Picturing counting reductions with the ZH-calculus

Counting the solutions to Boolean formulae defines the problem #SAT, which is complete for the complexity class #P. We use the ZH-calculus, a universal and complete graphical language for linear maps which naturally encodes counting problems in terms of diagrams, to give graphical reductions from #SAT to several related counting problems. Some of these graphical reductions, like to #2SAT, are substantially simpler than known reductions via the matrix permanent. Additionally, our approach allows us to consider the case of counting solutions modulo an integer on equal footing. Finally, since the ZH-calculus was originally introduced to reason about quantum computing, we show that the problem of evaluating ZH-diagrams in the fragment corresponding to the Clifford+T gateset, is in $FP^{\#P}$. Our results show that graphical calculi represent an intuitive and useful framework for reasoning about counting problems

#3-regular bipartite planar vertex cover is #p-complete

Chinese Acad Sci, Inst Software, Univ Leeds, Univ Wisconsin, Natl Nat Sci Fdn ChinaWe generalize the polynomial interpolation method by giving a sufficient condition, which guarantees that the coefficients of a polynomial are uniquely determined by its values on a recurrence sequence. Using this method, we show that #3-Regu

Algorithms and hardness results for geometric problems on stochastic datasets

University of Minnesota Ph.D. dissertation.July 2019. Major: Computer Science. Advisor: Ravi Janardan. 1 computer file (PDF); viii, 121 pages.Traditionally, geometric problems are studied on datasets in which each data object exists with probability 1 at its location in the underlying space. However, in many scenarios, there may be some uncertainty associated with the existence or the locations of the data points. Such uncertain datasets, called \textit{stochastic datasets}, are often more realistic, as they are more expressive and can model the real data more precisely. For this reason, geometric problems on stochastic datasets have received significant attention in recent years. This thesis studies three sets of geometric problems on stochastic datasets equipped with existential uncertainty. The first set of problems addresses the linear separability of a bichromatic stochastic dataset. Specifically, these problems are concerned with how to compute the probability that a realization of a bichromatic stochastic dataset is linearly separable as well as how to compute the expected separation-margin of such a realization. The second set of problems deals with the stochastic convex hull, i.e., the convex hull of a stochastic dataset. This includes computing the expected measures of a stochastic convex hull, such as the expected diameter, width, and combinatorial complexity. The third set of problems considers the dominance relation in a colored stochastic dataset. These problems involve computing the probability that a realization of a colored stochastic dataset does not contain any dominance pair consisting of two different-colored points. New algorithmic and hardness results are provided for the three sets of problems