Gomory-Hu Tree Research Articles

Approximate Gomory–Hu Tree is Faster than \(\boldsymbol{n}\,\boldsymbol{-\, 1}\) Maximum Flows

Approximate Gomory–Hu Tree is Faster than \(\boldsymbol{n}\,\boldsymbol{-\, 1}\) Maximum Flows

Generalized cut trees for edge-connectivity

We present three cut trees of graphs, each of them giving insights into the edge-connectivity structure. All three cut trees have in common that they are defined with respect to a given binary symmetric relation R on the vertex set of the graph, which generalizes Gomory-Hu trees. Applying these cut trees, we prove the following:•A pair of vertices {v,w} of a graph G is pendant if λ(v,w)=min⁡{d(v),d(w)}. Mader showed in 1974 that every simple graph with minimum degree δ contains at least δ(δ+1)/2 pendant pairs. We improve this lower bound to δn/24 for every simple graph G on n vertices with δ≥5 or λ≥4 or vertex connectivity κ≥3, and show that this is optimal up to a constant factor with regard to every parameter.•Every simple graph G satisfying δ>0 has O(n/δ)δ-edge-connected components. Moreover, every simple graph G that satisfies 0<λ<δ has O((n/δ)2) cuts of size less than min⁡{32λ,δ}, and O((n/δ)⌊2α⌋) cuts of size at most min⁡{α⋅λ,δ−1} for any given real number α≥1.•A cut is trivial if it or its complement in V(G) is a singleton. We provide an alternative proof of the following recent result of Lo et al.: Given a simple graph G on n vertices that satisfies δ>0, we can compute vertex subsets of G in near-linear time such that contracting these vertex subsets separately preserves all non-trivial min-cuts of G and leaves a graph having O(n/δ) vertices and O(n) edges.

P-Edge/vertex-connected vertex cover: Parameterized and approximation algorithms

We introduce and study two natural generalizations of the Connected Vertex Cover (VC) problem: the p-Edge-Connected and p-Vertex-Connected VC problem (where p≥2 is a fixed integer). We obtain an 2O(pk)nO(1)-time algorithm for p-Edge-Connected VC and an 2O(k2)nO(1)-time algorithm for p-Vertex-Connected VC. Thus, like Connected VC, both constrained VC problems are FPT. Furthermore, like Connected VC, neither problem admits a polynomial kernel unless NP ⊆ coNP/poly, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. Finally, we describe a 2(p+1)-approximation algorithm for the p-Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning p-vertex/edge-connected subgraphs of a p-vertex/edge-connected graph by Nishizeki and Poljak (1994) [30] and Nagamochi and Ibaraki (1992) [27].

Identifying Critical Isolation Valves in a Water Distribution Network: A Socio-Technical Approach

Isolation valves are critical for the reliable functioning of water distribution networks (WDNs). However, it is challenging for utilities to prioritize valve rehabilitation and replacement given it is often unclear if certain valves are operable in a given WDN. This study uses the Gomory–Hu tree of the segment-valve representation (or dual representation) of WDNs to obtain the logical implications of inoperable valves (i.e., which segments should be isolated and merged unnecessarily due to valve inoperability). Multi-objective optimization is then used to identify the critical valves based on selected attributes (e.g., social vulnerability, flow volume) of segments that would be unnecessarily isolated as a result. This study developed three multi-objective formulations: first, deterministic; second, accounting for uncertainty; and third, accounting for both uncertainty and the likelihood of failure of pipes within segments. Identified critical valves are compared between the three developed formulations and a method considering only a single objective. Results demonstrated that multi-objective optimization provided additional information which can be used to discern valve importance for utilities in comparison with using a single objective. Further, though there was overlap between the results from the three formulations, the third formulation provided the most insight without overwhelming decision-makers with a large number of identified valves.

Gomory-Hu Trees Over Wireless

The Gomory-Hu tree is a popular optimization algorithm that enables to efficiently find a min-cut (or equivalently, max-flow) for every pair of nodes in a graph. However, graphs cannot capture broadcasting and interference in wireless: over wireless networks we need to resort to the information-theoretical cut-set to bound the max-flow. Leveraging the submodularity of mutual information, we show that the Gomory-Hu algorithm can be used to efficiently find information-theoretic rate characterizations such as the capacity or an approximation to the capacity, over a number of network scenarios including wireless Gaussian networks and deterministic relay networks.

