We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out of a set of exponentially small measure in the space of data the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity which is doubly exponential (and this is so for almost all data).

One powerful theme in complexity theory and pseudorandomness in the past few decades has been the use of lower bounds to give pseudorandom generators (PRGs). However, the general results using this hardness vs. randomness paradigm suffer a quantitative loss in parameters, and hence do not give nontrivial implications for models where we dont know super-polynomial lower bounds but do know lower bounds of a fixed polynomial. We show that when such lower bounds are proved using random restrictions, we can construct PRGs which are essentially best possible without in turn improving the lower bounds.
More specifically, say that a circuit family has shrinkage exponent if a random restriction leaving a p fraction of variables unset shrinks the size of any circuit in the family by a factor of p^{+o(1)}. Our PRG uses a seed of length s^{1/(+1)+o(1)} to fool circuits in the family of size s. By using this generic construction, we get PRGs with polynomially small error for the following classes of circuits of size s and with the following seed lengths:
1. For de Morgan formulas, seed length s^{1/3+o(1)};
2. For formulas over an arbitrary basis, seed length s^{1/2+o(1)};
3. For read-once de Morgan formulas, seed length s^{.234...};
4. For branching programs of size s, seed length s^{1/2+o(1)}.
The previous best PRGs known for these classes used seeds of length bigger than n/2 to output n bits, and worked only for size s = O(n) (Bogdanov, Papakonstantinou, & Wan).

In this paper we introduce a new approach for approximately counting in bounded degree systems with higher-order constraints. Our main result is an algorithm to approximately count the number of solutions to a CNF formula ¦ when the width is logarithmic in the maximum degree. This closes an exponential gap between the known upper and lower bounds. Moreover our algorithm extends straightforwardly to approximate sampling, which shows that under Lovasz Local Lemma-like conditions it is not only possible to find a satisfying assignment, it is also possible to generate one approximately uniformly at random from the set of all satisfying assignments. Our approach is a significant departure from earlier techniques in approximate counting, and is based on a framework to bootstrap an oracle for computing marginal probabilities on individual variables. Finally, we give an application of our results to show that it is algorithmically possible to sample from the posterior distribution in an interesting class of graphical models.

We construct near optimal linear decision trees for a variety of decision problems in combinatorics and discrete geometry. For example, for any constant $k$, we construct linear decision trees that solve the $k$-SUM problem on $n$ elements using $O(n \log^2 n)$ linear queries. Moreover, the queries we use are comparison queries, which compare the sums of two $k$-subsets; when viewed as linear queries, comparison queries are $2k$-sparse and have only $\{-1,0,1\}$ coefficients. We give similar constructions for sorting sumsets $A+B$ and for solving the SUBSET-SUM problem, both with optimal number of queries, up to poly-logarithmic terms. Our constructions are based on the notion of ``inference dimension", recently introduced by the authors in the context of active classification with comparison queries. This can be viewed as another contribution to the fruitful link between machine learning and discrete geometry, which goes back to the discovery of the VC dimension.

We develop a systematic approach, based on convex programming and real analysis, for obtaining upper bounds on the capacity of the binary deletion channel and, more generally, channels with i.i.d. insertions and deletions. Other than the classical deletion channel, we give a special attention to the Poisson-repeat channel introduced by Mitzenmacher and Drinea (IEEE Transactions on Information Theory, 2006). Our framework can be applied to obtain capacity upper bounds for any repetition distribution (the deletion and Poisson-repeat channels corresponding to the special cases of Bernoulli and Poisson distributions). Our techniques essentially reduce the task of proving capacity upper bounds to maximizing a univariate, real-valued, and often concave function over a bounded interval. The corresponding univariate function is carefully designed according to the underlying distribution of repetitions and the choices vary depending on the desired strength of the upper bounds as well as the desired simplicity of the function (e.g., being only efficiently computable versus having an explicit closed-form expression in terms of elementary, or common special, functions). Among our results, we show the following: 1. The capacity of the binary deletion channel with deletion probability $d$ is at most $(1-d) \log \varphi$ for $d \geq 1/2$, and, assuming the capacity function is convex, is at most $1-d \log(4/\varphi)$ for $d<1/2$, where $\varphi=(1+\sqrt{5})/2$ is the golden ratio. This is the first nontrivial capacity upper bound for any value of $d$ outside the limiting case $d \to 0$ that is fully explicit and proved without computer assistance. 2. We derive the first set of capacity upper bounds for the Poisson-repeat channel. Our results uncover further striking connections between this channel and the deletion channel, and suggest, somewhat counter-intuitively, that the Poisson-repeat channel is actually analytically simpler than the deletion channel and may be of key importance to a complete understanding of the deletion channel. 3. We derive several novel upper bounds on the capacity of the deletion channel. All upper bounds are maximums of efficiently computable, and concave, univariate real functions over a bounded domain. In turn, we upper bound these functions in terms of explicit elementary and standard special functions, whose maximums can be found even more efficiently (and sometimes, analytically, for example for $d=1/2$). Along the way, we develop several new techniques of potentially independent interest. For example, we develop systematic techniques to study channels with mean constraints over the reals. Furthermore, we motivate the study of novel probability distributions over non-negative integers, as well as novel special functions which could be of interest to mathematical analysis.

