Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas

We investigate the approximability of several classes of real-valued functions by functions of a small number of variables ({\em juntas}). Our main results are tight bounds on the number of variables required to approximate a function $f:\{0,1\}^n \rightarrow [0,1]$ within $\ell_2$-error $\epsilon$ over the uniform distribution: 1. If $f$ is submodular, then it is $\epsilon$-close to a function of $O(\frac{1}{\epsilon^2} \log \frac{1}{\epsilon})$ variables. This is an exponential improvement over previously known results. We note that $\Omega(\frac{1}{\epsilon^2})$ variables are necessary even for linear functions. 2. If $f$ is fractionally subadditive (XOS) it is $\epsilon$-close to a function of $2^{O(1/\epsilon^2)}$ variables. This result holds for all functions with low total $\ell_1$-influence and is a real-valued analogue of Friedgut's theorem for boolean functions. We show that $2^{\Omega(1/\epsilon)}$ variables are necessary even for XOS functions. As applications of these results, we provide learning algorithms over the uniform distribution. For XOS functions, we give a PAC learning algorithm that runs in time $2^{poly(1/\epsilon)} poly(n)$. For submodular functions we give an algorithm in the more demanding PMAC learning model (Balcan and Harvey, 2011) which requires a multiplicative $1+\gamma$ factor approximation with probability at least $1-\epsilon$ over the target distribution. Our uniform distribution algorithm runs in time $2^{poly(1/(\gamma\epsilon))} poly(n)$. This is the first algorithm in the PMAC model that over the uniform distribution can achieve a constant approximation factor arbitrarily close to 1 for all submodular functions. As follows from the lower bounds in (Feldman et al., 2013) both of these algorithms are close to optimal. We also give applications for proper learning, testing and agnostic learning with value queries of these classes.Comment: Extended abstract appears in proceedings of FOCS 201

Set Selection under Explorable Stochastic Uncertainty via Covering Techniques

Given subsets of uncertain values, we study the problem of identifying the subset of minimum total value (sum of the uncertain values) by querying as few values as possible. This set selection problem falls into the field of explorable uncertainty and is of intrinsic importance therein as it implies strong adversarial lower bounds for a wide range of interesting combinatorial problems such as knapsack and matchings. We consider a stochastic problem variant and give algorithms that, in expectation, improve upon these adversarial lower bounds. The key to our results is to prove a strong structural connection to a seemingly unrelated covering problem with uncertainty in the constraints via a linear programming formulation. We exploit this connection to derive an algorithmic framework that can be used to solve both problems under uncertainty, obtaining nearly tight bounds on the competitive ratio. This is the first non-trivial stochastic result concerning the sum of unknown values without further structure known for the set. Further, we handle for the first time uncertainty in the constraints in a value-query model. With our novel methods, we lay the foundations for solving more general problems in the area of explorable uncertainty

Covering minimum spanning trees of random subgraphs

We consider the problem of finding a sparse set of edges containing the minimum spanning tree (MST) of a random subgraph of G with high probability. The two random models that we consider are subgraphs induced by a random subset of vertices, each vertex included independently with probability p, and subgraphs generated as a random subset of edges, each edge with probability p. Let n denote the number of vertices, choose p ∈ (0, 1) possibly depending on n and let b = 1/(1 − p). We show that in both random models, for any weighted graph G, there is a set of edges Q of cardinality O(n log b n) which contains the minimum spanning tree of a random subgraph of G with high probability. This result is asymptotically optimal. As a consequence, we also give a bound of O(kn) on the size of the union of all minimum spanning trees of G with some k vertices (or edges) removed. More generally, we show a bound of O(n log b n) on the size of a covering set in a matroid of rank n, which contains the minimum-weight basis of a random subset with high probability. Also, we give a randomized algorithm which calls an MST subroutine only a polylogarithmic number of times, and finds the covering set with high probability

Probabilistic methods in combinatorial and stochastic optimization

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (leaves 103-106).(cont.) Packing/Covering problems, we prove upper and lower bounds on the adaptivity gap depending on the dimension. We also design polynomial-time algorithms achieving near-optimal approximation guarantees with respect to the adaptive optimum. Finally, we prove complexity-theoretic results regarding optimal adaptive policies. These results are based on a connection between adaptive policies and Arthur-Merlin games which yields PSPACE-hardness results for numerous questions regarding adaptive policies.In this thesis we study a variety of combinatorial problems with inherent randomness. In the first part of the thesis, we study the possibility of covering the solutions of an optimization problem on random subgraphs. The motivation for this approach is a situation where an optimization problem needs to be solved repeatedly for random instances. Then we seek a pre-processing stage which would speed-up subsequent queries by finding a fixed sparse subgraph covering the solution for a random subgraph with high probability. The first problem that we investigate is the minimum spanning tree. Our essential result regarding this problem is that for every graph with edge weights, there is a set of O(n log n) edges which contains the minimum spanning tree of a random subgraph with high probability. More generally, we extend this result to matroids. Further, we consider optimization problems based on the shortest path metric and we find covering sets of size 0(n(Ì1+2/c) log2Ì n) that approximate the shortest path metric of a random vertex-induced subgraph within a constant factor of c with high probability. In the second part, we turn to a model of stochastic optimization, where a solution is built sequentially by selecting a collection of "items". We distinguish between adaptive and non-adaptive strategies, where adaptivity means being able to perceive the precise characteristics of chosen items and use this knowledge in subsequent decisions. The benefit of adaptivity is our central concept which we investigate for a variety of specific problems. For the Stochastic Knapsack problem, we prove constant upper and lower bounds on the "adaptivity gap" between optimal adaptive and non-adaptive policies. For more general Stochasticby Jan Vondrák.Ph.D

A tight bound on the collection of edges in MSTs of induced subgraphs

Let G=(V,E) be a complete n-vertex graph with distinct positive edge weights. We prove that for k{1,2,…,n−1}, the set consisting of the edges of all minimum spanning trees (MSTs) over induced subgraphs of G with n−k+1 vertices has at most elements. This proves a conjecture of Goemans and Vondrák [M.X. Goemans, J. Vondrák, Covering minimum spanning trees of random subgraphs, Random Structures Algorithms 29 (3) (2005) 257–276]. We also show that the result is a generalization of Mader's Theorem, which bounds the number of edges in any edge-minimal k-connected graph