Showing papers on "Minimum cut published in 2016"

Distributed algorithms for planar networks II: low-congestion shortcuts, MST, and Min-Cut

Abstract: This paper introduces the concept of low-congestion shortcuts for (near-)planar networks, and demonstrates their power by using them to obtain near-optimal distributed algorithms for problems such as Minimum Spanning Tree (MST) or Minimum Cut, in planar networks.Consider a graph G = (V, E) and a partitioning of V into subsets of nodes S1, . . ., SN, each inducing a connected subgraph G[Si]. We define an α-congestion shortcut with dilation β to be a set of subgraphs H1, . . ., HN ⊆ G, one for each subset Si, such that1. For each i ∈ [1, N], the diameter of the subgraph G[Si] + Hi is at most β.2. For each edge e ∈ E, the number of subgraphs G[Si] + Hi containing e is at most α.We prove that any partition of a D-diameter planar graph into individually-connected parts admits an O(D log D)-congestion shortcut with dilation O(D log D), and we also present a distributed construction of it in O(D) rounds. We moreover prove these parameters to be near-optimal; i.e., there are instances in which, unavoidably, max{α, β} = Ω(D[EQUATION]).Finally, we use low-congestion shortcuts, and their efficient distributed construction, to derive O(D)-round distributed algorithms for MST and Min-Cut, in planar networks. This complexity nearly matches the trivial lower bound of Ω(D). We remark that this is the first result bypassing the well-known Ω(D + [EQUATION]) existential lower bound of general graphs (see Peleg and Rubinovich [FOCS'99]; Elkin [STOC'04]; and Das Sarma et al. [STOC'11]) in a family of graphs of interest.

On Sketching Quadratic Forms

Abstract: We undertake a systematic study of sketching a quadratic form: given an n x n matrix A, create a succinct sketch sk(A) which can produce (without further access to A) a multiplicative (1+e)-approximation to xT A x for any desired query x ∈ Rn. While a general matrix does not admit non-trivial sketches, positive semi-definite (PSD) matrices admit sketches of size θ(e{-2n), via the Johnson-Lindenstrauss lemma, achieving the "for each" guarantee, namely, for each query x, with a constant probability the sketch succeeds. (For the stronger "for all" guarantee, where the sketch succeeds for all x's simultaneously, again there are no non-trivial sketches.)We design significantly better sketches for the important subclass of graph Laplacian matrices, which we also extend to symmetric diagonally dominant matrices. A sequence of work culminating in that of Batson, Spielman, and Srivastava (SIAM Review, 2014), shows that by choosing and reweighting O(e{-2n) edges in a graph, one achieves the "for all" guarantee. Our main results advance this front. For the "for all" guarantee, we prove that Batson et al.'s bound is optimal even when we restrict to "cut queries" x ∈ (0,1}n. Specifically, an arbitrary sketch that can (1+e)-estimate the weight of all cuts (S, bar S) in an n-vertex graph must be of size Ω(e{-2n) bits. Furthermore, if the sketch is a cut-sparsifier (i.e., itself a weighted graph and the estimate is the weight of the corresponding cut in this graph), then the sketch must have Ω(e{-2n) edges.In contrast, previous lower bounds showed the bound only for spectral-sparsifiers.For the "for each" guarantee, we design a sketch of size O(e{-1n) bits for "cut queries" x ∈{0,1}n. We apply this sketch to design an algorithm for the distributed minimum cut problem. We prove a nearly-matching lower bound of Ω(e{-1n) bits. For general queries x ∈ Rn, we construct sketches of size O(e{-1.6n) bits.Our results provide the first separation between the sketch size needed for the "for all" and "for each" guarantees for Laplacian matrices.

Automatic Forest Mapping at Individual Tree Levels from Terrestrial Laser Scanning Point Clouds with a Hierarchical Minimum Cut Method

Abstract: Laser scanning technology plays an important role in forest inventory, as it enables accurate 3D information capturing in a fast and environmentally-friendly manner. The goal of this study is to develop methods for detecting and discriminating individual trees from TLS point clouds of five plots in a boreal coniferous forest. The proposed hierarchical minimum cut method adopts the detected trunk points that are recognized according to pole like shape segmentation as foreground seed points and other points as background seed points, respectively. It constructs the undirected weighted graph of the foreground and background seed points to deduce a cost function for tree crown point segmentation with the decreasing ranking of tree trunk heights. The intermediate results lead to global optimization segmentation of individual trees in a hierarchical order. Finally, the structure metrics of the detected individual trees are calculated and checked with field observations. Plots with different attributes were selected to verify the proposed method, and the experimental studies show that the proposed method is efficient and robust for extracting individual trees from TLS point clouds in terms of the recall of 90.42%.

