We consider the problem of finding shortest paths in a graph with independent randomly distributed edge lengths. Our goal is to maximize the probability that the path length does not exceed a given threshold value (deadline). We give a surprising exact n Θ(log n) algorithm for the case of normally distributed edge lengths, which is based on quasi-convex maximization. We then prove average and smoothed polynomial bounds for this algorithm, which also translate to average and smoothed bounds for the parametric shortest path problem, and extend to a more general non-convex optimization setting. We also consider a number other edge length distributions, giving a range of exact and approximation schemes.

Abstract—We give the first black-box reduction from arbitrary approximation algorithms to truthful approximation mechanisms for a non-trivial class of multiparameter problems. Specifically, we prove that every packing problem that admits an FPTAS also admits a truthful-in-expectation randomized mechanism that is an FPTAS. Our reduction makes novel use of smoothed analysis, by employing small perturbations as a tool in algorithmic mechanism design. We develop a “duality” between linear perturbations of the objective function of an optimization problem and of its feasible set, and use the “primal ” and “dual ” viewpoints to prove the running time bound and the truthfulness guarantee, respectively, for our mechanism.

Many algorithms and heuristics work well on real data, despite having poor complexity under the standard worst-case measure. Smoothed analysis [36] is a step towards a theory that explains the behavior of algorithms in practice. It is based on the assumption that inputs to algorithms are subject to random perturbation and modification in their formation. A concrete example of such a smoothed analysis is a proof that the simplex algorithm for linear programming usually runs in polynomial time, when its input is subject to modeling or measurement noise. 1. MODELING REAL DATA “My experiences also strongly confirmed my previous opinion that the best theory is inspired by practice and the best practice is inspired by theory. ” [Donald E. Knuth: “Theory and Practice”, Theoretical Computer Science, 90 (1), 1–15, 1991.] Algorithms are high-level descriptions of how computational tasks are performed. Engineers and experimentalists design and implement algorithms, and generally consider them a success if they work in practice. However, an algorithm that works well in one practical domain might perform poorly in another. Theorists also design and analyze algorithms, with the goal of providing provable guarantees about their performance. The traditional goal of theoretical computer science is to prove that an algorithm performs well This material is based upon work supported by the National

Abstract Algorithms based on convex optimization, especially linear and semidefinite programming, are ubiquitous in Computer Science. While there are polynomial time algorithms known to solve such problems, quite often the running time of these algorithms is very high. Designing simpler and more efficient algorithms is important for practical impact. In this thesis, we explore applications of the Multiplicative Weights method in the design of efficient algorithms for various optimization problems. This method, which was repeatedly discovered in quite diverse fields, is an algorithmic technique which maintains a distribution on a certain set of interest, and updates it iteratively by multiplying the probability mass of elements by suitably chosen factors based on feedback obtained by running another algorithm on the distribution. We present a single meta-algorithm which unifies all known applications of this method in a common framework. Next, we generalize the method to the setting of symmetric matrices rather than real numbers. We derive the following applications of the resulting Matrix Multiplicative Weights algorithm: 1. The first truly general, combinatorial, primal-dual method for designing efficient algorithms for semidefinite programming. Using these techniques, we obtain significantly faster algorithms for obtaining O(plog n) approximations to various graph partitioning problems, such as Sparsest Cut, Balanced Separator in both directed and undirected weighted graphs, and constraint satisfaction problems such as Min UnCut and Min 2CNF Deletion. 2. An ~O(n3) time derandomization of the Alon-Roichman construction of expanders using Cayley graphs. The algorithm yields a set of O(log n) elements which generates an expanding Cayley graph in any group of n elements. 3. An ~O(n3) time deterministic O(log n) approximation algorithm for the quantum hypergraph covering problem. 4. An alternative proof of a result of Aaronson that the fl-fat-shattering dimension of quantum states on n qubits is O ( nfl2).

In this paper, we resolve the smoothed and approximative complexity of low-rank quasi-concave minimization, providing both upper and lower bounds. As an upper bound, we provide the first smoothed analysis of quasi-concave minimization. The analysis is based on a smoothed bound for the number of extreme points of the projection of the feasible polytope onto a k-dimensional subspace, where k is the rank (informally, the dimension of nonconvexity) of the quasi-concave function. Our smoothed bound is polynomial in the original dimension of the problem n and the perturbation size ρ, and it is exponential in the rank of the function k. From this, we obtain the first randomized fully polynomialtime approximation scheme for low-rank quasi-concave minimization under broad conditions. In contrast with this, we prove log n-hardness of approximation for general quasi-concave minimization. This shows that our smoothed bound is essentially tight, in that no polynomial smoothed bound is possible for quasi-concave functions of general rank k. The tools that we introduce for the smoothed analysis may be of independent interest. All previous smoothed analyses of polytopes analyzed projections onto two-dimensional subspaces and studied them using trigonometry to examine the angles between vectors and 2-planes in R n. In this paper, we provide what is, to our knowledge, the first smoothed analysis of the projection of polytopes onto higher-dimensional subspaces. To do this, we replace the trigonometry with tools from random matrix theory and differential geometry on the Grassmannian. Our hardness reduction is based on entirely different proofs that may also be of independent interest: we show that the stochastic 2-stage minimum spanning tree problem has a supermodular objective and that su-

