Showing papers on "Minimum cut published in 2012"

Unsupervised Salient Object Segmentation Based on Kernel Density Estimation and Two-Phase Graph Cut

Abstract: In this paper, we propose an unsupervised salient object segmentation approach based on kernel density estimation (KDE) and two-phase graph cut. A set of KDE models are first constructed based on the pre-segmentation result of the input image, and then for each pixel, a set of likelihoods to fit all KDE models are calculated accordingly. The color saliency and spatial saliency of each KDE model are then evaluated based on its color distinctiveness and spatial distribution, and the pixel-wise saliency map is generated by integrating likelihood measures of pixels and saliency measures of KDE models. In the first phase of salient object segmentation, the saliency map based graph cut is exploited to obtain an initial segmentation result. In the second phase, the segmentation is further refined based on an iterative seed adjustment method, which efficiently utilizes the information of minimum cut generated using the KDE model based graph cut, and exploits a balancing weight update scheme for convergence of segmentation refinement. Experimental results on a dataset containing 1000 test images with ground truths demonstrate the better segmentation performance of our approach.

Designing FPT Algorithms for Cut Problems Using Randomized Contractions

Abstract: We introduce a new technique for designing fixed-parameter algorithms for cut problems, namely randomized contractions. With our framework: * We obtain the first FPT algorithm for the parameterized version of the UNIQUE LABEL COVER problem, with single exponential dependency on the size of the cutset and the size of the alphabet. As a consequence, we extend the set of the polynomial time solvable instances of UNIQUE GAMES to those with at most O(sqrt{log n}) violated constraints. * We obtain a new FPT algorithm for the STEINER CUT problem with exponential speed-up over the recent work of Kawarabayashi and Thorup (FOCS'11). * We show how to combine considering 'cut' and 'uncut' constraints at the same time. We define a robust problem NODE MULTIWAY CUT-UNCUT that can serve as an abstraction of introducing uncut constraints, and show that it admits an FPT algorithm with single exponential dependency on the size of the cutset. To the best of our knowledge, the only known way of tackling uncut constraints was via the approach of Marx, O'Sullivan and Razgon (STACS'10), which yields algorithms with double exponential running time. An interesting aspect of our algorithms is that they can handle real weights, to the best of our knowledge, the technique of important separators does not work in the weighted version.

Parameterized tractability of multiway cut with parity constraints

Abstract: In this paper, we study a parity based generalization of the classical Multiway Cut problem. Formally, we study the Parity Multiway Cut problem, where the input is a graph G, vertex subsets Te and To (T=Te∪To) called terminals, a positive integer k and the objective is to test whether there exists a k-sized vertex subset S such that S intersects all odd paths from v∈To to T∖{v} and all even paths from v∈Te to T∖{v}. When Te=To, this is precisely the classical Multiway Cut problem. If To=∅ then this is the Even Multiway Cut problem and if Te=∅ then this is the Odd Multiway Cut problem. We remark that even the problem of deciding whether there is a set of at most k vertices that intersects all odd paths between a pair of vertices s and t is NP-complete. Our primary motivation for studying this problem is the recently initiated parameterized study of parity versions of graphs minors (Kawarabayashi, Reed and Wollan, FOCS 2011) and separation problems similar to Multiway Cut. The area of design of parameterized algorithms for graph separation problems has seen a lot of recent activity, which includes algorithms for Multi-Cut on undirected graphs (Marx and Razgon, STOC 2011, Bousquet, Daligault and Thomasse, STOC 2011), k-way cut (Kawarabayashi and Thorup, FOCS 2011), and Multiway Cut on directed graphs (Chitnis, Hajiaghayi and Marx, SODA 2012). A second motivation is that this problem serves as a good example to illustrate the application of a generalization of important separators which we introduce, and can be applied even when most of the recently develped tools fail to apply. We believe that this could be a useful tool for several other separation problems as well. We obtain this generalization by dividing the graph into slices with small boundaries and applying a divide and conquer paradigm over these slices. We show that Parity Multiway Cut is fixed parameter tractable (FPT) by giving an algorithm that runs in time $f(k)n^{{\mathcal{O}}(1)}$. More precisely, we show that instances of this problem with solutions of size ${\cal O}(\log \log n)$ can be solved in polynomial time. Along with this new notion of generalized important separators, our algorithm also combines several ideas used in previous parameterized algorithms for graph separation problems including the notion of important separators and randomized selection of important sets to simplify the input instance.

