We present randomized approximation algorithms for the maximum cut (MAX CUT) and maximum 2-satisfiability (MAX 2SAT) problems that always deliver solutions of expected value at least .87856 times the optimal value. These algorithms use a simple and elegant technique that randomly rounds the solution to a nonlinear programming relaxation. This relaxation can be interpreted both as a semidefinite program and as an eigenvalue minimization problem. The best previously known approximation algorithms for these problems had performance guarantees of ...

This paper presents fast algorithms that find approximate solutions for a general class of problems, which we call fractional packing and covering problems. The only previously known algorithms for solving these problems are based on general linear programming techniques. The techniques developed in this paper greatly outperform the general methods in many applications, and are extensions of a method previously applied to find approximate solutions to multicommodity flow problems. Our algorithm is a Lagrangean relaxation technique; an important aspect of our results is that we obtain a theoretical analysis of the running time of a Lagrangean relaxation-based algorithm. We give several applications of our algorithms. The new approach yields several orders of magnitude of improvement over the best previously known running times for algorithms for the scheduling of unrelated parallel machines in both the preemptive and the non-preemptive models, for the job shop problem, for th...

We present a factor 2 approximation algorithm for finding a minimum-cost subgraph having at least a specified number of edges in each cut. This class of problems includes, among others, the generalized Steiner network problem, which is also known as the survivable network design problem. Our algorithm first solves the linear relaxation of this problem, and then iteratively rounds off the solution. The key idea in rounding off is that in a basic solution of the LP relaxation, at least one edge gets included at least to the extent of half. We include this edge into our integral solution and solve the residual problem. 1 Introduction We consider the problem of finding a minimum-cost subgraph of a given graph such that the number of edges crossing each cut is at least a specified requirement. Formally, given an undirected multigraph G = (V; E), a non-negative cost function c : E ! Q+ , and a requirement function f : 2 V ! Z , solve the following integer program (IP): min X e2E c e x...

We describe a randomized approximation algorithm which takes an instance of MAX 3SAT as input. If the instance---a collection of clauses each of length at most three---is satisfiable, then the expected weight of the assignment found is at least 7=8 of optimal. We provide strong evidence (but not a proof) that the algorithm performs equally well on arbitrary MAX 3SAT instances. Our algorithm uses semidefinite programming and may be seen as a sequel to the MAXCUT algorithm of Goemans and Williamson and the MAX 2SAT algorithm of Feige and Goemans. Though the algorithm itself is fairly simple, its analysis is quite complicated as it involves the computation of volumes of spherical tetrahedra. Hastad has recently shown that, assuming P 6= NP , no polynomial-time algorithm for MAX 3SAT can achieve a performance ratio exceeding 7=8, even when restricted to satisfiable instances of the problem. Our algorithm is therefore optimal in this sense. We also describe a method of obtaining direct semi...

This paper presents a comprehensive survey of existing techniques for interconnect optimization during the VLSI physical design process, with emphasis on recent studies on interconnect design and optimization for high-performance VLSI circuit design under the deep submicron fabrication technologies. First, we present a number of interconnect delay models and driver/gate delay models of various degrees of accuracy and efficiency which are most useful to guide the circuit design and interconnect optimization process. Then, we classify the existing work on optimization of VLSI interconnect into the following three categories and discuss the results in each category in detail: (i) topology optimization for highperformance interconnects, including the algorithms for total wire length minimization, critical path length minimization, and delay minimization; (ii) device and interconnect sizing, including techniques for efficient driver, gate, and transistor sizing, optimal wire sizing, and simultaneous topology construction, buffer insertion, buffer and wire sizing; (iii) highperfbrmance clock routing, including abstract clock net topology generation and embedding, planar clock routing, buffer and wire sizing for clock nets, non-tree clock routing, and clock schedule optimization. For each method, we discuss its effectiveness, its advantages and limitations, as well as its computational efficiency. We group the related techniques according to either their optimization techniques or optimization objectives so that the reader can easily compare the quality and efficiency of different solutions.

this paper. Given the MOS circuit topology, the delay can be controlled byvarying the sizes of transistors in the circuit. Here, the size of a transistor is measured in terms of its channel width, since the channel lengths in a digital circuit are generally uniform. Roughly speaking, the sizes of certain transistors can be increased to reduce the circuit delay at the expense of additional chip area

In the last decade important relations between Laplace eigenvalues and eigenvectors of graphs and several other graph parameters were discovered. In these notes we present some of these results and discuss their consequences. Attention is given to the partition and the isoperimetric properties of graphs, the max-cut problem and its relation to semidefinite programming, rapid mixing of Markov chains, and to extensions of the results to infinite graphs.

We consider a class of network design problems in which one needs to find a minimum-cost network satisfying certain connectivity requirements. For example, in the survivable network design problem, the requirements specify that there should be at least r(v; w) edge-disjoint paths between each pair of vertices v and w. We present an approximation algorithm with a performance guarantee of 2H(fmax ) = 2(1 + 2 + 3 + \Delta \Delta \Delta + fmax ) where fmax is the maximum requirement. This improves upon the best previously known performance guarantee of 2fmax . We also show

Abstract. In an online convex optimization problem a decision-maker makes a sequence of decisions, i.e., choose a sequence of points in Euclidean space, from a fixed feasible set. After each point is chosen, it encounters an sequence of (possibly unrelated) convex cost functions. Zinkevich [Zin03] introduced this framework, which models many natural repeated decision-making problems and generalizes many existing problems such as Prediction from Expert Advice and Cover’s Universal Portfolios. Zinkevich showed that a simple online gradient descent algorithm achieves additive regret O ( √ T), for an arbitrary sequence of T convex cost functions (of bounded gradients), with respect to the best single decision in hindsight. In this paper, we give algorithms that achieve regret O(log(T)) for an arbitrary sequence of strictly convex functions (with bounded first and second derivatives). This mirrors what has been done for the special cases of prediction from expert advice by Kivinen and Warmuth [KW99], and Universal Portfolios by Cover [Cov91]. We propose several algorithms achieving logarithmic regret, which besides being more general are also much more efficient to implement. The main new ideas give rise to an efficient algorithm based on the Newton method for optimization, a new tool in the field. Our analysis shows a surprising connection to follow-the-leader method, and builds on the recent work of Agarwal and Hazan [AH05]. We also analyze other algorithms, which tie together several different previous approaches including follow-the-leader, exponential weighting, Cover’s algorithm and gradient descent. 1

This paper addresses the familiar problem of predicting with a linear classifier . The