This paper presents a new method and a constraint-based objective function to solve two problemsrelated to the design of optical telecommunication networks, namely the Synchronous Optical Net-work Ring Assignment Problem (SRAP) and the Intra-ring Synchronous Optical Network DesignProblem (IDP). These network topology problems can be represented as a graph partitioning withcapacity constraints as shown in previous works. We present here a new objective function and anew local search algorithm to solve these problems. Experiments conducted in C
Y. Lee, H. Sherali, J. Han and S. Kim in 2000 ( @cite_7 ), have studied the IDP problem with an additional constraint such that for each ring @math , @math . The authors present a mixed-integer programming model for the problem, and develop a branch-and-cut algorithm. They also introduce a heuristic to generate an initial feasible solution, and another one to improve the initial solution. To initialize a ring, the heuristic first, adds the node @math with the maximum graph degree, with respect to unassigned edges, and then adds to the partition the edge @math such that the graph degree of @math is maximum. It iteratively increases the partition by choosing a node such that the total traffic does not exceed the limit @math . A set of 40 instances is generated to test these heuristics and the branch-and-cut.
@cite_7
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We study computationally hard combinatorial problems arising from the important engineering question of how to maximize the number of connections that can be simultaneously served in a wavelength division ultiplexing (WDM) optical network. In such networks, WDM technology can satisfy a set of connections by computing a route and assigning a wavelength to each connection so that no two connections routed through the same fiber are assigned the same wavelength. Each fiber supports a limited number of @math wavelengths and in order to fully exploit the parallelism provided by the technology, one should select a set of connections of maximum cardinality which can be satisfied using the available wavelengths. This is known as the maximum routing and path coloring (maxRPC) problem. Our main contribution is a general analysis method for a class of iterative algorithms for a more general coloring problem. A lower bound on the benefit of the solution computed by such an algorithm in terms of the optimal benefit and the number of available wavelengths is given by a benefit-revealing linear program. We apply this method to maxRPC in both undirected and bidirected rings to obtain upper bounds on the approximation ratio of several algorithms. The best bounds obtained are @math and @math for undirected and bidirected rings; these results improve known bounds of @math and @math , respectively. Our results extend to multifiber ring networks and also apply to the maximum path coloring (maxPC) problem, where paths instead of connection requests are given as part of the input. We also study the profit version of maxPC in rings where each path has a profit and the objective is to satisfy a set of paths of maximum total profit. We present an algorithm based on linear programming and randomized rounding with approximation ratio 1.49015, improving on the @math bound obtained by a simple iterative algorithm. This paper addresses the problem of finding the minimum number of vehicles required to visit a set of nodes subject to time window and capacity constraints. The fleet is homogeneous and is located at a common depot. Each node requires the same type of service. An exact method is introduced based on branch and cut. In the computations, ever increasing lower bounds on the optimal solution are obtained by solving a series of relaxed problems that incorporate newly found valid inequalities. Feasible solutions or upper bounds are obtained with the help of greedy randomized adaptive search procedure (GRASP). A wide variety of cuts is introduced to tighten the linear programming (LP) relaxation of the original mixed-integer program. To find violated cuts, it is necessary to solve a separation problem. A substantial portion of the paper is aimed at describing the heuristics developed for this purpose. A new approach for obtaining feasible solutions from the LP relaxation is also discussed. Numerical results for standard 50- and 100-node benchmark problems are reported. In this paper, we deal with a network design problem arising from the deployment of synchronous optical networks (SONET), a standard of transmission using optical fiber technology. The problem is to find an optimal clustering of traffic demands in the network such that the total number of node assignments (and, hence, add-drop multiplexer equipment requirements) is minimized, while satisfying the ring capacity and node cardinality constraints. This problem can be conceptualized as an edge-capacitated graph partitioning problem with node cardinality constraints. We formulate the problem as a mixed-integer programming model and develop a new branch-and-cut algorithm along with preprocessing routines for optimally solving the problem. We also prescribe an effective heuristic procedure. Promising computational results are obtained using the proposed method. We consider the problem of interconnecting a set of customer sites using bidirectional