Flow trees for vertex-capacitated networks

AbstractGiven a graph G=(V,E) with a cost function c(S)⩾0∀S⊆V, we want to represent all possible min-cut values between pairs of vertices i and j. We consider also the special case with an additive cost c where there are vertex capacities c(v)⩾0 ∀v∈V, and for a subset S⊆V, c(S)=∑v∈Sc(v). We consider two variants of cuts: in the first one, separation, {i} and {j} are feasible cuts that disconnect i and j. In the second variant, vertex-cut, a cut-set that disconnects i from j does not include i or j. We consider both variants for undirected and directed graphs. We prove that there is a flow-tree for separations in undirected graphs. We also show that a compact representation does not exist for vertex-cuts in undirected graphs, even with additive costs. For directed graphs, a compact representation of the cut-values does not exist even with additive costs, for neither the separation nor the vertex-cut cases

Tight Bounds for Gomory-Hu-like Cut Counting

By a classical result of Gomory and Hu (1961), in every edge-weighted graph $G=(V,E,w)$, the minimum $st$-cut values, when ranging over all $s,t\in V$, take at most $|V|-1$ distinct values. That is, these $\binom{|V|}{2}$ instances exhibit redundancy factor $\Omega(|V|)$. They further showed how to construct from $G$ a tree $(V,E',w')$ that stores all minimum $st$-cut values. Motivated by this result, we obtain tight bounds for the redundancy factor of several generalizations of the minimum $st$-cut problem. 1. Group-Cut: Consider the minimum $(A,B)$-cut, ranging over all subsets $A,B\subseteq V$ of given sizes $|A|=\alpha$ and $|B|=\beta$. The redundancy factor is $\Omega_{\alpha,\beta}(|V|)$. 2. Multiway-Cut: Consider the minimum cut separating every two vertices of $S\subseteq V$, ranging over all subsets of a given size $|S|=k$. The redundancy factor is $\Omega_{k}(|V|)$. 3. Multicut: Consider the minimum cut separating every demand-pair in $D\subseteq V\times V$, ranging over collections of $|D|=k$ demand pairs. The redundancy factor is $\Omega_{k}(|V|^k)$. This result is a bit surprising, as the redundancy factor is much larger than in the first two problems. A natural application of these bounds is to construct small data structures that stores all relevant cut values, like the Gomory-Hu tree. We initiate this direction by giving some upper and lower bounds.Comment: This version contains additional references to previous work (which have some overlap with our results), see Bibliographic Update 1.

Efficient algorithm for computing all low s-t edge connectivities in directed graphs

LNCS v. 9235 entitled: Mathematical Foundations of Computer Science 2015: 40th International Symposium, MFCS 2015, Milan, Italy, August 24-28, 2015, Proceedings, Part 2Given a directed graph with n nodes and m edges, the (strong) edge connectivity λ (u; v) between two nodes u and v is the minimum number of edges whose deletion makes u and v not strongly connected. The problem of computing the edge connectivities between all pairs of nodes of a directed graph can be done in O(m ω) time by Cheung, Lau and Leung (FOCS 2011), where ω is the matrix multiplication factor (≈ 2:373), or in Õ (mn1:5) time using O(n) computations of max-flows by Cheng and Hu (IPCO 1990). We consider in this paper the “low edge connectivity” problem, which aims at computing the edge connectivities for the pairs of nodes (u; v) such that λ (u; v) ≤ k. While the undirected version of this problem was considered by Hariharan, Kavitha and Panigrahi (SODA 2007), who presented an algorithm with expected running time Õ (m+nk3), no algorithm better than computing all-pairs edge connectivities was proposed for directed graphs. We provide an algorithm that computes all low edge connectivities in O(kmn) time, improving the previous best result of O (min(m ω, mn1:5)) when k ≤ √ n. Our algorithm also computes a minimum u-v cut for each pair of nodes (u; v) with λ (u; v) ≤ k.postprin