Given k+1 square matrices A_1, ..., A_k, C, all of the same dimension and with real algebraic coordinates, we examine the problem of deciding whether there exist matrices of the form exp(A_i t_j) whose product equals C. Our results have applications to reachability problems for linear hybrid automata. Our decidability proof relies on a number of theorems from algebraic and transcendental number theory, most notably those of Baker, Lindemann, and Masser, as well as some useful geometric and linear-algebraic results, including the Minkowski-Weyl theorem and a new (to the best of our knowledge) result about the uniqueness of strictly upper triangular matrix logarithms of upper unitriangular matrices. On the other hand, our undecidability results are shown by reduction from Hilbert's Tenth Problem, through a series of matrix constructions.

We present prior robust algorithms for a large class of resource allocation problems where requests arrive one-by-one (online), drawn independently from an {\em unknown} distribution at every step. We design a single algorithm that, for every possible underlying distribution, obtains a $1-\epsilon$ fraction of the profit obtained by an algorithm that knows the entire request sequence ahead of time. The factor $\epsilon$ approaches $0$ when no single request consumes/contributes a significant fraction of the global consumption/contribution by all requests together. We show that the trade-off we obtain here that determines how fast $\epsilon$ approaches $0$, is near optimal: we give a nearly matching lower bound showing that the trade-off cannot be improved much beyond what we obtain. Going beyond the model of a static underlying distribution, we introduce the {\em adversarial stochastic input} model, where an adversary, possibly in an adaptive manner, controls the distributions from which the requests are drawn at each step. Placing no restriction on the adversary, we design an algorithm that obtains a $1-\epsilon$ fraction of the optimal profit obtainable w.r.t. the worst distribution in the adversarial sequence. Further, if the algorithm is given one number per distribution, namely, the optimal profit possible for each of the adversary's distribution, we design an algorithm that achieves a $1-\epsilon$ fraction of the weighted average of the optimal profit of each distribution the adversary picks. In the offline setting we give a fast algorithm to solve very large LPs with both packing and covering constraints. We give algorithms to approximately solve (within a factor of $1+\epsilon$) the mixed packing-covering problem with $O(\frac{\gamma m \log (n/\delta)}{\epsilon^2})$ oracle calls where the constraint matrix of this LP has dimension $n\times m$, the success probability of the algorithm is $1-\delta$, and $\gamma$ quantifies how significant a single request is when compared to the sum total of all requests. We discuss implications of our results to several special cases including online combinatorial auctions, network routing and the adwords problem.

In a multi-armed bandit problem, an online algorithm chooses from a set of strategies in a sequence of trials so as to maximize the total payoff of the chosen strategies. While the performance of bandit algorithms with a small finite strategy set is quite well understood, bandit problems with large strategy sets are still a topic of very active investigation, motivated by practical applications such as online auctions and web advertisement. The goal of such research is to identify broad and natural classes of strategy sets and payoff functions which enable the design of efficient solutions. In this work we study a very general setting for the multi-armed bandit problem in which the strategies form a metric space, and the payoff function satisfies a Lipschitz condition with respect to the metric. We refer to this problem as the "Lipschitz MAB problem". We present a solution for the multi-armed bandit problem in this setting. That is, for every metric space we define an isometry invariant which bounds from below the performance of Lipschitz MAB algorithms for this metric space, and we present an algorithm which comes arbitrarily close to meeting this bound. Furthermore, our technique gives even better results for benign payoff functions. We also address the full-feedback (``best expert'') version of the problem, where after every round the payoffs from all arms are revealed.

We give a new general approach for designing exact exponential-time algorithms for subset prob- lems. In a subset problem the input implicitly describes a family of sets over a universe of size n and the task is to determine whether the family contains at least one set. A typical example of a subset problem is Weighted d-SAT. Here, the input is a CNF-formula with clauses of size at most d, and an integer W. The universe is the set of variables and the variables have integer weights. The family contains all the subsets S of variables such that the total weight of the vari- ables in S does not exceed W, and setting the variables in S to 1 and the remaining variables to 0 satisfies the formula. Our approach is based on monotone local search, where the goal is to extend a partial solution to a solution by adding as few elements as possible. More formally, in the extension problem we are also given as input a subset X of the universe and an integer k. The task is to determine whether one can add at most k elements to X to obtain a set in the (implicitly defined) family. Our main result is that a c^kn^{O(1)} time algorithm for the extension problem immediately yields a randomized algorithm for finding a solution of any size with running time O((2 1/c)^n). c In many cases, the extension problem can be reduced to simply finding a solution of size at most k. Furthermore, efficient algorithms for finding small solutions have been extensively studied in the field of parameterized algorithms. Directly applying these algorithms, our theorem yields in one stroke significant improvements over the best known exponential-time algorithms for several well-studied problems, including d-Hitting Set, Feedback Vertex Set, Node Unique Label Cover, and Weighted d-SAT. Our results demonstrate an interesting and very concrete connection between parameterized algorithms and exact exponential-time algorithms. We also show how to derandomize our algorithms at the cost of a subexponential multiplicative factor in the running time. Our derandomization is based on an efficient construction of a new pseudo-random object that might be of independent interest. Finally, we extend our methods to establish new combinatorial upper bounds and develop enumeration algorithms.

We show how to transform formulae written in the real-time temporal logic MITL into timed automata recognizing their satisfying models, taken to be continuous-time Boolean signals. This compositional construction is much simpler than previously known; it supports both past and future operators and can easily be extended.