We present fully polynomial approximation schemes for concurrent multicommodity flow prob-lems that run in time of minimum possible dependency on the number of commodities k. We showthat by modifying the algorithms by Garg & K"onemann [7] and Fleischer [5] we can reduce their running time on a graph with n vertices and m edges from ~O("-2(m2 + km)) to ~O("-2m2) foran implicit representation of the output, or ~ O("-2(m2 + kn)) for an explicit representation, where ~ O(f) denotes a quantity that is O(f logO(1) m). The implicit representation consists of a set oftrees rooted at sources (there can be more than one tree per source), and with sinks as their leaves, together with flow values for the flow directed from the source to the sinks in a particular tree.Given this implicit representation, the approximate value of the concurrent flow is known, but if we want the explicit flow per commodity per edge, we would have to combine all these trees together,and the cost of doing so may be prohibitive. In case we want to calculate explicitly the solution flow, we modify our schemes so that they run in time poly-logarithmic in nk (n is the numberof nodes in the network). This is within a poly-logarithmic factor of the trivial lower bound of time \Omega (nk) needed to explicitly write down a multicommodity flow of k commodities in a network of n nodes. Therefore our schemes are within a poly-logarithmic factor of the minimum possible dependency of the running time on the number of commodities k.

We consider the problem of scheduling n independent jobs on m unrelated parallel machines without preemption. Job i takes processing time pi j on machine j, and the total time used by a machine is the sum of the processing times for the jobs assigned to it. The objective is to minimize makespan. The best known approximation algorithms for this problem compute an optimum fractional solution and then use rounding techniques to get an integral 2-approximation. In this paper we present a combinatorial approximation algorithm that matches this approximation quality. It is much simpler than the previously known algorithms and its running time is better. This is the first time that a combinatorial algorithm always beats the interior point approach for this problem. Our algorithm is a generic minimum cost flow algorithm, without any complex enhancements, tailored to handle unsplittable flow. It pushes unsplittable jobs through a two-layered bipartite generalized network defined by the scheduling problem. In our analysis, we take advantage from addressing the approximation problem directly. In particular, we replace the classical technique of solving the LP-relaxation and rounding afterwards by a completely integral approach. We feel that this approach will be helpful also for other applications.

In this paper we address the problem of minimizing the response time of a multi-way join query using pipelined (inter-operator) parallelism, in a parallel or a distributed environment. We observe that in order to fully exploit the parallelism in the system, we must consider a new class of “interleaving” plans, where multiple query plans are used simultaneously to minimize the response time of a query (or maximize the tuple-throughput of the system). We cast the query planning problem in this environment as a “flow maximization problem”, and present polynomial-time algorithms that (statically) find the optimal set of plans to use for a large class of multi-way join queries. Our proposed algorithms also naturally extend to query optimization over web services. Finally we present an extensive experimental evaluation that demonstrates both the need to consider such plans in parallel query processing and the effectiveness of our proposed algorithms. 1

Pipelined filter ordering is a central problem in database query optimization. The problem is to determine the optimal order in which to apply a given set of commutative filters (predicates) to a set of elements (the tuples of a relation), so as to find, as efficiently as possible, the tuples that satisfy all of the filters. Optimization of pipelined filter ordering has recently received renewed attention in the context of environments such as the web, continuous high-speed data streams, and sensor networks. Pipelined filter ordering problems are also studied in areas such as fault detection and machine learning under names such as learning with attribute costs, minimumsum set cover, and satisficing search. We present algorithms for two natural extensions of the classical pipelined filter ordering problem: (1) a distributional type problem where the filters run in parallel and the goal is to maximize throughput, and (2) an adversarial type problem where the goal is to minimize the expected value of multiplicative regret. We present two related algorithms for solving (1), both running in time O(n²), which improve on the O(n³ log n) algorithm of Kodialam. We use techniques from our algorithms for (1) to obtain an algorithm for (2).