Computing minimum cuts in hypergraphs

Abstract: We study algorithmic and structural aspects of connectivity in hypergraphs. Given a hypergraph $H=(V,E)$ with $n = |V|$, $m = |E|$ and $p = \sum_{e \in E} |e|$ the best known algorithm to compute a global minimum cut in $H$ runs in time $O(np)$ for the uncapacitated case and in $O(np + n^2 \log n)$ time for the capacitated case. We show the following new results. 1. Given an uncapacitated hypergraph $H$ and an integer $k$ we describe an algorithm that runs in $O(p)$ time to find a subhypergraph $H'$ with sum of degrees $O(kn)$ that preserves all edge-connectivities up to $k$ (a $k$-sparsifier). This generalizes the corresponding result of Nagamochi and Ibaraki from graphs to hypergraphs. Using this sparsification we obtain an $O(p + \lambda n^2)$ time algorithm for computing a global minimum cut of $H$ where $\lambda$ is the minimum cut value. 2. We generalize Matula's argument for graphs to hypergraphs and obtain a $(2+\epsilon)$-approximation to the global minimum cut in a capacitated hypergraph in $O(\frac{1}{\epsilon} (p \log n + n \log^2 n))$ time. 3. We show that a hypercactus representation of all the global minimum cuts of a capacitated hypergraph can be computed in $O(np + n^2 \log n)$ time and $O(p)$ space. We utilize vertex ordering based ideas to obtain our results. Unlike graphs we observe that there are several different orderings for hypergraphs which yield different insights.

On Graphs of the Cone Decompositions for the Min-Cut and Max-Cut Problems

Abstract: We consider maximum and minimum cut problems with nonnegative weights of edges. We define the graphs of the cone decompositions and find a linear clique number for the min-cut problem and a superpolynomial clique number for the max-cut problem. These values characterize the time complexity in a broad class of algorithms based on linear comparisons.

A competitive study of the pseudoflow algorithm for the minimum s---t cut problem in vision applications

Abstract: Rapid advances in image acquisition and storage technology underline the need for real-time algorithms that are capable of solving large-scale image processing and computer-vision problems. The minimums---tcut problem, which is a classical combinatorial optimization problem, is a prominent building block in many vision and imaging algorithms such as video segmentation, co-segmentation, stereo vision, multi-view reconstruction, and surface fitting to name a few. That is why finding a real-time algorithm which optimally solves this problem is of great importance. In this paper, we introduce to computer vision the Hochbaum's pseudoflow (HPF) algorithm, which optimally solves the minimum s---t cut problem. We compare the performance of HPF, in terms of execution times and memory utilization, with three leading published algorithms: (1) Goldberg's and Tarjan's Push-Relabel; (2) Boykov's and Kolmogorov's augmenting paths; and (3) Goldberg's partial augment-relabel. While the common practice in computer-vision is to use either BK or PRF algorithms for solving the problem, our results demonstrate that, in general, HPF algorithm is more efficient and utilizes less memory than these three algorithms. This strongly suggests that HPF is a great option for many real-time computer-vision problems that require solving the minimum s---t cut problem.

Cut Tree Construction from Massive Graphs

Abstract: The construction of cut trees (also known as Gomory-Hu trees) for a given graph enables the minimum-cut size of the original graph to be obtained for any pair of vertices. Cut trees are a powerful back-end for graph management and mining, as they support various procedures related to the minimum cut, maximum flow, and connectivity. However, the crucial drawback with cut trees is the computational cost of their construction. In theory, a cut tree is built by applying a maximum flow algorithm for n times, where n is the number of vertices. Therefore, naive implementations of this approach result in cubic time complexity, which is obviously too slow for today's large-scale graphs. To address this issue, in the present study, we propose a new cut-tree construction algorithm tailored to real-world networks. Using a series of experiments, we demonstrate that the proposed algorithm is several orders of magnitude faster than previous algorithms and it can construct cut trees for billion-scale graphs.