SONET rings of equal capacity. Each site is assigned to exactly one ring and a special ring, called the federal ring, interconnects the other rings together. The objective is to minimize the total cost of the network subject to a ring capacity limit where the capacity of a ring is determined by the total bandwidth required between sites assigned to the same ring plus the total bandwidth request between these sites and sites assigned to other rings.We present exact, integer-programming based solution techniques and fast heuristic algorithms for this problem. We compare the results from applying the heuristic algorithms with those produced by the exact methods for real-world as well as randomly generated problem instances. We show that two of the heuristics find solutions that cost at most twice that of an optimal solution. Empirical evidence indicates that in practice the algorithms perform much better than their theoretical bound and often find optimal solutions. We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k ( ) ( ) ;, where k is the alphabet size of I, @math , and @math denotes the number of eigenvalues larger than @math in the normalized adjacency matrix of the constraint graph of @math . In the case that @math is a instance, the threshold @math is only a polynomial in @math , and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the @math constraints specified by the @math level of the Lasserre hierarchy, and in some cases @math rounds of our program can be evaluated in time @math . We study a fingerprinting game in which the number of colluders and the collusion channel are unknown. The encoder embeds fingerprints into a host sequence and provides the decoder with the capability to trace back pirated copies to the colluders. Fingerprinting capacity has recently been derived as the limit value of a sequence of maximin games with mutual information as their payoff functions. However, these games generally do not admit saddle-point solutions and are very hard to solve numerically. Here under the so-called Boneh-Shaw marking assumption, we reformulate the capacity as the value of a single two-person zero-sum game, and show that it is achieved by a saddle-point solution. If the maximal coalition size is k and the fingerprinting alphabet is binary, we show that capacity decays quadratically with k. Furthermore, we prove rigorously that the asymptotic capacity is 1 (k221n2) and we confirm our earlier conjecture that Tardos' choice of the arcsine distribution asymptotically maximizes the mutual information payoff function while the interleaving attack minimizes it. Along with the asymptotics, numerical solutions to the game for small k are also presented. In the 0-extension problem, we are given a weighted graph with some nodes marked as terminals and a semimetric on the set of terminals. Our goal is to assign the rest of the nodes to terminals so as to minimize the sum, over all edges, of the product of the edge's weight and the distance between the terminals to which its endpoints are assigned. This problem generalizes the multiway cut problem of [SIAM J. Comput. , 23 (1994), pp. 864--894] and is closely related to the metric labeling problem introduced by Kleinberg and Tardos [Proceedings of the 40th IEEE Annual Symposium on Foundations of Computer Science, New York, 1999, pp. 14--23]. We present approximation algorithms for 0-Extension . In arbitrary graphs, we present a O(log k)-approximation algorithm, k being the number of terminals. We also give O(1)-approximation guarantees for weighted planar graphs. Our results are based on a natural metric relaxation of the problem previously considered by Karzanov [European J. Combin., 19 (1998), pp. 71--101]. It is similar in flavor to the linear programming relaxation of Garg, Vazirani, and Yannakakis [SIAM J. Comput. , 25 (1996), pp. 235--251] for the multicut problem, and similar to relaxations for other graph partitioning problems. We prove that the integrality ratio of the metric relaxation is at least @math for a positive c for infinitely many k. Our results improve some of the results of Kleinberg and Tardos, and they further our understanding on how to use metric relaxations. This article proposes a linear integer programming formulation and several heuristics based on GRASP and path relinking for the antibandwidth problem. In the antibandwidth problem, one is given an undirected graph with n nodes and must label the nodes in a way that each node receives a unique label from the set 1, 2,…,n , such that, among all adjacent node pairs, the minimum difference between the node labels is maximized. Computational results show that only small instances of this problem can be solved exactly (to optimality) with a commercial integer programming solver and that the heuristics find high-quality solutions in much less time than the commercial solver. © 2010 Wiley Periodicals, Inc. NETWORKS, Vol. 58(3), 171–189 2011 © 2011 Wiley Periodicals, Inc. The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weight sum of the cut edges. Here, we introduce a linear-time algorithm to compute near-minimum cuts. Our algorithm is based on cluster contraction using label propagation and Padberg and Rinaldi’s contraction heuristics [SIAM Review, 1991]. We give both sequential and shared-memory parallel implementations of our algorithm. Extensive