We introduce a new technique called oblivious rounding --- a variant of randomized rounding that avoids the bottleneck of first solving the linear program. Avoiding this bottleneck yields more efficient algorithms and brings probabilistic methods to bear on a new class of problems. We give oblivious rounding algorithms that approximately solve general packing and covering problems, including a parallel algorithm to find sparse strategies for matrix games.

We significantly improve known time bounds for solving the minimum cut problem on undirected graphs. We use a "semi-duality" between minimum cuts and maximum spanning tree packings combined with our previously developed random sampling techniques. We give a randomized (Monte Carlo) algorithm that finds a minimum cut in an m-edge, n-vertex graph with high probability in O(m log 3 n) time. We also give a simpler randomized algorithm that finds all minimum cuts with high probability in O(n 2 log n) time. This variant has an optimal RNC parallelization. Both variants improve on the previous best time bound of O(n 2 log 3 n). Other applications of the tree-packing approach are new, nearly tight bounds on the number of near minimum cuts a graph may have and a new data structure for representing them in a space-efficient manner. 1 Introduction The minimum cut problem has been studied for many years as a fundamental graph optimization problem with numerous applications. Initially, th...

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

The minimum-cost multicommodity flow problem involves simultaneously shipping multiple commodities through a single network so that the total flow obeys arc capacity constraints and has minimum cost.

We develop an algorithm for resolving a conic linear system (FP d ), which is a system of the form (FP d ): b Ax 2 C Y x 2 CX ; where CX and C Y are closed convex cones, and the data for the system is d = (A; b).

Generalized network flow problems generalize normal network flow problems by specifying a flow multiplier (a) for each arc a. For every unit of flow entering the arc, (a) units of flow exit. Flow multipliers permit modelling transforming one type into another and modification of the amount of flow.

We present a combinatorial polynomial time algorithm to compute a maximum stable set of a t-perfect graph. The algorithm rests on an e-approximation algorithm for general set covering and packing problems and is combinatorial in the sense that it does not use an explicit linear programming algorithm or methods from linear algebra or convex geometry. Instead our algorithm is based on basic arithmetic operations and comparisons of rational numbers which are of polynomial binary encoding size in the input.

This paper describes an NP-hard combinatorial optimization problem arising in the Sloan

We show how to compute O ( √ log n)-approximations to Sparsest Cut and Balanced Separator problems in Õ(n2) time, thus improving upon the recent algorithm of Arora, Rao and Vazirani (2004). Their algorithm uses semidefinite programming and required Õ(n4.5) time. Our algorithm relies on efficiently finding expander flows in the graph and does not solve semidefinite programs. The existence of expander flows was also established by Arora, Rao, and Vazirani. 1

We introduce a new algorithm and a new analysis technique that is applicable to a variety of online optimization scenarios, including regret minimization for Lipschitz regret functions, universal portfolio management, online convex optimization and online utility maximization. In addition to being more efficient and deterministic, our algorithm applies to a more general setting (e.g. when the payoff function is unknown). For the general online game playing setting it is the first to attain logarithmic regret, as opposed to previous algorithms attaining polynomial regret. The algorithm extends a natural online method studied in the 1950’s, called “follow the leader”, thus answering in the affirmative a conjecture about universal portfolios made by Cover and Ordentlich and independently by Kalai and Vempala. The techniques also leads to derandomization of an algorithm by Hannan, and Kalai and Vempala. Our analysis shows a surprising connection between interior point methods and online optimization by using the follow the leader method.