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109
The Complexity of Multiterminal Cuts
 SIAM Journal on Computing
, 1994
"... In the Multiterminal Cut problem we are given an edgeweighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, maxflow problem, and ..."
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Cited by 190 (0 self)
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In the Multiterminal Cut problem we are given an edgeweighted graph and a subset of the vertices called terminals, and asked for a minimum weight set of edges that separates each terminal from all the others. When the number k of terminals is two, this is simply the mincut, maxflow problem, and can be solved in polynomial time. We show that the problem becomes NPhard as soon as k = 3, but can be solved in polynomial time for planar graphs for any fixed k. The planar problem is NPhard, however, if k is not fixed. We also describe a simple approximation algorithm for arbitrary graphs that is guaranteed to come within a factor of 2  2/k of the optimal cut weight.
Optimum communication spanning trees
 SIAM J. Comput
, 1974
"... Abstract. Given a set of nodes N (i 1, 2,..., n) which may represent cities and a set of requirements ria which may represent the number of telephone calls between N and N j, the problem is to build a spanning tree connecting these n nodes such that the total cost of communication of the spanning tr ..."
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Cited by 90 (1 self)
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Abstract. Given a set of nodes N (i 1, 2,..., n) which may represent cities and a set of requirements ria which may represent the number of telephone calls between N and N j, the problem is to build a spanning tree connecting these n nodes such that the total cost of communication of the spanning tree is a minimum among all spanning trees. The cost of communication for a pair of nodes is r;a multiplied by the sum of the distances of arcs which form the unique path connecting Ni and N in the spanning tree. Summing over all () pairs of nodes, we have the total cost of communication of the spanning tree. Note that the problem is different from the minimum spanning tree problem solved by Kruskal and Prim. Key words, communication spanning trees, cuttree
Minimizing Register Requirements under ResourceConstrained RateOptimal Software Pipelining
, 1995
"... The rapid advances in highperformance computer architecture and compilation techniques provide both challenges and opportunities to exploit the rich solution space of software pipelined loop schedules. In this paper, we develop a framework to construct a software pipelined loop schedule which runs ..."
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Cited by 80 (15 self)
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The rapid advances in highperformance computer architecture and compilation techniques provide both challenges and opportunities to exploit the rich solution space of software pipelined loop schedules. In this paper, we develop a framework to construct a software pipelined loop schedule which runs on the given architecture (with a fixed number of processor resources) at the maximum possible iteration rate (`a la rateoptimal) while minimizing the number of buffers  a close approximation to minimizing the number of registers. The main contributions of this paper are: ffl First, we demonstrate that such problem can be described by a simple mathematical formulation with precise optimization objectives under a periodic linear scheduling framework. The mathematical formulation provides a clear picture which permits one to visualize the overall solution space (for rateoptimal schedules) under different sets of constraints. ffl Secondly, we show that a precise mathematical formulation...
Optimal capacity placement for path restoration in STM or ATM meshsurvivable networks
 IEEE/ACM Trans. Networking
, 1988
"... Abstract—The total transmission capacity required by a transport network to satisfy demand and protect it from failures contributes significantly to its cost, especially in longhaul networks. Previously, the spare capacity of a network with a given set of working span sizes has been optimized to ..."
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Cited by 80 (19 self)
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Abstract—The total transmission capacity required by a transport network to satisfy demand and protect it from failures contributes significantly to its cost, especially in longhaul networks. Previously, the spare capacity of a network with a given set of working span sizes has been optimized to facilitate span restoration [11], [12]. Path restorable networks can, however, be even more efficient by defining the restoration problem from an end to end rerouting viewpoint. We provide a method for capacity optimization of pathrestorable networks which is applicable to both synchronous transfer mode (STM) and asynchronous transfer mode (ATM) virtual path (VP)based restoration. Lower bounds on spare capacity requirements in span and pathrestorable networks are first compared, followed by an integer program formulation based on flow constraints which solves the spare and/or working capacity placement problem in either spanor pathrestorable networks. The benefits of path and span restoration, and of jointly optimizing working path routing and spare capacity placement, are then analyzed. Index Terms—Capacity placement, mesh restoration, survivable networks. I.