Dynamic Graph Clustering Using Minimum-Cut Trees

Abstract: Algorithms or target functions for graph clustering rarely admit quality guarantees or optimal results in general. Based on properties of minimum-cut trees, a clustering algorithm by Flake et al. does however yield such a provable guarantee. We show that the structure of minimum-s -t -cuts in a graph allows for an efficient dynamic update of minimum-cut trees, and present a dynamic graph clustering algorithm that maintains a clustering fulfilling this quality quarantee, and that effectively avoids changing the clustering. Experiments on real-world dynamic graphs complement our theoretical results.

A polynomial-time approximation scheme for planar multiway cut

Abstract: Given an undirected graph with edge lengths and a subset of nodes (called the terminals), the multiway cut (also called the multi-terminal cut) problem asks for a subset of edges, with minimum total length, whose removal disconnects each terminal from all others. The problem generalizes minimum s-t cut, but is NP-hard for planar graphs and APX-hard for general graphs [11]. In this paper, we present a PTAS for multiway cut on planar graphs.

First-principles multiway spectral partitioning of graphs

Abstract: We consider the minimum-cut partitioning of a graph into more than two parts using spectral methods. While there exist well-established spectral algorithms for this problem that give good results, they have traditionally not been well motivated. Rather than being derived from first principles by minimizing graph cuts, they are typically presented without direct derivation and then proved after the fact to work. In this paper, we take a contrasting approach in which we start with a matrix formulation of the minimum cut problem and then show, via a relaxed optimization, how it can be mapped onto a spectral embedding defined by the leading eigenvectors of the graph Laplacian. The end result is an algorithm that is similar in spirit to, but different in detail from, previous spectral partitioning approaches. In tests of the algorithm we find that it outperforms previous approaches on certain particularly difficult partitioning problems.

Global minimum cuts in surface embedded graphs

Abstract: We give a deterministic algorithm to find the minimum cut in a surface-embedded graph in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in gO(g)n log log n time, matching the running time of the fastest algorithm known for planar graphs, due to Lacki and Sankowski, for any constant g. Indeed, our algorithm calls Lacki and Sankowski's recent O(n log log n) time planar algorithm as a subroutine.Previously, the best time bounds known for this problem followed from two algorithms for general sparse graphs: a randomized algorithm of Karger that runs in O(n log3n) time and succeeds with high probability, and a deterministic algorithm of Nagamochi and Ibaraki that runs in O(n2 log n) time. We can also achieve a deterministic gO(g)n2 log log n time bound by repeatedly applying the best known algorithm for minimum (s, t)-cuts in surface graphs. The bulk of our work focuses on the case where the dual of the minimum cut splits the underlying surface into multiple components with positive genus.

On Mimicking Networks Representing Minimum Terminal Cuts

Abstract: Given a capacitated undirected graph $G=(V,E)$ with a set of terminals $K \subset V$, a mimicking network is a smaller graph $H=(V_H,E_H)$ that exactly preserves all the minimum cuts between the terminals. Specifically, the vertex set of the sparsifier $V_H$ contains the set of terminals $K$ and for every bipartition $U, K-U $ of the terminals $K$, the size of the minimum cut separating $U$ from $K-U$ in $G$ is exactly equal to the size of the minimum cut separating $U$ from $K-U$ in $H$. This notion of a mimicking network was introduced by Hagerup, Katajainen, Nishimura and Ragde (1995) who also exhibited a mimicking network of size $2^{2^{k}}$ for every graph with $k$ terminals. The best known lower bound on the size of a mimicking network is linear in the number of terminals. More precisely, the best known lower bound is $k+1$ for graphs with $k$ terminals (Chaudhuri et al. 2000). In this work, we improve both the upper and lower bounds reducing the doubly-exponential gap between them to a single-exponential gap. Specifically, we obtain the following upper and lower bounds on mimicking networks: 1) Given a graph $G$, we exhibit a construction of mimicking network with at most $(|K|-1)$'th Dedekind number ($\approx 2^{{(k-1)} \choose {\lfloor {{(k-1)}/2} \rfloor}}$) of vertices (independent of size of $V$). Furthermore, we show that the construction is optimal among all {\it restricted mimicking networks} -- a natural class of mimicking networks that are obtained by clustering vertices together. 2) There exists graphs with $k$ terminals that have no mimicking network of size smaller than $2^{\frac{k-1}{2}}$. We also exhibit improved constructions of mimicking networks for trees and graphs of bounded tree-width.