Measuring Network Robustness by Average Network Flow

Infrastructure networks such as the Internet backbone and power grids are essential for our everyday lives. With the prevalence of cyber-attacks on them, measuring their robustness has become an important issue. To date, many robustness metrics have been proposed. It is desirable for a robustness metric to possess the following three properties: considering global network topologies, strictly increasing upon link additions, and having a quadratic complexity in terms of the number of nodes on sparse networks. This paper proposes to use Average Network Flow (ANF) with unit edge capacity as a robustness metric, and shows that it satisfies these three properties. Especially, an algorithm by leveraging Gomory-Hu trees is proposed to compute ANF. The Gomory-Hu tree for a network embeds the all-pairs maximum flows of this network into its edges, thus enabling our algorithm to achieve the quadratic complexity. Moreover, this paper compares ANF with seven existing representative metrics. By experimenting on the scenario in which network topologies preserve the numbers of nodes and links, several new phenomena of robustness metrics are reported.

Vulnerability analysis of cut-capacity structure and OD demand using Gomory-Hu tree method

Vulnerability analysis of transportation networks has rapidly become important in recent decades given the increasing numbers of transportation disasters. In this paper, we describe a method for calculating the minimum cuts between all pairs of nodes in a transportation network and develop two indices for analyzing vulnerability; these are derived from topology-based vulnerability/demand-accountable analyses and enable evaluation of cuts without assuming route-choice behaviors. We show that such analyses can be performed using a Gomory–Hu tree to reduce the computational load. This method is efficient, requiring only N − 1 calculations of the maximum flow problem even though the number of node pairs is N2, where N is the number of nodes. In addition, we show that the total demand passing each minimum cut, which is necessary for demand-accountable analysis, can be obtained by network loading onto the tree using the Gomory–Hu tree features. We apply the proposed method to the central region of Japan to illustrate the applicability of the two indices for identifying vulnerable links in the road network.

Cut Star of an Undirected Graph

Suppose that [Formula: see text] is an undirected graph. An ordered pair [Formula: see text] of the vertices of the graph [Formula: see text] is called a pendant pair for the graph if [Formula: see text] is a minimum cut separating [Formula: see text] and [Formula: see text] Stoer and Wagner obtained a global minimum cut of [Formula: see text] by using pendant pairs of [Formula: see text] and its contractions. A Gomory Hu tree of the graph [Formula: see text] is a very useful data structure which gives us all the minimum s-t cuts of [Formula: see text] for every pair of distinct vertices [Formula: see text] and [Formula: see text] In this paper, we construct a new type of tree for the graph [Formula: see text] called cut star, by using pendant pairs of [Formula: see text] and its contractions. A cut star of the graph [Formula: see text] is constructed more quickly than a Gomory Hu tree of [Formula: see text] We characterize a class of graphs for which a cut star of a graph of this class is also a Gomory Hu tree.

Symmetric submodular system: Contractions and Gomory-Hu tree

Let S=(V,f), be a symmetric submodular system. For two distinct elements s and l of V, let Γ(s,l) denote the set of all subsets of V which separate s from l. By using any Gomory-Hu tree of S we can obtain an element of Γ(s,l) which has minimum value among all the elements of Γ(s,l). This tree can be constructed iteratively by solving |V|−1 minimum sl−separator problem. Pendant pairs of a symmetric submodular system play a key role in finding a minimizer of this system. In this paper, we obtain a Gomory-Hu tree of a contraction of S with respect to some subsets of V only by using contraction in a Gomory-Hu tree of S. Furthermore, we obtain some pendant pairs of S and of its contractions by using a Gomory-Hu tree of S.

Efficient and Optimal Service Component Distribution across Multiple Clouds

&lt;span class="fontstyle0"&gt;Cloud computing technology is one of the key considerations for business willing to access to different cloud services over the Internet and to benefit from the diversity of IaaS offers and pricing models. Although several solutions are available in the market, there are still some issues to solve. The main important aspect to address is the user’s request complexity, the vendor lock-in risk and the SLA fulfillment. In this paper, we propose a Multi-Cloud Broker called MCB that allows an efficient and optimal service component distribution among different clouds in flexible and dynamic infrastructure provisioning environment, in order to achieve better Quality of Service and cost efficiency. The request partitioning is the main step of our approach, this step is performed using Gomory-Hu tree based algorithm. Our simulation results show how our algorithm is better than existing partitioning algorithms in terms of running time.&lt;/span&gt;

