Abstract. This paper seeks to bridge the two major algorithmic approaches to sparse signal recovery from an incomplete set of linear measurements – L1-minimization methods and iterative methods (Matching Pursuits). We find a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L1-minimization. Our algorithm ROMP reconstructs a sparse signal in a number of iterations linear in the sparsity, and the reconstruction is exact provided the linear measurements satisfy the Uniform Uncertainty Principle. 1.

We advance significantly beyond the recent progress on the algorithmic complexity of Nash equilibria by solving two major open problems in the approximation of Nash equilibria and in the smoothed analysis of algorithms. • We show that no algorithm with complexity poly(n, 1 ɛ) can compute an ɛ-approximate Nash equilibrium in a two-player game, in which each player has n pure strategies, unless PPAD ⊆ P. In other words, the problem of computing a Nash equilibrium in a twoplayer game does not have a fully polynomial-time approximation scheme unless PPAD ⊆ P. • We prove that no algorithm for computing a Nash equilibrium in a two-player game can have smoothed complexity poly(n, 1 σ) under input perturbation of magnitude σ, unless PPAD ⊆ RP. In particular, the smoothed complexity of the classic Lemke-Howson algorithm is not polynomial unless PPAD ⊆ RP. Instrumental to our proof, we introduce a new discrete fixed-point problem on a high-dimensional hypergrid with constant side-length, and show that it can host the embedding of the proof structure of any PPAD problem. We prove a key geometric lemma for finding a discrete fixed-point, a new concept defined on n +1vertices of a unit hypercube. This lemma enables us to overcome the curse of dimensionality in reasoning about fixed-points in high dimensions. 1

The Border Gateway Protocol (BGP) for interdomain routing is designed to allow autonomous systems (ASes) to express policy preferences over alternative routes. We model these preferences as arising from an AS's underlying utility for each route and study the problem of finding a set of routes that maximizes the overall welfare (i.e., the sum of all ASes' utilities for their selected routes).

We investigate variants of Lloyd’s heuristic for clustering high dimensional data in an attempt to explain its popularity (a half century after its introduction) among practitioners, and in order to suggest improvements in its application. We propose and justify a clusterability criterion for data sets. We present variants of Lloyd’s heuristic that quickly lead to provably near-optimal clustering solutions when applied to well-clusterable instances. This is the first performance guarantee for a variant of Lloyd’s heuristic. The provision of a guarantee on output quality does not come at the expense of speed: some of our algorithms are candidates for being faster in practice than currently used variants of Lloyd’s method. In addition, our other algorithms are faster on well-clusterable instances than recently proposed approximation algorithms, while maintaining similar guarantees on clustering quality. Our main algorithmic contribution is a novel probabilistic seeding process for the starting configuration of a Lloyd-type iteration. 1.

Hashing is fundamental to many algorithms and data structures widely used in practice. For theoretical analysis of hashing, there have been two main approaches. First, one can assume that the hash function is truly random, mapping each data item independently and uniformly to the range. This idealized model is unrealistic because a truly random hash function requires an exponential number of bits to describe. Alternatively, one can provide rigorous bounds on performance when explicit families of hash functions are used, such as 2-universal or O(1)-wise independent families. For such families, performance guarantees are often noticeably weaker than for ideal hashing. In practice, however, it is commonly observed that weak hash functions, including 2-universal hash functions, perform as predicted by the idealized analysis for truly random hash functions. In this paper, we try to explain this phenomenon. We demonstrate that the strong performance of universal hash functions in practice can arise naturally from a combination of the randomness of the hash function and the data. Specifically, following the large body of literature on random sources and randomness extraction, we model the data as coming from a “block source, ” whereby