We describe the optimization problem associated with the concurrent routing of demands with guaranteed shared recovery in case of network failures. This problem arises in routing with protection in meshes and is known to be hard. We describe the problem in the context of the efficient design of restorable MPS tunnels in optical networks. The underlying design gives rise to a stochastic optimization problem that is equivalent to a (very) largescale linear programming (LP) problem that explicitly incorporates the network failure scenarios. The feasible region for this LP is given by combined packing and covering constraints for concurrent and optimal multicommodity flows. We develop a novel ffl-approximation procedure for this problem and demonstrate its performance for a variety of real network sizes. An attraction of our approach is that its main computation consists of routing flow along a pair of short paths and these paths are easily found. Commercial general-purpose LP solvers are typically unable to solve these problems once they become large enough, while our approach scales for large networks. We conclude that the proposed scheme provides guaranteed approximation to the design of restorable MPS tunnels with shared protection within realistic network settings.

Abstract — We study the resource allocation problem in OFDMA based 802.16 broadband wireless access systems. Frequency and time resources must be allocated by a central controller (Base Station) to a number of users. We consider variations of a resource allocation problem, some of which are difficult to solve. Situations in which only the objective of the Base Station need to be maximized are easily dealt with as are cases where all the users perceive the same channel conditions. Scenarios where both the objectives of the BS as well as those of the end users must be met simultaneously require more complicated solutions since individual users experience different channel conditions. We present linear programming relaxations for the resource allocation problem. While solving the LP using standard techniques like ellipsoidal algorithm can provide optimal allocations for all users, it can be expensive in terms of computing overhead as the number of users in the system increase. Therefore we present an efficient algorithm which performs well even as the number of clients

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A D-polyhedron is a polyhedron P such that if x, y are in P then so are their componentwise max and min. In other words, the point set of a D-polyhedron forms a distributive lattice with the dominance order. We provide a full characterization of the bounding hyperplanes of D-polyhedra. Aside from being a nice combination of geometric and order theoretic concepts, D-polyhedra are a unifying generalization of several distributive lattices which arise from graphs. In fact every D-polyhedron corresponds to a directed graph with arc-parameters, such that every point in the polyhedron corresponds to a vertex potential on the graph. Alternatively, an edge-based description of the point set can be given. The objects in this model are dual to generalized flows, i.e., dual to flows with gains and losses. These models can be specialized to yield some cases of distributive lattices that have been studied previously. Particular specializations are: lattices of flows of planar digraphs (Khuller, Naor and Klein), of α-orientations of planar graphs (Felsner), of c-orientations (Propp) and of ∆-bonds of digraphs (Felsner and Knauer). As an additional application we exhibit a distributive lattice structure on generalized flow of breakeven planar digraphs. 1

In this paper we relate the problem of finding structures related to perfect matchings in bipartite graphs to a stochastic process similar to throwing balls into bins. Given a bipartite graph with n nodes on each side, we view each node on the left as having balls that it can throw into nodes on the right (bins) to which it is adjacent. If each node on the left throws exactly one ball and each bin on the right gets exactly one ball, then the edges represented by the ball-placement form a perfect matching. Further, if each thrower is allowed to throw a large but equal number of balls, and each bin on the right receives an equal number of balls, then the set of ball-placements corresponds to a perfect fractional matching – a weighted subgraph on all nodes with nonnegative weights on edges so that the total weight incident at each node is 1. We show that several simple algorithms based on throwing balls into bins deliver a near-perfect fractional matching. For example, we show that by iteratively picking a random node on the left and throwing a ball into its least-loaded neighbor, the distribution of balls obtained is no worse than randomly throwing kn balls into n bins. Another algorithm is based on the d-choice load-balancing of balls and bins. By picking a constant number of nodes on the left and appropriately inserting a ball into the least-loaded of their neighbors, we achieve a smoother load distribution on both sides – maximum load is at most log log n / log d + O(1). When each vertex on the left throws k balls, we obtain an algorithm that achieves a load within k ± 1 on the right vertices. 1

We present a fully polynomial-time approximation scheme for a multicommodity flow problem that yields lower bounds of the graph bisection problem. We compare