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55
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Fast Approximation Algorithms for Multicommodity Flow Problems
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1991
"... All previously known algorithms for solving the multicommodity flow problem with capacities are based on linear programming. The best of these algorithms [15] uses a fast matrix multiplication algorithm and takes O(k 3:5 n 3 m :5 log(nDU )) time for the multicommodity flow problem with inte ..."
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Cited by 172 (21 self)
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All previously known algorithms for solving the multicommodity flow problem with capacities are based on linear programming. The best of these algorithms [15] uses a fast matrix multiplication algorithm and takes O(k 3:5 n 3 m :5 log(nDU )) time for the multicommodity flow problem with integer demands and at least O(k 2:5 n 2 m :5 log(nffl \Gamma1 DU )) time to find an approximate solution, where k is the number of commodities, n and m denote the number of nodes and edges in the network, D is the largest demand, and U is the largest edge capacity. Substantially more time is needed to find an exact solution. As a consequence, even multicommodity flow problems with just a few commodities are believed to be much harder than singlecommodity maximumflow or minimumcost flow problems. In this paper, we describe the first polynomialtime combinatorial algorithms for approximately solving the multicommodity flow problem. The running time of our randomized algorithm i...
Minimum Interference Routing with Applications to MPLS Traffic Engineering
, 2000
"... This paper presents a new algorithm for dynamic routing of bandwidth guaranteed tunnels where tunnel routing requests arrive onebyone and there is no a priori knowledge regarding future requests. This problem is motivated by service provider needs for fast deployment of bandwidth guaranteed servic ..."
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Cited by 143 (4 self)
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This paper presents a new algorithm for dynamic routing of bandwidth guaranteed tunnels where tunnel routing requests arrive onebyone and there is no a priori knowledge regarding future requests. This problem is motivated by service provider needs for fast deployment of bandwidth guaranteed services and the consequent need in backbone networks for fast provisioning of bandwidth guaranteed paths. Offline routing algorithms cannot be used since they require a priori knowledge of all tunnel requests that are to be routed. Instead, online algorithms that handle requests arriving onebyone and that satisfy as many potential future demands as possible are needed. The newly developed algorithm is an online algorithm and is based on the idea that a newly routed tunnel must follow a route that does not "interfere too much" with a route that may be critical to satisfy a future demand. We show that this problem is NPhard. We then develop a path selection heuristic that is based on the idea ...
A FASTER STRONGLY POLYNOMIAL MINIMUM COST FLOW ALGORITHM
, 1991
"... In this paper, we present a new strongly polynomial time algorithm for the minimum cost flow problem, based on a refinement of the EdmondsKarp scaling technique. Our algorithm solves the uncapacitated minimum cost flow problem as a sequence of O(n log n) shortest path problems on networks with n no ..."
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Cited by 116 (10 self)
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In this paper, we present a new strongly polynomial time algorithm for the minimum cost flow problem, based on a refinement of the EdmondsKarp scaling technique. Our algorithm solves the uncapacitated minimum cost flow problem as a sequence of O(n log n) shortest path problems on networks with n nodes and m arcs and runs in O(n log n (m + n log n)) time. Using a standard transformation, thjis approach yields an O(m log n (m + n log n)) algorithm for the capacitated minimum cost flow problem. This algorithm improves the best previous strongly polynomial time algorithm, due to Z. Galil and E. Tardos, by a factor of n 2 /m. Our algorithm for the capacitated minimum cost flow problem is even more efficient if the number of arcs with finite upper bounds, say n', is much less than m. In this case, the running time of the algorithm is O((m ' + n)log n(m + n log n)).
Faster approximation algorithms for the unit capacity concurrent flow problem with applications to routing and finding sparse cuts
 SIAM Journal on Computing
, 1994
"... Abstract. This paper describes new algorithms for approximately solving the concurrent multicommodity flow problem with uniform capacities. These algorithms are much faster than algorithms discovered previously. Besides being an important problem in its own right, the uniformcapacity concurrent flo ..."
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Cited by 84 (20 self)
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Abstract. This paper describes new algorithms for approximately solving the concurrent multicommodity flow problem with uniform capacities. These algorithms are much faster than algorithms discovered previously. Besides being an important problem in its own right, the uniformcapacity concurrent flow problem has many interesting applications. Leighton and Rao used uniformcapacity concurrent flow to find an approximately "sparsest cut " in a graph and thereby approximately solve a wide variety of graph problems, including minimum feedback arc set, minimum cut linear arrangement, and minimum area layout. However, their method appeared to be impractical as it required solving a large linear program. This paper shows that their method might be practical by giving an O(m log m) expectedtime randomized algorithm for their concurrent flow problem on an medge graph. Raghavan and Thompson used uniformcapacity concurrent flow to solve approximately a channel width minimization problem in very large scale integration. An O (k 3/2 (m + n log n)) expectedtime randomized algorithm and an O (k min {n, k} (m + n log n) log k) deterministic algorithm is given for this problem when the channel width is f2 (log n), where k denotes the number of wires to be routed in an nnode, medge network. Key words, multicommodity flow, approximation, concurrent flow, graph separators, VLSI routing AMS subject classification. 68Q25, 90C08, 90C27 1. Introduction. The
Auction algorithms for network flow problems: A tutorial introduction
 Comput. Optim. Appl
, 1992
"... by ..."
A Simple LocalControl Approximation Algorithm for Multicommodity Flow
 In Proceedings of the 34th Annual Symposium on Foundations of Computer Science
, 1993
"... In this paper, we describe a very simple (1 + ") approximation algorithm for the multicommodity flow problem. The algorithm runs in time that is polynomial in N (the number of nodes in the network) and ffl \Gamma1 (the closeness of the approximation to optimal). The algorithm is remarkable in th ..."