Artificial Intelligence, to appear Maximum Boolean satisfiability (max-SAT) is the optimization counterpart of Boolean satisfiability (SAT), in which a variable assignment is sought to satisfy the maximum number of clauses in a Boolean formula. A branch and bound algorithm based on the Davis-Putnam-Logemann-Loveland procedure (DPLL) is one of the most competitive exact algorithms for solving max-SAT. In this paper, we propose and investigate a number of strategies for max-SAT. The first strategy is a set of unit propagation or unit resolution rules for max-SAT. We summarize three existing unit propagation rules and propose a new one based on a nonlinear programming formulation of max-SAT. The second strategy is an effective lower bound based on linear programming (LP). We show that the LP lower bound can be made effective as the number of clauses increases. The third strategy consists of a a binary-clause first rule and a dynamicweighting variable ordering rule, which are motivated by a thorough analysis of two existing well-known variable orderings. Based on the analysis of these strategies, we develop an exact solver for both max-SAT and weighted max-SAT. Our experimental results on random problem instances and many instances from the max-SAT libraries show that our new solver outperforms most of the existing exact max-SAT solvers, with orders of magnitude of improvement in many cases.

We present the first average-case analysis proving a polynomial upper bound on the expected running time of an exact algorithm for the 0/1 knapsack problem. In particular, we prove for various input distributions, that the number of Pareto-optimal knapsack fillings is polynomially bounded in the number of available items. An algorithm by Nemhauser and Ullmann can enumerate these solutions very efficiently so that a polynomial upper bound on the number of Pareto-optimal solutions implies an algorithm with expected polynomial running time. The random input model underlying our analysis is quite general and not restricted to a particular input distribution. We assume adversarial weights and randomly drawn profits (or vice versa). Our analysis covers general probability distributions with finite mean and, in its most general form, can even handle different probability distributions for the profits of different items. This feature enables us to study the effects of correlations between profits and weights. Our analysis confirms and explains practical studies showing that so-called strongly correlated instances are harder to solve than weakly correlated ones.

We prove that Bimatrix, the problem of finding a Nash equilibrium in a two-player game, is complete for the complexity class PPAD (Polynomial Parity Argument, Directed version) introduced by Papadimitriou in 1991. Our result, building upon the work of Daskalakis, Goldberg, and Papadimitriou on the complexity of four-player Nash equilibria [21], settles a long standing open problem in algorithmic game theory. It also serves as a starting point for a series of results concerning the complexity of two-player Nash equilibria. In particular, we prove the following theorems: • Bimatrix does not have a fully polynomial-time approximation scheme unless every problem in PPAD is solvable in polynomial time. • The smoothed complexity of the classic Lemke-Howson algorithm and, in fact, of any algorithm for Bimatrix is not polynomial unless every problem in PPAD is solvable in randomized polynomial time. Our results also have a complexity implication in mathematical economics: • Arrow-Debreu market equilibria are PPAD-hard to compute.

Smoothed analysis combines elements over worst-case and average case analysis. For an instance x the smoothed complexity is the average complexity of an instance obtained from x by a perturbation. The smoothed complexity of a problem is the worst smoothed complexity of any instance. Spielman and Teng introduced this notion for continuous problems. We apply the concept to combinatorial problems and study the smoothed complexity of three classical discrete problems: quicksort, left-to-right maxima counting, and shortest paths. This opens a vast eld of nice analyses (using for example generating functions in the discrete case) which should lead to a better understanding of complexity landscapes of algorithms.

We present a probabilistic analysis for a large class of combinatorial optimization problems containing, e.g., all binary optimization problems defined by linear constraints and a linear objective function over {0,1} n. By parameterizing which constraints are of stochastic and which are of adversarial nature, we obtain a semirandom input model that enables us to do a general average-case analysis for a large class of optimization problems while at the same time taking care for the combinatorial structure of individual problems. Our analysis covers various probability distributions for the choice of the stochastic numbers and includes smoothed analysis with Gaussian and other kinds of perturbation models as a special case. In fact, we can exactly characterize the smoothed complexity of optimization problems in terms of their random worstcase complexity. A binary optimization problem has a polynomial smoothed