Emerging Graph Queries in Linked Data

Abstract: In a wide array of disciplines, data can be modeled as an interconnected network of entities, where various attributes could be associated with both the entities and the relations among them. Knowledge is often hidden in the complex structure and attributes inside these networks. While querying and mining these linked datasets are essential for various applications, traditional graph queries may not be able to capture the rich semantics in these networks. With the advent of complex information networks, new graph queries are emerging, including graph pattern matching and mining, similarity search, ranking and expert finding, graph aggregation and OLAP. These queries require both the topology and content information of the network data, and hence, different from classical graph algorithms such as shortest path, reach ability and minimum cut, which depend only on the structure of the network. In this tutorial, we shall give an introduction of the emerging graph queries, their indexing and resolution techniques, the current challenges and the future research directions.

Structural and algorithmic properties for parametric minimum cuts

Abstract: We consider the minimum s, t-cut problem in a network with parametrized arc capacities. Following the seminal work of Gallo et al. (SIAM J. Comput. 18(1):30---55, 1989), classes of this parametric problem have been shown to enjoy the nice Structural Property that minimum cuts are nested, and the nice Algorithmic Property that all minimum cuts can be computed in the same asymptotic time as a single minimum cut by using a clever Flow Update step to move from one value of the parameter to the next. We present a general framework for parametric minimum cuts that extends and unifies such results. We define two conditions on parametrized arc capacities that are necessary and sufficient for (strictly) decreasing differences of the parametric cut function. Known results in parametric submodular optimization then imply the Structural Property. We show how to construct appropriate Flow Updates in linear time under the above conditions, implying that the Algorithmic Property also holds under these conditions. We then consider other classes of parametric minimum cut problems, without decreasing differences, for which we establish the Structural and/or the Algorithmic Property, as well as other cases where nested minimum cuts arise.

On the feasibility of precoding-based network alignment for three unicast sessions

Abstract: We consider the problem of network coding across three unicast sessions over a directed acyclic graph, when each session has min-cut one. Previous work by Das et al. adapted a precoding-based interference alignment technique, originally developed for the wireless interference channel, specifically to this problem. We refer to this approach as precoding-based network alignment (PBNA). Similar to the wireless setting, PBNA asymptotically achieves half the minimum cut; different from the wireless setting, its feasibility depends on the graph structure. Das et al. provided a set of feasibility conditions for PBNA with respect to a particular precoding matrix. However, the set consisted of an infinite number of conditions, which is impossible to check in practice. Furthermore, the conditions were purely algebraic, without interpretation with regards to the graph structure. In this paper, we first prove that the set of conditions provided by Das. et al are also necessary for the feasibility of PBNA with respect to any precoding matrix. Then, using two graph-related properties and a degree-counting technique, we reduce the set to just four conditions. This reduction enables an efficient algorithm for checking the feasibility of PBNA on a given graph.