In this paper we study the homogeneous conic system F: Ax =0, x ∈ C \{0}. We choose a point ¯s ∈ intC ∗ that serves as a normalizer and consider computational properties of the normalized system F¯s: Ax = 0, ¯s T x =1, x ∈ C. We show that the computational complexity of solving F via an interior-point method depends only on the complexity value ϑ of the barrier for C and on the symmetry of the origin in the image set H¯s: = {Ax: ¯s T x =1, x ∈ C}, where the symmetry of 0 in H¯s is sym(0,H¯s):=max{α: y ∈ H¯s ⇒−αy ∈ H¯s}. We show that a solution of F can be computed in O ( √ ϑ ln(ϑ/sym(0,H¯s)) interior-point iterations. In order to improve the theoretical and practical computation of a solution of F, we next present a general theory for projective re-normalization of the feasible region F¯s and the image set H¯s and prove the existence of a normalizer ¯s such that sym(0,H¯s) ≥ 1/m provided that F has an interior solution. We develop a methodology for constructing a normalizer ¯s such that sym(0,H¯s) ≥ 1/m with high probability, based on sampling on a geometric random walk with associated probabilistic complexity analysis. While such a normalizer is not itself computable in strongly-polynomialtime, the normalizer will yield a conic system that is solvable in O ( √ ϑ ln(mϑ)) iterations, which is strongly-polynomialtime. Finally, we implement this methodology on randomly generated homogeneous linear programming feasibility problems, constructed to be poorly behaved. Our computational results indicate that the projective re-normalization methodology holds the promise to markedly reduce the overall computation time for conic feasibility problems; for instance we observe a 46 % decrease in average IPM iterations for 100 randomly generated poorly-behaved problem instances of dimension 1000 × 5000.

Abstract. The simplex method in Linear Programming motivates several problems of asymptotic convex geometry. We discuss some conjectures and known results in two related directions – computing the size of projections of high dimensional polytopes and estimating the norms of random matrices and their inverses. 1. Asyptotic convex geometry and Linear Programming Linear Programming studies the problem of maximizing a linear functional subject to linear constraints. Given an objective vector z ∈ R d and constraint vectors a1,...,an ∈ R d, we consider the linear program (LP) maximize 〈z, x〉 subject to 〈ai, x 〉 ≤ 1, i = 1,...,n. This linear program has d unknowns, represented by x, and n constraints. Every linear program can be reduced to this form by a simple interpolation argument [36]. The feasible set of the linear program is the polytope P: = {x ∈ R d: 〈ai, x 〉 ≤ 1, i = 1,..., n}. The solution of (LP) is then a vertex of P. We can thus look at (LP) from a geometric viewpoint: for a polytope P in R d given by n faces, and for a vector z, find the vertex that maximizes the linear functional 〈z, x〉. The oldest and still the most popular method to solve this problem is the simplex method. It starts at some vertex of P and generates a walk on the edges of P toward the solution vertex. At each step, a pivot rule determines a choice of the next vertex; so there are many variants of the simplex method with different pivot rules. (We are not concerned here with how to find the initial vertex, which is a nontrivial problem in itself).

The term “Black Box ” often refers to a device whose functionality we understand, but whose inner workings we don’t, or choose to ignore. This term appears a lot in a large variety of contexts within theoretical computer science, and happens to be extremely convenient to capture computations with restricted knowledge about or access to certain information.

Dan Spielman has made groundbreaking contributions in theoretical computer science and mathematical programming and his work has profound connections to the study of polytopes and convex bodies, to error-correcting codes, expanders, and numerical analysis. Many of Spielman’s achievements came with a beautiful collaboration spanned over two decades with Shang-Hua Teng. This paper describes some of Spielman’s main achievements. Section 1 describes smoothed analysis of algorithms, which is a new paradigm for the analysis of algorithms introduced by Spielman and Teng. Section 2 describes Spielman and Teng’s explanation for the excellent practical performance of the simplex algorithm via smoothed analysis. Spielman and Teng’s theorem asserts that the simplex algorithm takes a polynomial number of steps for a random Gaussian perturbation of every linear programming problem. Section 3 is devoted to Spielman’s works on error-correcting codes and in particular his construction of linear-time encodable and decodable high-rate codes based

This article is a survey of developments in algorithmic convex geometry over the past decade. These include algorithms for sampling, optimization, integration, rounding and learning, as well as mathematical tools such as isoperimetric and concentration inequalities. Several open problems and conjectures are discussed on the way.