is a faster algorithm for hypergraph maximal matching, a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of local graph algorithms. This gives an algorithm for $(2 \Delta - 1)$-edge-list coloring in $\tilde O(\log^2 \Delta \log n)$ rounds deterministically or $\tilde O( (\log \log n)^3 )$ rounds randomly. Another consequence (with additional optimizations) is an algorithm which generates an edge-orientation with out-degree at most $\lceil (1+\epsilon) \lambda \rceil$ for a graph of arboricity $\lambda$; for fixed $\epsilon$ this runs in $\tilde O(\log^6 n)$ rounds deterministically or $\tilde O(\log^3 n )$ rounds randomly

Polynomial Delay Algorithm for Listing Minimal Edge Dominating sets in Graphs

The Transversal problem, i.e, the enumeration of all the minimal transversals of a hypergraph in output-polynomial time, i.e, in time polynomial in its size and the cumulated size of all its minimal transversals, is a fifty years old open problem, and up to now there are few examples of hypergraph classes where the problem is solved. A minimal dominating set in a graph is a subset of its vertex set that has a non empty intersection with the closed neighborhood of every vertex. It is proved in [M. M. Kant\'e, V. Limouzy, A. Mary, L. Nourine, On the Enumeration of Minimal Dominating Sets and Related Notions, In Revision 2014] that the enumeration of minimal dominating sets in graphs and the enumeration of minimal transversals in hypergraphs are two equivalent problems. Hoping this equivalence can help to get new insights in the Transversal problem, it is natural to look inside graph classes. It is proved independently and with different techniques in [Golovach et al. - ICALP 2013] and [Kant\'e et al. - ISAAC 2012] that minimal edge dominating sets in graphs (i.e, minimal dominating sets in line graphs) can be enumerated in incremental output-polynomial time. We provide the first polynomial delay and polynomial space algorithm that lists all the minimal edge dominating sets in graphs, answering an open problem of [Golovach et al. - ICALP 2013]. Besides the result, we hope the used techniques that are a mix of a modification of the well-known Berge's algorithm and a strong use of the structure of line graphs, are of great interest and could be used to get new output-polynomial time algorithms.Comment: proofs simplified from previous version, 12 pages, 2 figure

A Polynomial Delay Algorithm for Enumerating Minimal Dominating Sets in Chordal Graphs

An output-polynomial algorithm for the listing of minimal dominating sets in graphs is a challenging open problem and is known to be equivalent to the well-known Transversal problem which asks for an output-polynomial algorithm for listing the set of minimal hitting sets in hypergraphs. We give a polynomial delay algorithm to list the set of minimal dominating sets in chordal graphs, an important and well-studied graph class where such an algorithm was open for a while.Comment: 13 pages, 1 figure, submitte

Overlap properties of geometric expanders

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of the simplices induced by the images of its hyperedges. In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence $\{H_n\}_{n=1}^\infty$ of arbitrarily large $(d+1)$-uniform hypergraphs with bounded degree, for which $\inf_{n\ge 1} c(H_n)>0$. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of $(d+1)$-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant $c=c(d)$. We also show that, for every $d$, the best value of the constant $c=c(d)$ that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete $(d+1)$-uniform hypergraphs with $n$ vertices, as $n\rightarrow\infty$. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any $d$ and any $\epsilon>0$, there exists $K=K(\epsilon,d)\ge d+1$ satisfying the following condition. For any $k\ge K$, for any point $q \in \mathbb{R}^d$ and for any finite Borel measure $\mu$ on $\mathbb{R}^d$ with respect to which every hyperplane has measure $0$, there is a partition $\mathbb{R}^d=A_1 \cup \ldots \cup A_{k}$ into $k$ measurable parts of equal measure such that all but at most an $\epsilon$-fraction of the $(d+1)$-tuples $A_{i_1},\ldots,A_{i_{d+1}}$ have the property that either all simplices with one vertex in each $A_{i_j}$ contain $q$ or none of these simplices contain $q$

Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure

Essential edges in Poisson random hypergraphs

Consider a random hypergraph on a set of N vertices in which, for k between 1 and N, a Poisson(N beta_k) number of hyperedges is scattered randomly over all subsets of size k. We collapse the hypergraph by running the following algorithm to exhaustion: pick a vertex having a 1-edge and remove it; collapse the hyperedges over that vertex onto their remaining vertices; repeat until there are no 1-edges left. We call the vertices removed in this process "identifiable". Also any hyperedge all of whose vertices are removed is called "identifiable". We say that a hyperedge is "essential" if its removal prior to collapse would have reduced the number of identifiable vertices. The limiting proportions, as N tends to infinity, of identifiable vertices and hyperedges were obtained by Darling and Norris. In this paper, we establish the limiting proportion of essential hyperedges. We also discuss, in the case of a random graph, the relation of essential edges to the 2-core of the graph, the maximal sub-graph with minimal vertex degree 2.Comment: 12 pages, 3 figures. Revised version with minor corrections/clarifications and slightly expanded introductio