Parallel cut tree algorithms

A cut tree is a combinatorial structure that represents the edge-connectivity between all pairs of vertices of an undirected graph. Cut trees solve the all pairs minimum s–t-cut problem efficiently. Cut trees have a large number of applications including the solution of important combinatorial problems in fields such as graph clustering and graph connectivity. They have also been applied to scheduling problems, social network analysis, biological data analysis, among others. Two sequential algorithms to compute a cut tree of a capacitated undirected graph are well known: the Gomory–Hu algorithm and the Gusfield algorithm. In this work three parallel cut tree algorithms are presented, including parallel versions of Gusfield and Gomory–Hu algorithms. A hybrid algorithm that combines techniques from both algorithms is proposed which provides a more robust performance for arbitrary instances. Experimental results show that the three algorithms achieve significant speedups on real and synthetic graphs. We discuss the trade-offs between the alternatives, each of which presents better results given the characteristics of the input graph. On several instances the hybrid algorithm outperformed both other algorithms, being faster than the parallel Gomory–Hu algorithm on most instances.

Min st -Cut Oracle for Planar Graphs with Near-Linear Preprocessing Time

For an undirected n -vertex planar graph G with nonnegative edge weights, we consider the following type of query: given two vertices s and t in G , what is the weight of a min st -cut in G ? We show how to answer such queries in constant time with O ( n log 4 n ) preprocessing time and O ( n log n ) space. We use a Gomory-Hu tree to represent all the pairwise min cuts implicitly. Previously, no subquadratic time algorithm was known for this problem. Since all-pairs min cut and the minimum-cycle basis are dual problems in planar graphs, we also obtain an implicit representation of a minimum-cycle basis in O ( n log 4 n ) time and O ( n log n ) space. Additionally, an explicit representation can be obtained in O ( C ) time and space where C is the size of the basis. These results require that shortest paths are unique. This can be guaranteed either by using randomization without overhead or deterministically with an additional log 2 n factor in the preprocessing times.

Retrieve Main Content using Vision-base Web Page Segmentation with Gomory-Hu Tree

The world wide (www) serves a huge, widely distributed global information services. A huge amount of data have been accumulated and stored on the web. The information on Web is usually presented via Hypertext Markup Language (HTML) to make its perception easier for humans. Web pages usually contain various contents, which are relevant or irrelevant to the main topic. Irrelevant contents are called noise. A web page usually contains the number of noise which is not related to the main information of the page such as navigation bar, advertisements, and related articles and so on. Noise on the web pages tends to problem mining the main content of these pages. This paper is proposed wed page segmentation using Gomory-Hu tree based Vision-based Page Segmentation (VIPS) algorithm. General Terms Web Content Mining

ON HALF-INTEGRALITY OF NETWORK SYNTHESIS PROBLEM

Network synthesis problem is the problem of constructing a minimum cost network satisfying a given ow-requirement. A classical result of Gomory and Hu is that if the cost is uniform and the ow requirement is integer-valued, then there exists a half-integral optimal solution. They also gave a simple algorithm to nd a half-integral optimal solution. In this paper, we show that this half-integrality and the Gomory-Hu algorithm can be extended to a class of fractional cut-covering problems dened by skew-supermodular functions. Application to approximation algorithm is also given.

Optimizing compatible sets in wireless networks through integer programming

In wireless networks, the notion of compatible set refers to a set of radio links that can be simultaneously active with a tolerable interference. Finding a compatible set with maximum weighted revenue from the parallel transmissions is an important optimization subproblem for resource management in wireless networks. For this subproblem, two basic ways of expressing the signal-to-interference-plus-noise-ratio within integer programming are used, differing in the number of variables and the quality of the upper bound given by their linear relaxations. To our knowledge, there is no systematic study comparing the effectiveness of these two approaches to the compatible set optimization problem. The contribution of the article is twofold. First, we present a comparison of the two basic approaches, and, second, we introduce matching inequalities that improve the upper bounds achievable with the two basic approaches. The matching inequalities are generated within the branch-and-cut process using a minimum odd-cut generation procedure based on the Gomory–Hu tree algorithm. The article presents results of a numerical study illustrating our statements and findings.

