After several decades of sustained research and testing, linear programming has evolved into a remarkably reliable, accurate and useful tool for handling industrial optimization problems. Yet, large problems arising from several concrete applications routinely defeat the very best linear programming codes, running on the fastest computing hardware. Moreover, this is a trend that may well continue and intensify, as problem sizes escalate and the need for fast algorithms becomes more stringent. Traditionally, the focus in optimization algorithms, and in particular, in algorithms for linear programming, has been to solve problems "to optimality." In concrete implementations, this has always meant the solution ofproblems to some finite accuracy (for example, eight digits). An alternative approach would be to explicitly, and rigorously, trade o# accuracy for speed. One motivating factor is that in many practical applications, quickly obtaining a partially accurate solution is much preferable to obtaining a very accurate solution very slowly. A secondary (and independent) consideration is that the input data in many practical applications has limited accuracy to begin with. During the last ten years, a new body ofresearch has emerged, which seeks to develop provably good approximation algorithms for classes of linear programming problems. This work both has roots in fundamental areas of mathematical programming and is also framed in the context ofthe modern theory ofalgorithms. The result ofthis work has been a family ofalgorithms with solid theoretical foundations and with growing experimental success. In this manuscript we will study these algorithms, starting with some ofthe very earliest examples, and through the latest theoretical and computational developments.

Achieving a global goal based on local information is challenging, especially in complex and large-scale networks such as the Internet or even the human brain. In this paper, we provide an almost tight classification of the possible trade-off between the amount of local information and the quality of the global solution for general covering and packing problems. Specifically, we give a distributed algorithm using only small messages which obtains an (ρ∆) 1/k-approximation for general covering and packing problems in time O(k 2), where ρ depends on the LP’s coefficients. If message size is unbounded, we present a second algorithm that achieves an O(n 1/k) approximation in O(k) rounds. Finally, we prove that these algorithms are close to optimal by giving a lower bound on the approximability of packing problems given that each node has to base its decision on information from its k-neighborhood. 1

Semidefinite programs (SDP) have been used in many recent approximation algorithms. We develop a general primal-dual approach to solve SDPs using a generalization of the well-known multiplicative weights update rule to symmetric matrices. For a number of problems, such as Sparsest Cut and Balanced Separator in undirected and directed weighted graphs, and the Min UnCut problem, this yields combinatorial approximation algorithms that are significantly more efficient than interior point methods. The design of our primal-dual algorithms is guided by a robust analysis of rounding algorithms used to obtain integer solutions from fractional ones. 1

Algorithms in varied fields use the idea of maintaining a distribution over a certain set and use the multiplicative update rule to iteratively change these weights. Their analysis are usually very similar and rely on an exponential potential function. We present a simple meta algorithm that unifies these disparate algorithms and drives them as simple instantiations of the meta algorithm. 1

We describe sequential and parallel algorithms that approximately solve linear programs with no negative coefficients (a.k.a. mixed packing and covering problems). For explicitly given problems, our fastest sequential algorithm returns a solution satisfying all constraints within a ¦ ¯ factor in Ç Ñ � ÐÓ � Ñ � ¯ time, where Ñ is the number of constraints and � is the maximum number of constraints any variable appears in. Our parallel algorithm runs in time polylogarithmic in the input size times ¯   � and uses a total number of operations comparable to the sequential algorithm. The main contribution is that the algorithms solve mixed packing and covering problems (in contrast to pure packing or pure covering problems, which have only “� ” or only “� ” inequalities, but not both) and run in time independent of the so-called width of the problem. 1. Background Packing and covering problems are problems that can be formulated as linear programs using only non-negative coefficients and non-negative variables. Special cases include pure packing problems, which are of the form Ñ�Ü � ¡ Ü � �Ü � � � and pure covering problems, which are of the form Ñ�Ò � ¡ Ü � �Ü � ��. Lagrangian-relaxation algorithms are based on the following basic idea. Given an optimization problem specified as a collection of constraints, modify the problem by selecting some of the constraints and replacing them by a continuous “penalty ” function that, given a partial solution Ü, measures how close Ü is to violating the removed constraints. Construct a solution iteratively in small steps, making each choice to maintain the remaining constraints while minimizing the increase in the penalty function. While Lagrangian-relaxation algorithms have the disadvantage of producing only approximately optimal (or approximately feasible) solutions, the algorithms have the fol-

