NP-hardness of circuit minimization for multi-output functions

Can we design efficient algorithms for finding fast algorithms? This question is captured by various circuit minimization problems, and algorithms for the corresponding tasks have significant practical applications. Following the work of Cook and Levin in the early 1970s, a central question is whether minimizing the circuit size of an explicitly given function is NP-complete. While this is known to hold in restricted models such as DNFs, making progress with respect to more expressive classes of circuits has been elusive. In this work, we establish the first NP-hardness result for circuit minimization of total functions in the setting of general (unrestricted) Boolean circuits. More precisely, we show that computing the minimum circuit size of a given multi-output Boolean function f : {0,1}^n ? {0,1}^m is NP-hard under many-one polynomial-time randomized reductions. Our argument builds on a simpler NP-hardness proof for the circuit minimization problem for (single-output) Boolean functions under an extended set of generators. Complementing these results, we investigate the computational hardness of minimizing communication. We establish that several variants of this problem are NP-hard under deterministic reductions. In particular, unless ? = ??, no polynomial-time computable function can approximate the deterministic two-party communication complexity of a partial Boolean function up to a polynomial. This has consequences for the class of structural results that one might hope to show about the communication complexity of partial functions

Approximation Algorithms for Stochastic Boolean Function Evaluation and Stochastic Submodular Set Cover

Stochastic Boolean Function Evaluation is the problem of determining the value of a given Boolean function f on an unknown input x, when each bit of x_i of x can only be determined by paying an associated cost c_i. The assumption is that x is drawn from a given product distribution, and the goal is to minimize the expected cost. This problem has been studied in Operations Research, where it is known as "sequential testing" of Boolean functions. It has also been studied in learning theory in the context of learning with attribute costs. We consider the general problem of developing approximation algorithms for Stochastic Boolean Function Evaluation. We give a 3-approximation algorithm for evaluating Boolean linear threshold formulas. We also present an approximation algorithm for evaluating CDNF formulas (and decision trees) achieving a factor of O(log kd), where k is the number of terms in the DNF formula, and d is the number of clauses in the CNF formula. In addition, we present approximation algorithms for simultaneous evaluation of linear threshold functions, and for ranking of linear functions. Our function evaluation algorithms are based on reductions to the Stochastic Submodular Set Cover (SSSC) problem. This problem was introduced by Golovin and Krause. They presented an approximation algorithm for the problem, called Adaptive Greedy. Our main technical contribution is a new approximation algorithm for the SSSC problem, which we call Adaptive Dual Greedy. It is an extension of the Dual Greedy algorithm for Submodular Set Cover due to Fujito, which is a generalization of Hochbaum's algorithm for the classical Set Cover Problem. We also give a new bound on the approximation achieved by the Adaptive Greedy algorithm of Golovin and Krause

String Matching: Communication, Circuits, and Learning

String matching is the problem of deciding whether a given n-bit string contains a given k-bit pattern. We study the complexity of this problem in three settings. - Communication complexity. For small k, we provide near-optimal upper and lower bounds on the communication complexity of string matching. For large k, our bounds leave open an exponential gap; we exhibit some evidence for the existence of a better protocol. - Circuit complexity. We present several upper and lower bounds on the size of circuits with threshold and DeMorgan gates solving the string matching problem. Similarly to the above, our bounds are near-optimal for small k. - Learning. We consider the problem of learning a hidden pattern of length at most k relative to the classifier that assigns 1 to every string that contains the pattern. We prove optimal bounds on the VC dimension and sample complexity of this problem