Improved Hardness for Cut, Interdiction, and Firefighter Problems

Abstract: We study variants of the classic $s$-$t$ cut problem and prove the following improved hardness results assuming the Unique Games Conjecture (UGC). - For any constant $k \geq 2$ and $\epsilon > 0$, we show that Directed Multicut with $k$ source-sink pairs is hard to approximate within a factor $k - \epsilon$. This matches the trivial $k$-approximation algorithm. By a simple reduction, our result for $k = 2$ implies that Directed Multiway Cut with two terminals (also known as $s$-$t$ Bicut) is hard to approximate within a factor $2 - \epsilon$, matching the trivial $2$-approximation algorithm. Previously, the best hardness factor for these problems (for constant $k$) was $1.5 - \epsilon$ under the UGC. - For Length-Bounded Cut and Shortest Path Interdiction, we show that both problems are hard to approximate within any constant factor, even if we allow bicriteria approximation. If we want to cut vertices or the graph is directed, our hardness factor for Length-Bounded Cut matches the best approximation ratio up to a constant. Previously, the best hardness factor was $1.1377$ for Length-Bounded Cut and $2$ for Shortest Path Interdiction. - Assuming a variant of the UGC (implied by another variant of Bansal and Khot), we prove that it is hard to approximate Resource Minimization Fire Containment within any constant factor. Previously, the best hardness factor was $2$. Our results are based on a general method of converting an integrality gap instance to a length-control dictatorship test for variants of the $s$-$t$ cut problem, which may be useful for other problems.

Hardness of Online Sleeping Combinatorial Optimization Problems

Abstract: We show that several online combinatorial optimization problems that admit efficient no-regret algorithms become computationally hard in the sleeping setting where a subset of actions becomes unavailable in each round. Specifically, we show that the sleeping versions of these problems are at least as hard as PAC learning DNF expressions, a long standing open problem. We show hardness for the sleeping versions of Online Shortest Paths, Online Minimum Spanning Tree, Online k-Subsets, Online k-Truncated Permutations, Online Minimum Cut, and Online Bipartite Matching. The hardness result for the sleeping version of the Online Shortest Paths problem resolves an open problem presented at COLT 2015 [Koolen et al., 2015].

Incremental Exact Min-Cut in Poly-logarithmic Amortized Update Time

Abstract: We present a deterministic incremental algorithm for exactly maintaining the size of a minimum cut with ~O(1) amortized time per edge insertion and O(1) query time. This result partially answers an open question posed by Thorup [Combinatorica 2007]. It also stays in sharp contrast to a polynomial conditional lower-bound for the fully-dynamic weighted minimum cut problem. Our algorithm is obtained by combining a recent sparsification technique of Kawarabayashi and Thorup [STOC 2015] and an exact incremental algorithm of Henzinger [J. of Algorithm 1997]. We also study space-efficient incremental algorithms for the minimum cut problem. Concretely, we show that there exists an O(n log n/epsilon^2) space Monte-Carlo algorithm that can process a stream of edge insertions starting from an empty graph, and with high probability, the algorithm maintains a (1+epsilon)-approximation to the minimum cut. The algorithm has ~O(1) amortized update-time and constant query-time.

Energy Minimization of Discrete Protein Titration State Models Using Graph Theory.

Abstract: There are several applications in computational biophysics that require the optimization of discrete interacting states, for example, amino acid titration states, ligand oxidation states, or discrete rotamer angles. Such optimization can be very time-consuming as it scales exponentially in the number of sites to be optimized. In this paper, we describe a new polynomial time algorithm for optimization of discrete states in macromolecular systems. This algorithm was adapted from image processing and uses techniques from discrete mathematics and graph theory to restate the optimization problem in terms of "maximum flow-minimum cut" graph analysis. The interaction energy graph, a graph in which vertices (amino acids) and edges (interactions) are weighted with their respective energies, is transformed into a flow network in which the value of the minimum cut in the network equals the minimum free energy of the protein and the cut itself encodes the state that achieves the minimum free energy. Because of its deterministic nature and polynomial time performance, this algorithm has the potential to allow for the ionization state of larger proteins to be discovered.

On the Complexity of Matching Cut in Graphs of Fixed Diameter

Abstract: In a graph, a matching cut is an edge cut that is a matching. Matching Cut is the problem of deciding whether or not a given graph has a matching cut, which is known to be NP-complete even when restricted to bipartite graphs. It has been proved that Matching Cut is polynomially solvable for graphs of diameter two. In this paper, we show that, for any fixed integer d geq 4, Matching Cut is NP-complete in the class of graphs of diameter d. This almost resolves an open problem posed by Borowiecki and Jesse-Jozefczyk in [Matching cutsets in graphs of diameter 2, Theoretical Computer Science 407 (2008) 574-582]. We then show that, for any fixed integer d geq 5, Matching Cut is NP-complete even when restricted to the class of bipartite graphs of diameter d. Complementing the hardness results, we show that Matching Cut is in polynomial-time solvable in the class of bipartite graphs of diameter at most three, and point out a new and simple polynomial-time algorithm solving Matching Cut in graphs of diameter 2.