Approximating minimum label s - t cut via linear programming

Abstract: We consider the Minimum Label s-t Cut problem. Given an undirected graph G=(V,E) with a label set L, in which each edge has a label from L, and a source s∈V together with a sink t∈V, the goal of the Minimum Label s-t Cut problem is to pick a subset of labels of minimized cardinality, such that the removal of all edges with these labels from G disconnects s and t. We present a min { O((m/OPT)1/2), O(n2/3/OPT1/3) }-approximation algorithm for the Minimum Label s-t Cut problem using linear programming technique, where n=|V|, m=|E|, and OPT is the optimal value of the input instance. This result improves the previously best known approximation ratio O(m1/2) for this problem (Zhang et al., JOCO 21(2), 192---208 (2011)), and gives the first approximation ratio for this problem in terms of n. Moreover, we show that our linear program relaxation for the Minimum Label s-t Cut problem, even in a stronger form, has integrality gap Ω((m/OPT)1/2−e).

On the minimum cut separator problem

Abstract: Given G = (V,E) an undirected graph and two specified nonadjacent nodes a and b of V, a cut separator is a subset F =δ (C) ⊆ E such that a,b∈V / C and a and b belong to different connected components of the graph induced by V / C. Given a non-negative cost vector , the cut separator problem is to find a cut separator of minimum cost. This new problem can be seen as a generalization of the vertex separator problem. In this article, we give a polynomial time algorithm for this problem. We also present six equivalent linear programming formulations, and we show their tightness. Using these results we obtain an explicit short polyhedral description of the dominant of the cut separator polytope. © 2011 Wiley Periodicals, Inc. NETWORKS, 2012

Event Coreference Resolution using Mincut based Graph Clustering

Abstract: To extract participants of an event instance, it is necessary to identify all the sentences that describe the event instance. The set of all sentences referring to the same event instance are said to be corefering each other. Our proposed approach formulates the event coreference resolution as a graph based clustering model. It identifies the corefering sentences using minimum cut (mincut) based on similarity score between each pair of sentences at various levels such as trigger word similarity, time stamp similarity, entity similarity and semantic similarity. It achieves good B-Cubed F-measure score with some loss in recall.

Combinatiorial algorithms for wireless information flow

Abstract: A long-standing open question in information theory is to characterize the unicast capacity of a wireless relay network. The difficulty arises due to the complex signal interactions induced in the network, since the wireless channel inherently broadcasts the signals and there is interference among transmissions. Recently, Avestimehr et al. [2007b] proposed a linear deterministic model that takes into account the shared nature of wireless channels, focusing on the signal interactions rather than the background noise. They generalized the min-cut max-flow theorem for graphs to networks of deterministic channels and proved that the capacity can be achieved using information theoretical tools. They showed that the value of the minimum cut is in this case the minimum rank of all the adjacency matrices describing source-destination cuts.In this article, we develop a polynomial-time algorithm that discovers the relay encoding strategy to achieve the min-cut value in linear deterministic (wireless) networks, for the case of a unicast connection. Our algorithm crucially uses a notion of linear independence between channels to calculate the capacity in polynomial time. Moreover, we can achieve the capacity by using very simple one-symbol processing at the intermediate nodes, thereby constructively yielding finite-length strategies that achieve the unicast capacity of the linear deterministic (wireless) relay network.

Method and system for determining, contracting to exchange, and accounting for matched sets of offsetting cash flows

Abstract: Among other embodiments, methods and systems are disclosed for determining one or more sets of structured cash flows corresponding to a graph having one or more nodes corresponding to one or more exchange definitions or swap transactions. The net present value of a structured cash flow may be substantially zero, and may correspond to a maximum notional amount, a maximum flow, or a minimum cut.

Fast and Simple Fully-Dynamic Cut Tree Construction

Abstract: A cut tree of an undirected weighted graph G = (V,E) encodes a minimum s-t-cut for each vertex pair {s,t} ⊆ V and can be iteratively constructed by n − 1 maximum flow computations. They solve the multiterminal network flow problem, which asks for the all-pairs maximum flow values in a network and at the same time they represent n − 1 non-crossing, linearly independent cuts that constitute a minimum cut basis of G. Hence, cut trees are resident in at least two fundamental fields of network analysis and graph theory, which emphasizes their importance for many applications. In this work we present a fully-dynamic algorithm that efficiently maintains a cut tree for a changing graph. The algorithm is easy to implement and has a high potential for saving cut computations under the assumption that a local change in the underlying graph does rarely affect the global cut structure. We document the good practicability of our approach in a brief experiment on real world data.