An improved approximation algorithm for multiway cut
 Journal of Computer and System Sciences
, 1998
"... Given an undirected graph with edge costs and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest. Multiway Cut is the problem of finding a multiway cut of minimum cost. Previously, a very simple combinatorial algorithm due ..."
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Cited by 74 (5 self)
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Given an undirected graph with edge costs and a subset of k nodes called terminals, a multiway cut is a subset of edges whose removal disconnects each terminal from the rest. Multiway Cut is the problem of finding a multiway cut of minimum cost. Previously, a very simple combinatorial algorithm due to Dahlhaus, � Johnson, Papadimitriou, Seymour, and Yannakakis gave a performance guarantee of 2 1 − 1 k. In this paper, we present a new linear programming relaxation for Multiway Cut and a new approximation algorithm based on it. The algorithm breaks the threshold of 2 for approximating Multiway Cut, achieving a. This improves the previous result for every value of k. performance ratio of at most 1.5 − 1 k In particular, for k = 3 we get a ratio of 7
Globally minimal surfaces by continuous Maximal Flows
 IEEE Transactions on Pattern Analysis and Machine Intelligence
, 2006
"... Globally minimal surfaces by continuous maximal flows In this paper we address the computation of globally minimal curves and surfaces for image segmentation and stereo reconstruction. We present a solution, simulating a continuous maximal flow by a novel system of partial differential equations. Ex ..."
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Cited by 52 (5 self)
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Globally minimal surfaces by continuous maximal flows In this paper we address the computation of globally minimal curves and surfaces for image segmentation and stereo reconstruction. We present a solution, simulating a continuous maximal flow by a novel system of partial differential equations. Existing methods are either gridbiased (graphbased methods) or suboptimal (active contours and surfaces). The solution simulates the flow of an ideal fluid with isotropic velocity constraints. Velocity constraints are defined by a metric derived from image data. An auxiliary potential function is introduced to create a system of partial differential equations. It is proven that the algorithm produces a globally maximal continuous flow at convergence, and that the globally minimal surface may be obtained trivially from the auxiliary potential. The bias of minimal surface methods toward small objects is also addressed. An efficient implementation is given for the flow simulation. The globally minimal surface algorithm is applied to segmentation in 2D and 3D as well as to stereo matching. Results in 2D agree with an existing minimal contour algorithm for planar images. Results in 3D segmentation and stereo matching demonstrate that the new algorithm is robust and free from grid bias. I.
Continuous global optimization in multiview 3d reconstruction
 In International Conference on Energy Minimization Methods in Computer Vision and Pattern Recognition
, 2007
"... Abstract. In this work, we introduce a robust energy model for multiview 3D reconstruction that fuses silhouette and stereobased image information. It allows to cope with significant amounts of noise without manual presegmentation of the input images. Moreover, we suggest a method that can global ..."
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Cited by 46 (11 self)
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Abstract. In this work, we introduce a robust energy model for multiview 3D reconstruction that fuses silhouette and stereobased image information. It allows to cope with significant amounts of noise without manual presegmentation of the input images. Moreover, we suggest a method that can globally optimize this energy up to the visibility constraint. While similar global optimization has been presented in the discrete context in form of the maxflowmincut framework, we suggest the use of a continuous counterpart. In contrast to graph cut methods, discretizations of the continuous optimization technique are consistent and independent of the choice of the grid connectivity. Our experiments demonstrate that this leads to visible improvements. Moreover, memory requirements are reduced, allowing for global reconstructions at higher resolutions. 1
An Algorithm for Large Scale 01 Integer Programming With Application to Airline Crew Scheduling
, 1995
"... We present an approximation algorithm for solving large 01 integer programming problems where A is 01 and where b is integer. The method can be viewed as a dual coordinate search for solving the LPrelaxation, reformulated as an unconstrained nonlinear problem, and an approximation scheme working t ..."