experiments on both real-world and generated instances show that our algorithm finds the optimal cut on nearly all instances significantly faster than other state-of-the-art exact algorithms, and our error rate is lower than that of other heuristic algorithms. In addition, our parallel algorithm runs a factor 7.5× faster on average when using 32 threads. To further speed up computations, we also give a version of our algorithm that performs random edge contractions as preprocessing. This version achieves a lower running time and better parallel scalability at the expense of a higher error rate. In this paper, we consider the problem of minimizing the completion delay for instantly decodable network coding (IDNC) in wireless multicast and broadcast scenarios. We are interested in this class of network coding due to its numerous benefits, such as low decoding delay, low coding and decoding complexities, and simple receiver requirements. We first extend the IDNC graph, which represents all feasible IDNC coding opportunities, to efficiently operate in both multicast and broadcast scenarios. We then formulate the minimum completion delay problem for IDNC as a stochastic shortest path (SSP) problem. Although finding the optimal policy using SSP is intractable, we use this formulation to draw the theoretical guidelines for the policies that can minimize the completion delay in IDNC. Based on these guidelines, we design a maximum weight clique selection algorithm, which can efficiently reduce the IDNC completion delay in polynomial time. We also design a quadratic-time heuristic clique selection algorithm, which can operate in real-time applications. Simulation results show that our proposed algorithms significantly reduce the IDNC completion delay compared to the random and maximum-rate algorithms, and almost achieve the global optimal completion delay performance over all network codes in broadcast scenarios. Suppose that n processors are arranged in a ring and can communicate only with their immediate neighbors. It is shown that any probabilistic algorithm for 3 coloring the ring must take at least @math rounds, otherwise the probability that all processors are colored legally is less than @math . A similar time bound holds for selecting a maximal independent set. The bound is tight (up to a constant factor) in light of the deterministic algorithms of Cole and Vishkin [Inform, and Control, 70 (1986), pp. 32–53] and extends the lower bound for deterministic algorithms of Linial [Proc. 28th IEEE Foundations of Computer Science Symposium, 1987, pp. 331–335]. Mulmuley [Mul12a] recently gave an explicit version of Noether’s Normalization lemma for ring of invariants of matrices under simultaneous conjugation, under the conjecture that there are deterministic black-box algorithms for polynomial identity testing (PIT). He argued that this gives evidence that constructing such algorithms for PIT is beyond current techniques. In this work, we show this is not the case. That is, we improve Mulmuley’s reduction and correspondingly weaken the conjecture regarding PIT needed to give explicit Noether Normalization. We then observe that the weaker conjecture has recently been nearly settled by the authors ([FS12]), who gave quasipolynomial size hitting sets for the class of read-once oblivious algebraic branching programs (ROABPs). This gives the desired explicit Noether Normalization unconditionally, up to quasipolynomial factors. As a consequence of our proof we give a deterministic parallel polynomial-time algorithm for deciding if two matrix tuples have intersecting orbit closures, under simultaneous conjugation. We also study the strength of conjectures that Mulmuley requires to obtain similar results as ours. We prove that his conjectures are stronger, in the sense that the computational model he needs PIT algorithms for is equivalent to the well-known algebraic branching program (ABP) model, which is provably stronger than the ROABP model. Finally, we consider the depth-3 diagonal circuit model as defined by Saxena [Sax08], as PIT algorithms for this model also have implications in Mulmuley’s work. Previous work (such as [ASS12] and [FS12]) have given quasipolynomial size hitting sets for this model. In this work, we give a much simpler construction of such hitting sets, using techniques of Shpilka and Volkovich [SV09]. This paper studies a multi-facility network synthesis problem, called the Two-level Network Design (TLND) problem, that arises in the topological design of hierarchical communication, transportation, and electric power distribution networks. We are given an undirected network containing two types of nodes---primary and secondary---and fixed costs for installing either a primary or a secondary facility on each edge. Primary nodes require higher grade interconnections than secondary nodes, using the more expensive primary facilities. The TLND problem seeks a minimum cost connected design that spans all the nodes, and connects primary nodes via edges containing primary facilities; the design can use either primary or secondary edges to connect the secondary nodes. The TLND problem generalizes the well-known Steiner network problem and the hierarchical network design problem. In this paper, we study the relationship between alternative model formulations for this problem (e.g., directed and undirected models), and analyze the worst-case performance for a composite TLND heuristic based upon solving embedded subproblems (e.g., minimum spanning tree and either Steiner tree or shortest path subproblems). When the ratio of primary to secondary costs is the same for all edges and when we solve the embedded subproblems optimally, the worst-case performance ratio of the composite TLND heuristic is 4 3. This result applies to the hierarchical network design problem with constant primary-to-secondary cost ratio since its subproblems are shortest path and minimum spanning tree problems. For more general situations, we express the TLND heuristic worst-case ratio in terms of the performance of any heuristic used to solve the embedded Steiner tree subproblem. A companion paper develops and tests a dual ascent procedure that generates tight upper and lower bounds on the optimal value of a multi-level extension of this problem. LetH=(T,U) be a connected graph,V?Ta set, andca non-negative function on the unordered pairs of elements ofV. In theminimum0-extension problem(*), one is asked to minimize the inner productcmover all metricsmonVsuch that (i)mcoincides with the distance function ofHwithinT; and (ii) eachv?Vis at zero distance from somes?T, i.e.m(v,s)=0. This problem is known to be NP-hard ifH=K3(as being equivalent to the minimum 3-terminal cut problem), while it is polynomially solvable ifH=K2(the minimum cut problem) orH=K2,r(the minimum (2,r)-metric problem). We study problem (*) for all fixedH. More precisely, we consider the linear programming relaxation (**) of (*) that is obtained by dropping condition (ii) above, and callHminimizableif the minima in (*) and (**) coincide for allVandc. Note that for such anHproblem (*) is solvable in strongly polynomial time. Our main theorem asserts thatHis minimizable if and only ifHis bipartite, has no isometric circuit with six or more nodes, and is orientable in the sense thatHcan be oriented so that nonadjacent edges of any 4-circuit are oppositely directed along this circuit. The proof is based on a combinatorial and topological study of tight and extreme extensions of graph metrics. Based on the idea of the proof of the NP-hardness for the minimum 3-terminal cut problem in 4, we then show that the minimum 0-extension problem is strongly NP-hard for many non-minimizable graphsH. Other results are also presented. In this paper, we investigate the problem of Multicast Routing in Sparse Splitting Networks (MR-SSN). Given a network topology with the multicast capable nodes distributed uniformly throughout the network, and a multicast session, the MR-SSN problem is to find a route from the source node of the multicast session to all destinations of the multicast session such that the total number of fibers used in establishing the session is minimized. In this paper, we develop a rerouting algorithm for a given Steiner tree, which makes it feasible to route a multicast session using a tree-based solution in sparse light splitting optical networks. In addition, we present a heuristic based on Tabu Search (TS) that requires only one transmitter for the source node and one wavelength for each multicast session. To evaluate the performance of heuristics, we formulate the MR-SSP problem as an integer linear program (ILP), and optimally solve small instances using the commercially available linear solver, CPLEX. We test our heuristic on a wide range of network topologies. Our experimental results show that: (1) The difference between our solution and ILP optimal solution, in terms of the number of fibers used for establishing a multicast session, is within 10 in almost all the instances and within 5 in about half of the instances. (2) The average delay, taken over all destination nodes, falls within three times the optimal value. (3) A sparse light splitting all-optical network with 30 of multicast capable cross-connects has an acceptable low cost and relatively good performance. (4) The improvement achieved by TS heuristic increases considerably when the session size is large, the number of Splitter-and-Delivery cross-connects is small, and the network connectivity is dense. We consider the maximum independent set problem on sparse graphs with maximum degree d. The best known result for the problem is an SDP based O(d log log d log d) approximation due to Halperin. It is also known that no o(d log2 d) approximation exists assuming the Unique Games Conjecture. We show the following two results: (i) The natural LP formulation for the problem strengthened by O(log4(d)) levels of the mixed-hierarchy has an integrality gap of O(d log2 d), where O(·) ignores some log log d factors. However, our proof is non-constructive, in particular it uses an entropy based approach due to Shearer, and does not give a O(d log2 d) approximation algorithm with sub-exponential running time. (ii) We give an O(d log d) approximation based on polylog(d)-levels of the mixed hierarchy that runs in nO(1) exp(logO(1) d) time, improving upon Halperin's bound by a modest log log d factor. Our algorithm is based on combining Halperin's approach together with an idea used by Ajtai, Erdos, Komlos and Szemeredi to show that Kr-free, degree-d graphs have independent sets of size Ωr(n log log d d). We propose the use of incomplete dot products (IDP) to