On Matrices, Automata, and Double Counting

Matrix models are ubiquitous for constraint problems. Many such problems have a matrix of variables M, with the same constraint defined by a finite-state automaton A on each row of M and a global cardinality constraint gcc on each column of M. We give two methods for deriving, by double counting, necessary conditions on the cardinality variables of the gcc constraints from the automaton A. The first method yields linear necessary conditions and simple arithmetic constraints. The second method introduces the cardinality automaton, which abstracts the overall behaviour of all the row automata and can be encoded by a set of linear constraints. We evaluate the impact of our methods on a large set of nurse rostering problem instances

On the Reification of Global Constraints

We introduce a simple idea for deriving reified global constraints in a systematic way. It is based on the observation that most global constraints can be reformulated as a conjunction of pure functional dependency constraints together with a constraint that can be easily reified. We first show how the core constraints of the Global Constraint Catalogue can be reified and we then identify several reification categories that apply to at least 82% of the constraints in the Global Constraint Catalogue

Algebraic Methods in the Congested Clique

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an $O(n^{1-2/\omega})$ round matrix multiplication algorithm, where $\omega < 2.3728639$ is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: -- triangle and 4-cycle counting in $O(n^{0.158})$ rounds, improving upon the $O(n^{1/3})$ triangle detection algorithm of Dolev et al. [DISC 2012], -- a $(1 + o(1))$-approximation of all-pairs shortest paths in $O(n^{0.158})$ rounds, improving upon the $\tilde{O} (n^{1/2})$-round $(2 + o(1))$-approximation algorithm of Nanongkai [STOC 2014], and -- computing the girth in $O(n^{0.158})$ rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.Comment: This is work is a merger of arxiv:1412.2109 and arxiv:1412.266

A complete dichotomy for complex-valued holant^c

Holant problems are a family of counting problems on graphs, parametrised by sets of complex-valued functions of Boolean inputs. Holant^c denotes a subfamily of those problems, where any function set considered must contain the two unary functions pinning inputs to values 0 or 1. The complexity classification of Holant problems usually takes the form of dichotomy theorems, showing that for any set of functions in the family, the problem is either #P-hard or it can be solved in polynomial time. Previous such results include a dichotomy for real-valued Holant^c and one for Holant^c with complex symmetric functions, i.e. functions which only depend on the Hamming weight of the input. Here, we derive a dichotomy theorem for Holant^c with complex-valued, not necessarily symmetric functions. The tractable cases are the complex-valued generalisations of the tractable cases of the real-valued Holant^c dichotomy. The proof uses results from quantum information theory, particularly about entanglement. This full dichotomy for Holant^c answers a question that has been open for almost a decade.</p

Unbounded-error quantum computation with small space bounds

We prove the following facts about the language recognition power of quantum Turing machines (QTMs) in the unbounded error setting: QTMs are strictly more powerful than probabilistic Turing machines for any common space bound $s$ satisfying $s(n)=o(\log \log n)$. For "one-way" Turing machines, where the input tape head is not allowed to move left, the above result holds for $s(n)=o(\log n)$. We also give a characterization for the class of languages recognized with unbounded error by real-time quantum finite automata (QFAs) with restricted measurements. It turns out that these automata are equal in power to their probabilistic counterparts, and this fact does not change when the QFA model is augmented to allow general measurements and mixed states. Unlike the case with classical finite automata, when the QFA tape head is allowed to remain stationary in some steps, more languages become recognizable. We define and use a QTM model that generalizes the other variants introduced earlier in the study of quantum space complexity.Comment: A preliminary version of this paper appeared in the Proceedings of the Fourth International Computer Science Symposium in Russia, pages 356--367, 200

Fast Monotone Summation over Disjoint Sets

We study the problem of computing an ensemble of multiple sums where the summands in each sum are indexed by subsets of size $p$ of an $n$-element ground set. More precisely, the task is to compute, for each subset of size $q$ of the ground set, the sum over the values of all subsets of size $p$ that are disjoint from the subset of size $q$. We present an arithmetic circuit that, without subtraction, solves the problem using $O((n^p+n^q)\log n)$ arithmetic gates, all monotone; for constant $p$, $q$ this is within the factor $\log n$ of the optimal. The circuit design is based on viewing the summation as a "set nucleation" task and using a tree-projection approach to implement the nucleation. Applications include improved algorithms for counting heaviest $k$-paths in a weighted graph, computing permanents of rectangular matrices, and dynamic feature selection in machine learning

Differential Privacy of Aggregated DC Optimal Power Flow Data

We consider the problem of privately releasing aggregated network statistics obtained from solving a DC optimal power flow (OPF) problem. It is shown that the mechanism that determines the noise distribution parameters are linked to the topology of the power system and the monotonicity of the network. We derive a measure of "almost" monotonicity and show how it can be used in conjunction with a linear program in order to release aggregated OPF data using the differential privacy framework.Comment: Accepted by 2019 American Control Conference (ACC