Mimicking Networks and Succinct Representations of Terminal Cuts

Abstract: Given a large edge-weighted network $G$ with $k$ terminal vertices, we wish to compress it and store, using little memory, the value of the minimum cut (or equivalently, maximum flow) between every bipartition of terminals. One appealing methodology to implement a compression of $G$ is to construct a \emph{mimicking network}: a small network $G'$ with the same $k$ terminals, in which the minimum cut value between every bipartition of terminals is the same as in $G$. This notion was introduced by Hagerup, Katajainen, Nishimura, and Ragde [JCSS '98], who proved that such $G'$ of size at most $2^{2^k}$ always exists. Obviously, by having access to the smaller network $G'$, certain computations involving cuts can be carried out much more efficiently. We provide several new bounds, which together narrow the previously known gap from doubly-exponential to only singly-exponential, both for planar and for general graphs. Our first and main result is that every $k$-terminal planar network admits a mimicking network $G'$ of size $O(k^2 2^{2k})$, which is moreover a minor of $G$. On the other hand, some planar networks $G$ require $|E(G')| \ge \Omega(k^2)$. For general networks, we show that certain bipartite graphs only admit mimicking networks of size $|V(G')| \geq 2^{\Omega(k)}$, and moreover, every data structure that stores the minimum cut value between all bipartitions of the terminals must use $2^{\Omega(k)}$ machine words.

Non-Markovian dynamic time warping

Abstract: This paper proposes a new dynamic time warping (DTW) method, called non-Markovian DTW. In the conventional DTW, the warping function is optimized generally by dynamic programming (DP) subject to some Markovian constraints which restrict the relationship between neighboring time points. In contrast, the non-Markovian DTW can introduce non-Markovian constraints for dealing with the relationship between points with a large time interval. This new and promising ability of DTW is realized by using graph cut as the optimizer of the warping function instead of DP. Specifically, the conventional DTW problem is first converted as an equivalent minimum cut problem on a graph and then edges representing the non-Markovian constraints are added to the graph. An experiment on online character recognition showed the advantage of using non-Markovian constraints during DTW.

A branch-and-bound algorithm for the minimum cut linear arrangement problem

Abstract: Given an edge-weighted graph G of order n, the minimum cut linear arrangement problem (MCLAP) asks to find a one-to-one map from the vertices of G to integers from 1 to n such that the largest of the cut values c 1,?,c n?1 is minimized, where c i , i?{1,?,n?1}, is the total weight of the edges connecting vertices mapped to integers 1 through i with vertices mapped to integers i+1 through n In this paper, we present a branch-and-bound algorithm for solving this problem A salient feature of the algorithm is that it employs a dominance test which allows reducing the redundancy in the enumeration process drastically The test is based on the use of a tabu search procedure developed to solve the MCLAP We report computational results for both the unweighted and weighted graphs In particular, we focus on calculating the cutwidth of some well-known graphs from the literature

On the Feasibility of Precoding-Based Network Alignment for Three Unicast Sessions

Abstract: We consider the problem of network coding across three unicast sessions over a directed acyclic graph, when each session has min-cut one. Previous work by Das et al. adapted a precoding-based interference alignment technique, originally developed for the wireless interference channel, specifically to this problem. We refer to this approach as precoding-based network alignment (PBNA). Similar to the wireless setting, PBNA asymptotically achieves half the minimum cut; different from the wireless setting, its feasibility depends on the graph structure. Das et al. provided a set of feasibility conditions for PBNA with respect to a particular precoding matrix. However, the set consisted of an infinite number of conditions, which is impossible to check in practice. Furthermore, the conditions were purely algebraic, without interpretation with regards to the graph structure. In this paper, we first prove that the set of conditions provided by Das. et al are also necessary for the feasibility of PBNA with respect to any precoding matrix. Then, using two graph-related properties and a degree-counting technique, we reduce the set to just four conditions. This reduction enables an efficient algorithm for checking the feasibility of PBNA on a given graph.