Clustering and Community Detection with Imbalanced Clusters

Abstract: Spectral clustering methods which are frequently used in clustering and community detection applications are sensitive to the specific graph constructions particularly when imbalanced clusters are present. We show that ratio cut (RCut) or normalized cut (NCut) objectives are not tailored to imbalanced cluster sizes since they tend to emphasize cut sizes over cut values. We propose a graph partitioning problem that seeks minimum cut partitions under minimum size constraints on partitions to deal with imbalanced cluster sizes. Our approach parameterizes a family of graphs by adaptively modulating node degrees on a fixed node set, yielding a set of parameter dependent cuts reflecting varying levels of imbalance. The solution to our problem is then obtained by optimizing over these parameters. We present rigorous limit cut analysis results to justify our approach and demonstrate the superiority of our method through experiments on synthetic and real datasets for data clustering, semi-supervised learning and community detection.

Cut structures in zero-divisor graphs of commutative rings

Abstract: Zero-divisor graphs, and more recently, compressed zero-divisor graphs are well represented in the commutative ring literature. In this work, we consider various cut structures, sets of edges or vertices whose removal disconnects the graph, in both compressed and non-compressed zero-divisor graphs. In doing so, we connect these graph-theoretic concepts with algebraic notions and provide realization theorems of zero-divisor graphs for commutative rings with identity.

A min-cut approach to functional regionalization, with a case study of the Italian local labour market areas

Abstract: In several economical, statistical and geographical applications, a territory must be subdivided into functional regions. Such regions are not fixed and politically delimited, but should be identified by analyzing the interactions among all its constituent localities. This is a very delicate and important task, that often turns out to be computationally difficult. In this work we propose an innovative approach to this problem based on the solution of minimum cut problems over an undirected graph called here transitions graph. The proposed procedure guarantees that the obtained regions satisfy all the statistical conditions required when considering this type of problems. Results on real-world instances show the effectiveness of the proposed approach.

Tight Bounds for Gomory-Hu-like Cut Counting

Abstract: 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 $$\left {\begin{array}{c}|V|\\ 2\end{array}}\right $$ instances exhibit redundancy factor$${\varOmega }|V|$$. 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,i¾?B-cut, ranging over all subsets $$A,B\subseteq V$$ of given sizes $$|A|=\alpha $$ and $$|B|=\beta $$. The redundancy factor is $${\varOmega }_{\alpha ,\beta }|V|$$.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 $${\varOmega }_{k}|V|$$.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 $${\varOmega }_{k}|V|^k$$. 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, i la the Gomory-Hu tree. We initiate this direction by giving some upper and lower bounds.

Complex System Reliability Optimisation: Further Assessment of a Multicriteria Approach

Abstract: Optimal allocation of reliability to components of a system is one of the several ways of ensuring high system reliability in design. The problem which is typically formulated as single criterion optimisation encounters difficulty when the analytic system reliability expression is intractable usually as a result of complex system configuration. The current work re-examines a novel approach previously developed by the authors which takes a multicriteria view of the problem on the basis of the minimum cut sets of a system, and thus avoids finding the top level system reliability before optimisation. The model and methodology was applied to two complex system reliability allocation problems: a classical one and one associated with electricity transmission. The results were consistent with previously obtained ones by the authors and with the reliability importance of some notable components.

Network Flow Formulations for Learning Binary Hashing

Abstract: The problem of learning binary hashing seeks the identification of a binary mapping for a set of n examples such that the corresponding Hamming distances preserve high fidelity with a given \(n \times n\) matrix of distances (or affinities). This formulation has numerous applications in efficient search and retrieval of images (and other high dimensional data) on devices with storage/processing constraints. As a result, the problem has received much attention recently in vision and machine learning and a number of interesting solutions have been proposed. A common feature of most existing solutions is that they adopt continuous iterative optimization schemes which is then followed by a post-hoc rounding process to recover a feasible discrete solution. In this paper, we present a fully combinatorial network-flow based formulation for a relaxed version of this problem. The main maximum flow/minimum cut modules which drive our algorithm can be solved efficiently and can directly learn the binary codes. Despite its simplicity, we show that on most widely used benchmarks, our proposal yields competitive performance relative to a suite of nine different state of the art algorithms.