We present a routing paradigm called PB-routing that utilizes steepest gradient search methods to route data packets. More specifically, the PB-routing paradigm assigns scalar potentials to network elements and forwards packets in the direction of maximum positive force. We show that the family of PB-routing schemes are loop free and that the standard shortest path routing algorithms are a special case of the PB-routing paradigm. We then show how to design a potential function that accounts for traffic conditions at a node. The resulting routing algorithm routes around congested areas while preserving the key desirable properties of IP routing mechanisms including hop-byhop routing, local route computations and statistical multiplexing. Our simulations using the ns simulator indicate that the traffic aware routing algorithm shows significant improvements in end-to-end delay and jitter when compared to standard shortest path routing algorithms. The simulations also indicate that our algorithm does not incur too much control overheads and is fairly stable even when traffic conditions are dynamic.

To implement high-performance global interconnect without impacting the performance of existing blocks, the use of buffer blocks is increasingly popular in structured-custom and block-based ASIC/SOC methodologies. Recent works by Cong et al. [6] and Tang and Wong [25] give algorithms to solve the buffer block planning problem. In this paper we address the problem of how to perform buffering of global nets given an existing buffer block plan. Assuming as in [6, 25] that global nets have been already decomposed into two-pin connections, we give a provably good algorithm based on a recent approach of Garg and Könemann [8] and Fleischer [7]. Our method routes connections using available buffer blocks, such that required upper and lower bounds on buffer intervals – as well as wirelength upper bounds per connection – are satisfied. Unlike [6, 25], our model allows more than one buffer to be inserted into any given connection. In addition, our algorithm observes buffer parity constraints, i.e., it will choose to use an inverter or a buffer ( = co-located pair of inverters) according to source and destination signal parity. The algorithm outperforms previous approaches [6] and has been validated on top-level layouts extracted from a recent high-end microprocessor design. 1

We show that any rational polytope is polynomial-time representable as a “slim ” r × c × 3 three-way line-sum transportation polytope. This universality theorem has important consequences for linear and integer programming and for confidential statistical data disclosure. It provides polynomial-time embedding of arbitrary linear programs and integer programs in such slim transportation programs and in bipartite biflow programs. It resolves several standing problems on 3-way transportation polytopes. It demonstrates that the range of values an entry can attain in any slim 3-way contingency table with specified 2-margins can contain arbitrary gaps, suggesting that disclosure of k-margins of d-tables for 2 ≤ k<dis confidential. Our construction also provides a powerful tool in studying concrete questions about transportation polytopes and contingency tables; remarkably, it enables to automatically recover the famous “real-feasible integerinfeasible” 6×4×3 transportation polytope of M. Vlach, and to produce the first example of 2-margins for 6 × 4 × 3 contingency tables where the range of values a specified entry can attain has a gap.

Semidefinite programming (SDP) relaxations appear in many recent approximation algorithms but the only general technique for solving such SDP relaxations is via interior point methods. We use a Lagrangian-relaxation based technique (modified from the papers of Plotkin, Shmoys, and Tardos (PST), and Klein and Lu) to derive faster algorithms for approximately solving several families of SDP relaxations. The algorithms are based upon some improvements to the PST ideas — which lead to new results even for their framework — as well as improvements in approximate eigenvalue computations by using random sampling. 1.

We consider the problem of finding a lightpath assignment for a given set of communication requests on a multiber wdm optical network with wavelength translators. Given such a network, and w the number of wavelengths available on each ber, k the number of ber per link and c the number of partial wavelength translation available on each node, our problem stands for deciding whether it is possible to nd a w-lightpath for each request in the set such that there is no link carrying more that k lightpaths using the same wavelength nor node where more than c wavelength translations take place. Our main theoretical result is the writing of this problem as a particular instance of integral multicommodity flow, hence integrating routing and wavelength assignment in the same model. We then provide three heuristics mainly based upon randomized rounding of fractional multicommodity flow and enhancements that are three different answers to the trade-off between efficiency and tightness of approximation and discuss their practical performances on both theoretical and real-world instances.