The Structure of Minimum Vertex Cuts

In this paper we continue a long line of work on representing the cut structure of graphs. We classify the types of minimum vertex cuts, and the possible relationships between multiple minimum vertex cuts. As a consequence of these investigations, we exhibit a simple O(? n)-space data structure that can quickly answer pairwise (?+1)-connectivity queries in a ?-connected graph. We also show how to compute the "closest" ?-cut to every vertex in near linear O?(m+poly(?)n) time

Single Source - All Sinks Max Flows in Planar Digraphs

Let G = (V,E) be a planar n-vertex digraph. Consider the problem of computing max st-flow values in G from a fixed source s to all sinks t in V\{s}. We show how to solve this problem in near-linear O(n log^3 n) time. Previously, no better solution was known than running a single-source single-sink max flow algorithm n-1 times, giving a total time bound of O(n^2 log n) with the algorithm of Borradaile and Klein. An important implication is that all-pairs max st-flow values in G can be computed in near-quadratic time. This is close to optimal as the output size is Theta(n^2). We give a quadratic lower bound on the number of distinct max flow values and an Omega(n^3) lower bound for the total size of all min cut-sets. This distinguishes the problem from the undirected case where the number of distinct max flow values is O(n). Previous to our result, no algorithm which could solve the all-pairs max flow values problem faster than the time of Theta(n^2) max-flow computations for every planar digraph was known. This result is accompanied with a data structure that reports min cut-sets. For fixed s and all t, after O(n^{3/2} log^{3/2} n) preprocessing time, it can report the set of arcs C crossing a min st-cut in time roughly proportional to the size of C.Comment: 25 pages, 4 figures; extended abstract appeared in FOCS 201

Strong Connectivity in Directed Graphs under Failures, with Application

In this paper, we investigate some basic connectivity problems in directed graphs (digraphs). Let $G$ be a digraph with $m$ edges and $n$ vertices, and let $G\setminus e$ be the digraph obtained after deleting edge $e$ from $G$. As a first result, we show how to compute in $O(m+n)$ worst-case time: $(i)$ The total number of strongly connected components in $G\setminus e$, for all edges $e$ in $G$. $(ii)$ The size of the largest and of the smallest strongly connected components in $G\setminus e$, for all edges $e$ in $G$. Let $G$ be strongly connected. We say that edge $e$ separates two vertices $x$ and $y$, if $x$ and $y$ are no longer strongly connected in $G\setminus e$. As a second set of results, we show how to build in $O(m+n)$ time $O(n)$-space data structures that can answer in optimal time the following basic connectivity queries on digraphs: $(i)$ Report in $O(n)$ worst-case time all the strongly connected components of $G\setminus e$, for a query edge $e$. $(ii)$ Test whether an edge separates two query vertices in $O(1)$ worst-case time. $(iii)$ Report all edges that separate two query vertices in optimal worst-case time, i.e., in time $O(k)$, where $k$ is the number of separating edges. (For $k=0$, the time is $O(1)$). All of the above results extend to vertex failures. All our bounds are tight and are obtained with a common algorithmic framework, based on a novel compact representation of the decompositions induced by the $1$-connectivity (i.e., $1$-edge and $1$-vertex) cuts in digraphs, which might be of independent interest. With the help of our data structures we can design efficient algorithms for several other connectivity problems on digraphs and we can also obtain in linear time a strongly connected spanning subgraph of $G$ with $O(n)$ edges that maintains the $1$-connectivity cuts of $G$ and the decompositions induced by those cuts.Comment: An extended abstract of this work appeared in the SODA 201