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Cited by 65 (6 self)
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In this paper, we describe a very simple (1 + ") approximation algorithm for the multicommodity flow problem. The algorithm runs in time that is polynomial in N (the number of nodes in the network) and ffl \Gamma1 (the closeness of the approximation to optimal). The algorithm is remarkable in that it is much simpler than all known polynomial time flow algorithms (including algorithms for the special case of onecommodity flow). In particular, the algorithm does not rely on augmenting paths, shortest paths, mincost paths, or similar techniques to push flow through a network. In fact, no explicit attempt is ever made to push flow towards a sink during the algorithm. Because the algorithm is so simple, it can be applied to a variety of problems for which centralized decision making and flow planning is not possible. For example, the algorithm can be easily implemented with local control in a distributed network and it can be made tolerant to link failures. In addition, the algorithm ...
Minimum Interference Routing of Bandwidth Guaranteed Tunnels with MPLS Traffic Engineering Application
 IEEE Journal on Selected Areas in Communications
, 2000
"... Abstract—This paper presents new algorithms for dynamic routing of bandwidth guaranteed tunnels, where tunnel routing requests arrive one by one and there is no a priori knowledge regarding future requests. This problem is motivated by service provider needs for fast deployment of bandwidth guarante ..."
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Cited by 65 (4 self)
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Abstract—This paper presents new algorithms for dynamic routing of bandwidth guaranteed tunnels, where tunnel routing requests arrive one by one and there is no a priori knowledge regarding future requests. This problem is motivated by service provider needs for fast deployment of bandwidth guaranteed services. Offline routing algorithms cannot be used since they require a priori knowledge of all tunnel requests that are to be routed. Instead, online algorithms that handle requests arriving one by one and that satisfy as many potential future demands as possible are needed. The newly developed algorithms are online algorithms and are based on the idea that a newly routed tunnel must follow a route that does not “interfere too much ” with a route that may be critical to satisfy a future demand. We show that this problem is NPhard. We then develop path selection heuristics which are based on the idea of deferred loading of certain “critical ” links. These critical links are identified by the algorithm as links that, if heavily loaded, would make it impossible to satisfy future demands between certain ingress–egress pairs. Like minhop routing, the presented algorithm uses linkstate information and some auxiliary capacity information for path selection. Unlike previous algorithms, the proposed algorithm exploits any available knowledge of the network ingress–egress points of potential future demands, even though the demands themselves are unknown. If all nodes are ingress–egress nodes, the algorithm can still be used, particularly to reduce the rejection rate of requests between a specified subset of important ingress–egress pairs. The algorithm performs well in comparison to previously proposed algorithms on several metrics like the number of rejected demands and successful rerouting of demands upon link failure. Index Terms—Maximum flow, MPLS, optimization, quality of service routing, traffic engineering.
New scaling algorithms for the assignment and minimum mean cycle problems
, 1992
"... In this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems. Our assignment algorithm is based on applying scaling to a hybrid version of the recent auction algorithm of Bertsekas and the successive shortest path algorithm. The algorithm proceeds by relaxing th ..."
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Cited by 50 (4 self)
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In this paper we suggest new scaling algorithms for the assignment and minimum mean cycle problems. Our assignment algorithm is based on applying scaling to a hybrid version of the recent auction algorithm of Bertsekas and the successive shortest path algorithm. The algorithm proceeds by relaxing the optimality conditions, and the amount of relaxation is successively reduced to zero. On a network with 2n nodes, m arcs, and integer arc costs bounded by C, the algorithm runs in O(,/n m log(nC)) time and uses very simple data structures. This time bound is comparable to the time taken by Gabow and Tarjan's scaling algorithm, and is better than all other time bounds under the similarity assumption, i.e., C = O(n k) for some k. We next consider the minimum mean cycle problem. The mean cost of a cycle is defined as the cost of the cycle divided by the number of arcs it contains. The minimum mean cycle problem is to identify a cycle whose mean cost is minimum. We show that by using ideas of the assignment algorithm in an approximate binary search procedure, the minimum mean cycle problem can also be solved in O(~/n m log nC) time. Under the similarity assumption, this is the best available time bound to solve the minimum mean cycle problem.
Adding Multiple Cost Constraints to Combinatorial Optimization Problems, with Applications to Multicommodity Flows
 IN PROCEEDINGS OF THE 27TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING
, 1995
"... Minimum cost multicommodity flow is an instance of a simpler problem (multicommodity flow) to which a cost constraint has been added. In this paper we present a general scheme for solving a large class of such "costadded" problemseven if more than one cost is added. One of the main applicatio ..."
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Cited by 45 (5 self)
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Minimum cost multicommodity flow is an instance of a simpler problem (multicommodity flow) to which a cost constraint has been added. In this paper we present a general scheme for solving a large class of such "costadded" problemseven if more than one cost is added. One of the main applications of this method is a new deterministic algorithm for approximately solving the minimumcost multicommodity flow problem. Our algorithm finds a (1 + ffl) approximation to the minimum cost flow in ~ O(ffl \Gamma3 kmn) time, where k is the number of commodities, m is the number of edges, and n is the number vertices in the input problem. This improves the previous best deterministic bounds of O(ffl \Gamma4 kmn 2 ) [9] and ~ O(ffl \Gamma2 k 2 m 2 ) [15] by factors of n=ffl and fflkm=n respectively. In fact, it even dominates the best randomized bound of ~ O(ffl \Gamma2 km 2 ) [15]. The algorithm presented in this paper efficiently solves several other interesting generali...