This paper examines the computational and sample complexity of answering k-wise statistical queries, which were introduced by Felman and Ghazi [9] as a generalization to the standard statistical query model of Kearns [11]. In particular, our paper studies two sample reuse schemes: (1) reusing independent “pseudo-samples” for adaptive queries and (2) reusing dependent k-wise samples for non-adaptive queries. Comparing to a baseline non-reuse strategy, we show that the first reuse method offers a trade-off between k, the arity of the query, and M, the total number of queries to be answered. We also show that the second reuse method performs no worse than the baseline, and possibly better, from the perspective of variance reduction.
In this paper, we study a variety of NP-hard optimization problems, such as MAXCUT, MAXkSAT, and MAXNAE2SAT from the perspective of obtaining exact solutions. We derive differentiable functions for each of these problems using the dataless neural networks framework. Recently, it was shown that a single differentiable function on a dataless neural network can capture the Maximum Independent Set problem. Inspired by this design, we design dataless neural networks for a host of combinatorial optimization problems. We also establish the correctness of our derivations in a rigorous fashion.
Long-distance resolution for quantified boolean formula (QBF) solving was introduced in 2002 by Zhang and Malik, but has been controversial for the following decade, because it derives and uses tautologous clauses. Balabanov and Jiang (2012) gave a set of proof rules (called LDQ-resolution) that include long-distance resolution with conditions, but did not attach any “meaning” (i.e., semantic interpretation) to the derived tautologous clauses. Egly, Lonsing and Widl (2013) showed that the QBF certificate extraction algorithm of Goutltiaeva, Van Gelder and Bacchus (2011) could be applied with correct results to LDQ refutations. These results and others have brought LDQ-resolution back into the mainstream. This paper introduces partial boolean functions (pbfs) and develops a semantic interpretation for universal literals in QBF clauses. It develops LDP-resolution, which is refutationally complete for QBF formulas in prenex conjunction normal form (PCNF). LDP-resolution is shown to derive only logical consequences.
In computed tomography, the reconstructed object and the measured detector values are represented in a discrete domain. As the real world is not discrete, this restriction of the model usually creates artifacts (i.e., interpolation errors) on the reconstructed images. To formulate the projection model, we use so-called interpolation methods which enumerate the interactions between beams and the image as rays pass through the pixels of the image. The type of interpolation method we select strongly influences the artifacts in the reconstructed image. In this paper, we show a connection between pixel coverage to get a better understanding of interpolation errors, and we also propose an effective correction method to reduce the effects of interpolation errors in reconstructed images. We tested our proposed method in a comprehensive experiment, where we found that our proposed correction method can significantly improve the quality of the image.
In this paper, we establish the computational complexities of selected forms of refutations of linear programs. Linear programming is in the complexity class P and hence, it must have short affirmative and disqualifying certificates. One of the more celebrated lemmata in linear programming is Farkas’ lemma, which establishes that both “yes" and “no" certificates can be thought of as solutions to complementary linear programs. Since then, it has been established that if a linear program is feasible, then it must have a solution which is bounded by a polynomial function of the input size. The latter observation, coupled with Farkas’ lemma, immediately establishes that linear programming is in NP∩coNP. Our goal is to study the computational complexities of determining various constrained refutations for a given linear programming problem. This paper focuses on three distinct refutation forms, viz., read-once, tree-like and dag-like. We establish that checking if a linear program has a read-once refutation is NP-complete, even when it is defined by Binary Two Variable Per Inequality (BTVPI) constraints. Furthermore, the problems of finding the shortest tree-like and dag-like refutations are NPO-complete and NPO PB-complete respectively.
