On the complexity of strongly connected components in directed hypergraphs

We study the complexity of some algorithmic problems on directed hypergraphs and their strongly connected components (SCCs). The main contribution is an almost linear time algorithm computing the terminal strongly connected components (i.e. SCCs which do not reach any components but themselves). "Almost linear" here means that the complexity of the algorithm is linear in the size of the hypergraph up to a factor alpha(n), where alpha is the inverse of Ackermann function, and n is the number of vertices. Our motivation to study this problem arises from a recent application of directed hypergraphs to computational tropical geometry. We also discuss the problem of computing all SCCs. We establish a superlinear lower bound on the size of the transitive reduction of the reachability relation in directed hypergraphs, showing that it is combinatorially more complex than in directed graphs. Besides, we prove a linear time reduction from the well-studied problem of finding all minimal sets among a given family to the problem of computing the SCCs. Only subquadratic time algorithms are known for the former problem. These results strongly suggest that the problem of computing the SCCs is harder in directed hypergraphs than in directed graphs.Comment: v1: 32 pages, 7 figures; v2: revised version, 34 pages, 7 figure

Kernels for Below-Upper-Bound Parameterizations of the Hitting Set and Directed Dominating Set Problems

In the {\sc Hitting Set} problem, we are given a collection $\cal F$ of subsets of a ground set $V$ and an integer $p$, and asked whether $V$ has a $p$-element subset that intersects each set in $\cal F$. We consider two parameterizations of {\sc Hitting Set} below tight upper bounds: $p=m-k$ and $p=n-k$. In both cases $k$ is the parameter. We prove that the first parameterization is fixed-parameter tractable, but has no polynomial kernel unless coNP$\subseteq$NP/poly. The second parameterization is W[1]-complete, but the introduction of an additional parameter, the degeneracy of the hypergraph $H=(V,{\cal F})$, makes the problem not only fixed-parameter tractable, but also one with a linear kernel. Here the degeneracy of $H=(V,{\cal F})$ is the minimum integer $d$ such that for each $X\subset V$ the hypergraph with vertex set $V\setminus X$ and edge set containing all edges of $\cal F$ without vertices in $X$, has a vertex of degree at most $d.$ In {\sc Nonblocker} ({\sc Directed Nonblocker}), we are given an undirected graph (a directed graph) $G$ on $n$ vertices and an integer $k$, and asked whether $G$ has a set $X$ of $n-k$ vertices such that for each vertex $y\not\in X$ there is an edge (arc) from a vertex in $X$ to $y$. {\sc Nonblocker} can be viewed as a special case of {\sc Directed Nonblocker} (replace an undirected graph by a symmetric digraph). Dehne et al. (Proc. SOFSEM 2006) proved that {\sc Nonblocker} has a linear-order kernel. We obtain a linear-order kernel for {\sc Directed Nonblocker}

Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio

Solving the Canonical Representation and Star System Problems for Proper Circular-Arc Graphs in Log-Space

We present a logspace algorithm that constructs a canonical intersection model for a given proper circular-arc graph, where `canonical' means that models of isomorphic graphs are equal. This implies that the recognition and the isomorphism problems for this class of graphs are solvable in logspace. For a broader class of concave-round graphs, that still possess (not necessarily proper) circular-arc models, we show that those can also be constructed canonically in logspace. As a building block for these results, we show how to compute canonical models of circular-arc hypergraphs in logspace, which are also known as matrices with the circular-ones property. Finally, we consider the search version of the Star System Problem that consists in reconstructing a graph from its closed neighborhood hypergraph. We solve it in logspace for the classes of proper circular-arc, concave-round, and co-convex graphs.Comment: 19 pages, 3 figures, major revisio

Hypernetworks for reconstructing the dynamics of multilevel systems

Networks are fundamental for reconstructing the dynamics of many systems, but have the drawback that they are restricted to binary relations. Hypergraphs extend relational structure to multi-vertex edges, but are essentially set-theoretic and unable to represent essential structural properties. Hypernetworks are a natural multidimensional generalisation of networks, representing n-ary relations by simplices with n vertices. The assembly of vertices to make simplices is key for moving between levels in multilevel systems, and integrating dynamics between levels. It is argued that hypernetworks are necessary, if not sufficient, for reconstructing the dynamics of multilevel complex systems