Strong Connectivity in Directed Graphs under Failures, with Applications *

An extended abstract of this work appeared in the SODA '17: Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete AlgorithmsInternational audienceIn this paper, we investigate some basic connectivity problems in directed graphs (digraphs). Let G be a digraph with m edges and n vertices, and let G \ e (resp., G \ v) be the digraph obtained after deleting edge e (resp., vertex v) from G. As a first result, we show how to compute in O(m + n) worst-case time: • The total number of strongly connected components in G \ e (resp., G \ v), for all edges e (resp., for all vertices v) in G. • The size of the largest and of the smallest strongly connected components in G \ e (resp., G \ v), for all edges e (resp., for all vertices v) in G. Let G be strongly connected. We say that edge e (resp., vertex v) separates two vertices x and y, if x and y are no longer strongly connected in G \ e (resp., G \ v). As a second set of results, we show how to build in O(m + n) time O(n)-space data structures that can answer in optimal time the following basic connectivity queries on digraphs: • Report in O(n) worst-case time all the strongly connected components of G \ e (resp., G \ v), for a query edge e (resp., vertex v). • Test whether an edge or a vertex separates two query vertices in O(1) worst-case time. • Report all edges (resp., vertices) that separate two query vertices in optimal worst-case time, i.e., in time O(k), where k is the number of separating edges (resp., separating vertices). (For k = 0, the time is O(1)). All our bounds are tight and are obtained with a common algorithmic framework, based on a novel compact representation of the decompositions induced by the 1-connectivity (i.e., 1-edge and 1-vertex) cuts in digraphs, which might be of independent interest. With the help of our data structures we can design efficient algorithms for several other connectivity problems on digraphs and we can also obtain in linear time a strongly connected spanning subgraph of G with O(n) edges that maintains the 1-connectivity cuts of G and the decompositions induced by those cuts

Algoritmos paralelos para árvores de cortes e medidas de centralidade em grafos

Resumo: Uma árvore de cortes é uma representação compacta da aresta-conectividade de um grafo não orientado. As árvores de cortes resolvem de maneira eficiente o problema de calcular a arestaconectividade entre todos os pares de vértices do grafo. As árvores de cortes têm muitas aplicações como, por exemplo, no projeto de redes confiáveis, na partição de grafos, no agrupamento em grafos, na análise de redes sociais, dentre outras. Dois algoritmos para a construção de árvores de cortes de grafos não orientados e capacitados são bem conhecidos: o algoritmo de Gomory-Hu e o algoritmo de Gusfield. Este trabalho apresenta propostas de implementações paralelas de três algoritmos para encontrar uma árvore de cortes. Versões paralelas para os algoritmos de Gusfield e de Gomory-Hu são descritas e avaliadas experimentalmente. Um algoritmo híbrido que combina esses dois algoritmos e que busca tirar proveito das vantagens de cada um deles também é apresentado. Resultados experimentais mostram que os três algoritmos apresentam boas acelerações nos tempos de execução. Os experimentos também mostram que o algoritmo híbrido é quase sempre mais rápido do que o algoritmo de Gomory-Hu e em certas instâncias ele é muito mais rápido do que o algoritmo de Gusfield. Heurísticas para a melhoria do algoritmo de Gomory-Hu e do algoritmo híbrido são propostas e analisadas. Na segunda parte desta tese, são estudadas medidas de centralidade dos vértices de um grafo que são baseadas na conectividade - algumas delas podem ser calculadas a partir de árvores de cortes. As medidas de centralidade de vértices têm como objetivo quantificar a importância dos vértices de um grafo com base em diferentes critérios. Dentre as medidas de centralidade propostas, destaca-se a i-aresta-conectividade, que mede a aresta-conectividade dos vértices em relação ao grafo. Uma medida de conectividade baseada em cortes de vértices também é proposta. Um estudo experimental com as medidas de conectividade foi executado para avaliar a relação das medidas propostas com outras medidas de centralidade mais conhecidas. Esse estudo mostra empiricamente que vértices com alta conectividade tendem a ter baixa excentricidade. Além disso, experimentos mostram que as medidas de conectividade não são equivalentes ao grau como critério de ordenação dos vértices

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