dynamically adjust the number of input channels used in each layer of a convolutional neural network during feedforward inference. IDP adds monotonically non-increasing coefficients, referred to as a "profile", to the channels during training. The profile orders the contribution of each channel in non-increasing order. At inference time, the number of channels used can be dynamically adjusted to trade off accuracy for lowered power consumption and reduced latency by selecting only a beginning subset of channels. This approach allows for a single network to dynamically scale over a computation range, as opposed to training and deploying multiple networks to support different levels of computation scaling. Additionally, we extend the notion to multiple profiles, each optimized for some specific range of computation scaling. We present experiments on the computation and accuracy trade-offs of IDP for popular image classification models and datasets. We demonstrate that, for MNIST and CIFAR-10, IDP reduces computation significantly, e.g., by 75 , without significantly compromising accuracy. We argue that IDP provides a convenient and effective means for devices to lower computation costs dynamically to reflect the current computation budget of the system. For example, VGG-16 with 50 IDP (using only the first 50 of channels) achieves 70 in accuracy on the CIFAR-10 dataset compared to the standard network which achieves only 35 accuracy when using the reduced channel set. As the operation of our fiber-optic backbone networks migrates from interconnected SONET rings to arbitrary mesh topology, traffic grooming on wavelength-division multiplexing (WDM) mesh networks becomes an extremely important research problem. To address this problem, we propose a new generic graph model for traffic grooming in heterogeneous WDM mesh networks. The novelty of our model is that, by only manipulating the edges of the auxiliary graph created by our model and the weights of these edges, our model can achieve various objectives using different grooming policies, while taking into account various constraints such as transceivers, wavelengths, wavelength-conversion capabilities, and grooming capabilities. Based on the auxiliary graph, we develop an integrated traffic-grooming algorithm (IGABAG) and an integrated grooming procedure (INGPROC) which jointly solve several traffic-grooming subproblems by simply applying the shortest-path computation method. Different grooming policies can be represented by different weight-assignment functions, and the performance of these grooming policies are compared under both nonblocking scenario and blocking scenario. The IGABAG can be applied to both static and dynamic traffic grooming. In static grooming, the traffic-selection scheme is key to achieving good network performance. We propose several traffic-selection schemes based on this model and we evaluate their performance for different network topologies. We consider requests for capacity in a given tree network T e (V, E) where each edge e of the tree has some integer capacity ue. Each request f is a node pair with an integer demand df and a profit wf which is obtained if the request is satisfied. The objective is to find a set of demands that can be feasibly routed in the tree and which provides a maximum profit. This generalizes well-known problems, including the knapsack and b-matching problems. When all demands are 1, we have the integer multicommodity flow problem. [1997] had shown that this problem is NP-hard and gave a 2-approximation algorithm for the cardinality case (all profits are 1) via a primal-dual algorithm. Our main result establishes that the integrality gap of the natural linear programming relaxation is at most 4 for the case of arbitrary profits. Our proof is based on coloring paths on trees and this has other applications for wavelength assignment in optical network routing. We then consider the problem with arbitrary demands. When the maximum demand dmax is at most the minimum edge capacity umin, we show that the integrality gap of the LP is at most 48. This result is obtained by showing that the integrality gap for the demand version of such a problem is at most 11.542 times that for the unit-demand case. We use techniques of Kolliopoulos and Stein [2004, 2001] to obtain this. We also obtain, via this method, improved algorithms for line and ring networks. Applications and connections to other combinatorial problems are discussed. We study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of @math , a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the @math bound.Our main result is a @math -approximation online algorithm for stochastic bipartite matching, breaking this @math barrier. Furthermore, we show that no online algorithm can produce a @math approximation for an arbitrarily small @math for this problem. Our algorithms are based on computing an optimal offline solution to the expected instance, and using this solution as a guideline in the process of online allocation. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-) matchings. In addition to guiding the online decision making, these two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution. At the end, we discuss extensions of our results to more general bipartite allocations that are important in a display ad application.