Max-Sur-CSP on Two Elements

Abstract: Max-Sur-CSP is the following optimisation problem: given a set of constraints, find a surjective mapping of the variables to domain values that satisfies as many of the constraints as possible. Many natural problems, e.g. Minimum k-Cut (which has many different applications in a variety of fields) and Minimum Distance (which is an important problem in coding theory), can be expressed as Max-Sur-CSPs. We study Max-Sur-CSP on the two-element domain and determine the computational complexity for all constraint languages (families of allowed constraints). Our results show that the problem is solvable in polynomial time if the constraint language belongs to one of three classes, and NP-hard otherwise. An important part of our proof is a polynomial-time algorithm for enumerating all near-optimal solutions to a generalised minimum cut problem. This algorithm may be of independent interest.

Finding short vectors in a lattice of Voronoi's first kind

Abstract: We show that for those lattices of Voronoi's first kind with known obtuse superbasis, a vector of shortest nonzero Euclidean length can computed in polynomial time by computing a minimum cut in a graph.

Maximum Directed Cuts in Graphs with Degree Constraints

Abstract: The Max Cut problem is an NP-hard problem and has been studied extensively. Alon et al. (J Graph Theory 55:1–13, 2007) studied a directed version of the Max Cut problem and observed its connection to the Hall ratio of graphs. They proved, among others, that if an acyclic digraph has m edges and each vertex has indegree or outdegree at most 1, then it has a directed cut of size at least 2m/5. Lehel et al. (J Graph Theory 61:140–156, 2009) extended this result by replacing the “acyclic digraphs” with the “digraphs containing no directed triangles”. In this paper, we characterize the acyclic digraphs with m edges whose maximum dicuts have exactly 2m/5 edges, and our approach gives an alternative proof of the result of Lehel et al. We also show that there are infinitely many positive rational numbers β < 2/5 for which there exist digraphs D (with directed triangles) such that each vertex of D has indegree or outdegree at most 1, and any maximum directed cut in D has size precisely β|E(D)|.

Automated detection of pelvic fractures from volumetric CT images

Abstract: Pelvic fractures are a major cause of trauma patient mortality. Detection and management of pelvic injuries is challenging due to myriad injury patterns and associated complications such as hemorrhage and infection. In this paper, we propose an automated method of pelvic fracture detection from volumetric CT images. A coarse-to-fine strategy is adopted where a potential region containing the fracture is identified first using intensity and curvature information. The above region is modeled as a weighted graph and a fracture is modeled as a minimum cut in this graph. A second localizing algorithm models the same fracture as a valley, based on the signs of the mean and Gaussian curvature. The minimum cuts as well as the spatial consistent valleys, in isolation, generate a small number of false positives in addition to the true fracture. A joint decision process based on the volumetric graph cuts and the spatially consistent valleys eliminates the false positives. Experimental results indicate the effectiveness of the proposed scheme.

Finding short vectors in a lattice of Voronoi's first kind

Abstract: We show that for those lattices of Voronoi's first kind, a vector of shortest nonzero Euclidean length can computed in polynomial time by computing a minimum cut in a graph.

A Coding Theoretic Approach for Evaluating Accumulate Distribution on Minimum Cut Capacity of Weighted Random Graphs

Abstract: The multicast capacity of a directed network is closely related to the $s$-$t$ maximum flow, which is equal to the $s$-$t$ minimum cut capacity due to the max-flow min-cut theorem. If the topology of a network (or link capacities) is dynamically changing or have stochastic nature, it is not so trivial to predict statistical properties on the maximum flow. In this paper, we present a coding theoretic approach for evaluating the accumulate distribution of the minimum cut capacity of weighted random graphs. The main feature of our approach is to utilize the correspondence between the cut space of a graph and a binary LDGM (low-density generator-matrix) code with column weight 2. The graph ensemble treated in the paper is a weighted version of Erdős-Renyi random graph ensemble. The main contribution of our work is a combinatorial lower bound for the accumulate distribution of the minimum cut capacity. From some computer experiments, it is observed that the lower bound derived here reflects the actual statistical behavior of the minimum cut capacity.