The Weak Hyperedge Tenacity of the Hypercycles

Graphs play an important role in our daily life. For example, the urban transport network can be represented by a graph, as the intersections are the vertices and the streets are the edges of the graph. Suppose that some edges of the graph are removed, the question arises how damaged the graph is. There are some criteria for measuring the vulnerability of graph; the tenacity is the best criteria for measuring it. Since the hypergraph generalize the standard graph by defining any edge between multiple vertices instead of only two vertices, the above question is about the hypergraph. When a hyperedge is omitted from hypergraph, we have two kinds of deletion: strong deletion and weak deletion. Weak hyperedge deletion just deletes the connection between the vertices in the hyperedge and the vertices became in the hypergraph. In this paper, we obtain the tenacity of hypercycles by weak hyperedge deletion

Random Contractions and Sampling for Hypergraph and Hedge Connectivity

We initiate the study of hedge connectivity of undirected graphs, motivated by dependent edge failures in real-world networks. In this model, edges are partitioned into groups called hedges that fail together. The hedge connectivity of a graph is the minimum number of hedges whose removal disconnects the graph. We give a polynomial-time approximation scheme and a quasi-polynomial exact algorithm for hedge connectivity. This provides strong evidence that the hedge connectivity problem is tractable, which contrasts with prior work that established the intractability of the corresponding s−t min-cut problem. Our techniques also yield new combinatorial and algorithmic results in hypergraph connectivity. Next, we study the behavior of hedge graphs under uniform random sampling of hedges. We show that unlike graphs, all cuts in the sample do not converge to their expected value in hedge graphs. Nevertheless, the min-cut of the sample does indeed concentrate around the expected value of the original min-cut. This leads to a sharp threshold on hedge survival probabilities for graph disconnection. To the best of our knowledge, this is the first network reliability analysis under dependent edge failures

Cactus Representations in Polylogarithmic Max-flow via Maximal Isolating Mincuts

A cactus representation of a graph, introduced by Dinitz et al. in 1976, is an edge sparsifier of $O(n)$ size that exactly captures all global minimum cuts of the graph. It is a central combinatorial object that has been a key ingredient in almost all algorithms for the connectivity augmentation problems and for maintaining minimum cuts under edge insertions (e.g. [NGM97], [CKL+22], [Hen97]). This sparsifier was generalized to Steiner cactus for a vertex set $T$, which can be seen as a vertex sparsifier of $O(|T|)$ size that captures all partitions of $T$ corresponding to a $T$-Steiner minimum cut, and also hypercactus, an analogous concept in hypergraphs. These generalizations further extend the applications of cactus to the Steiner and hypergraph settings. In a long line of work on fast constructions of cactus and its generalizations, a near-linear time construction of cactus was shown by [Karger and Panigrahi 2009]. Unfortunately, their technique based on tree packing inherently does not generalize. The state-of-the-art algorithms for Steiner cactus and hypercactus are still slower than linear time by a factor of $\Omega(|T|)$ [DV94] and $\Omega(n)$ [CX17], respectively. We show how to construct both Steiner cactus and hypercactus using polylogarithmic calls to max flow, which gives the first almost-linear time algorithms of both problems. The constructions immediately imply almost-linear-time connectivity augmentation algorithms in the Steiner and hypergraph settings, as well as speed up the incremental algorithm for maintaining minimum cuts in hypergraphs by a factor of $n$. The key technique behind our result is a novel variant of the influential isolating mincut technique [LP20, AKL+21] which we called maximal isolating mincuts. This technique makes the isolating mincuts to be "more balanced" which, we believe, will likely be useful in future applications.Comment: To appear in SODA 202

Cuts and connectivity in graphs and hypergraphs

In this thesis, we consider cut and connectivity problems on graphs, digraphs, hypergraphs and hedgegraphs. The main results are the following: - We introduce a faster algorithm for finding the reduced graph in element-connectivity computations. We also show its application to node separation. - We present several results on hypergraph cuts, including (a) a near linear time algorithm for finding a (2+epsilon)-approximate min-cut, (b) an algorithm to find a representation of all min-cuts in the same time as finding a single min-cut, (c) a sparse subgraph that preserves connectivity for hypergraphs and (d) a near linear-time hypergraph cut sparsifier. - We design the first randomized polynomial time algorithm for the hypergraph k-cut problem whose complexity has been open for over 20 years. The algorithm generalizes to hedgegraphs with constant span. - We address the complexity gap between global vs. fixed-terminal cuts problems in digraphs by presenting a 2-1/448 approximation algorithm for the global bicut problem