Integer programming models for maximizing parallel transmissions in wireless networks

In radio communications, a set of links that can transmit in parallel with a tolerable interference is called a compatible set. Finding a compatible set with maximum weighted revenue of the parallel transmissions is an important subproblem in wireless network management. For the subproblem, there are two basic approaches to express the signal to interference plus noise ratio (SINR) within integer programming, differing in the number of variables and the quality of the upper bound given by linear relaxations. To our knowledge, there is no systematic study comparing the effectiveness of the two approaches. The contribution of the paper is two-fold. Firstly, we present such a comparison, and, secondly, we introduce matching inequalities improving the upper bounds as compared to the two basic approaches. The matching inequalities are generated within a branch-and-cut algorithm using a minimum odd-cut procedure based on the Gomory-Hu algorithm. The paper presents results of extensive numerical studies illustrating our statements and findings.

Properties of Gomory–Hu co-cycle bases

Let D = ( V , A ) be a directed graph with non-negative arc weights. We study the problem of computing certain special co-cycle bases of D , in particular, a minimum weight weakly fundamental co-cycle basis. A co-cycle in D corresponds to a cut in the underlying undirected graph and a { − 1 , 0 , 1 } arc incidence vector is associated with each co-cycle, where the ± 1 coordinates are used for the arcs crossing the cut. The weight of a co-cycle C is the sum of the weights of those arcs a such that C ( a ) = ± 1 . The vector space over Q generated by the arc incidence vectors of the co-cycles is the co-cycle space of D . The co-cycle space of D can also be defined as the orthogonal complement of the cycle space of D . A set of linearly independent co-cycles that span the co-cycle space is a co-cycle basis of D . The problem of computing a co-cycle basis { C 1 , … , C k } such that the sum of weights of the co-cycles in the basis is the least possible is the minimum co-cycle basis problem. A co-cycle basis { C 1 , … , C k } is weakly fundamental if for every i there is an arc a i such that C i ( a i ) = ± 1 while C j ( a i ) = 0 for j > i . The minimum cycle basis problem in directed and undirected graphs is a well-studied problem and while polynomial time algorithms are known for these problems, the problem of computing a minimum weight weakly fundamental cycle basis has recently been shown to be APX-hard. We show that the co-cycle basis corresponding to the cuts of a Gomory–Hu tree T of the underlying undirected graph of D is a minimum weight weakly fundamental co-cycle basis of D . This is, in fact, a minimum co-cycle basis of D and it is also totally unimodular. Thus this is a special co-cycle basis that simultaneously answers several questions in the domain of co-cycle bases. It is known that there is no such special cycle basis for general graphs.

Segmenting Webpage with Gomory-Hu Tree Based Clustering

We propose a novel web page segmentationalgorithm based on finding the Gomory-Hu tree in a planargraph. The algorithm firstly distills vision and structureinformation from a web page to construct a weightedundirected graph, whose vertices are the leaf nodes of theDOM tree and the edges represent the visible positionrelationship between vertices. Then it partitions the graphwith the Gomory-Hu tree based clustering algorithm.Experimental results show that, compared with VIPS andChakrabarti et al.’s graph theoretic algorithm, ouralgorithm improves upon the other two with much higherprecision and recall, and its running time is far lower thanthat of Chakrabarti et al.’s graph theoretic algorithm.

Effective implementation of image region segmentation based on Gomory-Hu algorithm

A graph-based image segmentation method was presented.Firstly,use region growing technique to find initial over-segmentation.Secondly,use these regions as nodes to create the graph.Finally,use minimum cut method to merge these regions.The proposed method has two advantages:one is using minimum cut criterion to merge regions,which can contain global information;the other is using regions as nodes to create the graph,which can greatly reduce nodes of the graph.The experimental results show the effectiveness of the approach.

Optimization in telecommunication networks

Network design and network synthesis have been the classical optimization problems in telecommunication for a long time. In the recent past, there have been many technological developments such as digitization of information, optical networks, internet, and wireless networks. These developments have led to a series of new optimization problems. This manuscript gives an overview of the developments in solving both classical and modern telecom optimization problems. The classical (still actual) network design and synthesis problems are described with an emphasis on the latest developments on modelling and solving them. Mathematical theorems will be related to the models described. This includes menger's disjoint paths theorem, the ford–fulkerson max-flow-min-cut theorem, and also gomory–hu trees and the okamura-seymour cut-condition finally, we describe recent optimization problems such as routing, wavelength assignment, and grooming in optical networks.