Polynomial-time Solvable #CSP Problems via Algebraic Models and Pfaffian Circuits

A Pfaffian circuit is a tensor contraction network where the edges are labeled with changes of bases in such a way that a very specific set of combinatorial properties are satisfied. By modeling the permissible changes of bases as systems of polynomial equations, and then solving via computation, we are able to identify classes of 0/1 planar #CSP problems solvable in polynomial-time via the Pfaffian circuit evaluation theorem (a variant of L. Valiant's Holant Theorem). We present two different models of 0/1 variables, one that is possible under a homogeneous change of basis, and one that is possible under a heterogeneous change of basis only. We enumerate a series of 1,2,3, and 4-arity gates/cogates that represent constraints, and define a class of constraints that is possible under the assumption of a ``bridge" between two particular changes of bases. We discuss the issue of planarity of Pfaffian circuits, and demonstrate possible directions in algebraic computation for designing a Pfaffian tensor contraction network fragment that can simulate a swap gate/cogate. We conclude by developing the notion of a decomposable gate/cogate, and discuss the computational benefits of this definition

Designed-in security for cyber-physical systems

An expert from academia, one from a cyber-physical system (CPS) provider, and one from an end asset owner and user offer their different perspectives on the meaning and challenges of 'designed-in security.' The academic highlights foundational issues and talks about emerging technology that can help us design and implement secure software in CPSs. The vendor's view includes components of the academic view but emphasizes the secure system development process and the standards that the system must satisfy. The user issues a call to action and offers ideas that will ensure progress

Measuring changes in the behavior and attitudes of street-corner groups of boys towards an agency with a detached worker service.

Thesis (M.S.)--Boston Universit

An Algebraic Exploration of Dominating Sets and Vizing's Conjecture

Systems of polynomial equations are commonly used to model combinatorial problems such as independent set, graph coloring, Hamiltonian path, and others. We formulate the dominating set problem as a system of polynomial equations in two di erent ways: rst, as a single, high-degree polynomial, and second as a collection of polynomials based on the complements of domination-critical graphs. We then provide a su cient criterion for demonstrating that a particular ideal representation is already the universal Gr obner bases of an ideal, and show that the second representation of the dominating set ideal in terms of domination-critical graphs is the universal Gr obner basis for that ideal. We also present the rst algebraic formulation of Vizing's conjecture, and discuss the theoretical and computational rami cations to this conjecture when using either of the two dominating set representations described above

On the Complexity of Hilbert Refutations for Partition

Given a set of integers W, the Partition problem determines whether W can be divided into two disjoint subsets with equal sums. We model the Partition problem as a system of polynomial equations, and then investigate the complexity of a Hilbert's Nullstellensatz refutation, or certificate, that a given set of integers is not partitionable. We provide an explicit construction of a minimum-degree certificate, and then demonstrate that the Partition problem is equivalent to the determinant of a carefully constructed matrix called the partition matrix. In particular, we show that the determinant of the partition matrix is a polynomial that factors into an iteration over all possible partitions of W.Comment: Final versio