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Cited by 44 (5 self)
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We present an approximation algorithm for solving large 01 integer programming problems where A is 01 and where b is integer. The method can be viewed as a dual coordinate search for solving the LPrelaxation, reformulated as an unconstrained nonlinear problem, and an approximation scheme working together with this method. The approximation scheme works by adjusting the costs as little as possible so that the new problem has an integer solution. The degree of approximation is determined by a parameter, and for different levels of approximation the resulting algorithm can be interpreted in terms of linear programming, dynamic programming, and as a greedy algorithm. The algorithm is used in the CARMEN system for airline crew scheduling used by several major airlines, and we show that the algorithm performs well for large set covering problems, in comparison to the CPLEX system, in terms of both time and quality. We also present results on some well known difficult set covering problems ...
Asymptotically Fast Computation of Hermite Normal Forms of Integer Matrices
 Proc. Int'l. Symp. on Symbolic and Algebraic Computation: ISSAC '96
, 1996
"... This paper presents a new algorithm for computing the Hermite normal form H of an A 2 ZZ n\Thetam of rank m together with a unimodular premultiplier matrix U such that UA = H. Our algorithm requires O~(m `\Gamma1 nM(m log jjAjj)) bit operations to produce both H and U . Here, jjAjj = max ij j ..."
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Cited by 40 (10 self)
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This paper presents a new algorithm for computing the Hermite normal form H of an A 2 ZZ n\Thetam of rank m together with a unimodular premultiplier matrix U such that UA = H. Our algorithm requires O~(m `\Gamma1 nM(m log jjAjj)) bit operations to produce both H and U . Here, jjAjj = max ij jA ij j, M(t) bit operations are sufficient to multiply two dtebit integers, and ` is the exponent for matrix multiplication over rings: two m \Theta m matrices over a ring R can be multiplied in O(m ` ) ring operations from R. The previously fastest algorithm of Hafner & McCurley requires O~(m 2 nM(m log jjAjj)) bit operations to produce H, but does not produce a unimodular matrix U which satisfies UA = H. Previous methods require on the order of O~(n 3 M(m log jjAjj)) bit operations to produce a U  our algorithm improves on this significantly in both a theoretical and practical sense. 1 Introduction A fundamental notion for matrices over rings is left equivalence. Two n \Th...
Information bounds are weak in the shortest distance problem
 J. ACM
, 1980
"... ASSTRACT. In the allpair shortest distance problem, one computes the matrix D = (du), where dq is the minimum weighted length of any path from vertex i to vertexj in a directed complete graph with a weight on each edge. In all the known algorithms, a shortest path p, ~ achieving di./is also implici ..."
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Cited by 28 (1 self)
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ASSTRACT. In the allpair shortest distance problem, one computes the matrix D = (du), where dq is the minimum weighted length of any path from vertex i to vertexj in a directed complete graph with a weight on each edge. In all the known algorithms, a shortest path p, ~ achieving di./is also implicitly computed. In fact, logs(f (n)) is an informationtheoretic lower bound, wheref(n) is the total number of distinct patterns (Po) for nvertex graphs. As f(n) potentially can be as large as 2&quot;:', it would appear possible that a nontrivial lower bound can be derived this way in the decision tree model. The characterization and enumeration of realizable patterns is studied, and it is shown thatf(n) < C &quot;~. Thus no lower bound greater than Cn 2 can be derived from this approach. It is proved as a corollary that the Triangular polyhedron T ~&quot;~, defined in E ¢~' ~ by d,j> 0 and the triangle inequalities d~j + dik> d,k, has at most C&quot; ' faces of all dimensions, thus resolving an open question in a similar information bound approach to the shortest distance problem.