A parallel min-cut algorithm using iteratively reweighted least squares targeting at problems with floating-point edge weights

Abstract: We formulate and derive a Parallel Iteratively Reweighted least squares Min-Cut solver (PIRMCut).We propose a novel two-level rounding procedure and prove a generalized Cheeger-type inequality.We developed an MPI based parallel implementation of PIRMCut.We demonstrate the parallel scalability of our PIRMCut implementation.We demonstrate PIRMCut significant speed improvement over a serial code. We present a parallel algorithm for the undirected st min-cut problem with floating-point valued edge weights. Our overarching algorithm uses an iteratively reweighted least squares framework. Specifically, this algorithm generates a sequence of Laplacian linear systems, which are solved in parallel. The iterative nature of our algorithm enables us to trade off solution quality for execution time, which is distinguished from those purely combinatorial algorithms that only produce solutions at optimum. We also propose a novel two-level rounding procedure that helps to enhance the quality of the approximate min-cut solution output by our algorithm. Our overall implementation, including the rounding procedure, demonstrates significant speed improvement over a state-of-the-art serial solver, where it could be up to 200 times faster on commodity platforms.

A linear time algorithm for a variant of the max cut problem in series parallel graphs

Abstract: Given a graph $G=(V, E)$, a connected sides cut $(U, V\backslash U)$ or $\delta (U)$ is the set of edges of E linking all vertices of U to all vertices of $V\backslash U$ such that the induced subgraphs $G[U]$ and $G[V\backslash U]$ are connected. Given a positive weight function $w$ defined on $E$, the maximum connected sides cut problem (MAX CS CUT) is to find a connected sides cut $\Omega$ such that $w(\Omega)$ is maximum. MAX CS CUT is NP-hard. In this paper, we give a linear time algorithm to solve MAX CS CUT for series parallel graphs. We deduce a linear time algorithm for the minimum cut problem in the same class of graphs without computing the maximum flow.

Research on Virtual Machine Cluster Deployment Algorithm in Cloud Computing Platform

Abstract: To address the virtual machine cluster deployment issues in cloud computing environment, a novel MCSA (Min-cut segmentation algorithm) of virtual machine cluster is proposed with resource and communication bandwidth constraints. In this paper, the basic idea is based on the fully consideration on the CPU, memory, hard-disk and other resource constraints between virtual machine cluster and physical host, as well as the communication bandwidth constraints between the virtual machine. We quantified the virtual machine cluster constructed an undirected graph. In the undirected graph, the nodes represent the virtual machine, so the weight of a node represents the value of resources, and the edges represent the communication bandwidth, so the weight of the edge represents the value of communication bandwidth. Base on the above transformations, the resources and bandwidth constrained optimization problem is transformed into the graph segmentation problem. Next we segment the undirected graph by minimum cut algorithm, and computing the matching degree of physical machines. Last we obtained the approximate solution. To validate the effectiveness of the new algorithm, we carried out extensive experiments based on the CloudSim platform.

A cut sequence set-based dynamic fault tree Monte-Carlo simulation quantitative calculation method

Abstract: The invention provides a cut sequence set-based dynamic fault tree Monte-Carlo simulation quantitative calculation method and relates to the technical field of reliability and safety (safe system engineering). The method comprises the steps of performing left-to-right depth-first traversal on different dynamic logic gates, and acquiring the minimum cut sequence set of a dynamic fault tree according to the cut sequence conversion rules of dynamic logic gates; employing the Monte-Carlo simulation method and comparing the sample failure time of events with the minimum cut sequence set, and recording a failure once if the time meets the occurrence order of events in the minimum cut sequence set and the occurrence time of the last event is within set system operation time. The method can provide important information for reliability design, improvement, enhancement and troubleshooting in engineering. The Monte-Carlo simulation is performed based on cut sequence sets, so that secondary conversion of a DFT, for example, to Markov, the Bayesian network or a failure time tree, is avoided and the quantitative calculation procedures are optimized.

Cheeger-Type Approximation for Sparsest st-Cut

Abstract: We introduce the st-cut version of the sparsest-cut problem, where the goal is to find a cut of minimum sparsity in a graph G(V, E) among those separating two distinguished vertices s, t ∈ V. Clearly, this problem is at least as hard as the usual (non-st) version. Our main result is a polynomial-time algorithm for the product-demands setting that produces a cut of sparsity O(√OPT), where OPT l 1 denotes the optimum when the total edge capacity and the total demand are assumed (by normalization) to be 1. Our result generalizes the recent work of Trevisan [arXiv, 2013] for the non-st version of the same problem (sparsest cut with product demands), which in turn generalizes the bound achieved by the discrete Cheeger inequality, a cornerstone of Spectral Graph Theory that has numerous applications. Indeed, Cheeger’s inequality handles graph conductance, the special case of product demands that are proportional to the vertex (capacitated) degrees. Along the way, we obtain an O(log vVv) approximation for the general-demands setting of sparsest st-cut.