A combinatorial approach to knot recognition

This is a report on our ongoing research on a combinatorial approach to knot recognition, using coloring of knots by certain algebraic objects called quandles. The aim of the paper is to summarize the mathematical theory of knot coloring in a compact, accessible manner, and to show how to use it for computational purposes. In particular, we address how to determine colorability of a knot, and propose to use SAT solving to search for colorings. The computational complexity of the problem, both in theory and in our implementation, is discussed. In the last part, we explain how coloring can be utilized in knot recognition

Data Reduction for Graph Coloring Problems

This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai's study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the NO-answer to an instance of q-List-Coloring on F.Comment: Author-accepted manuscript of the article that will appear in the FCT 2011 special issue of Information & Computatio

Constraint Expressions and Workflow Satisfiability

A workflow specification defines a set of steps and the order in which those steps must be executed. Security requirements and business rules may impose constraints on which users are permitted to perform those steps. A workflow specification is said to be satisfiable if there exists an assignment of authorized users to workflow steps that satisfies all the constraints. An algorithm for determining whether such an assignment exists is important, both as a static analysis tool for workflow specifications, and for the construction of run-time reference monitors for workflow management systems. We develop new methods for determining workflow satisfiability based on the concept of constraint expressions, which were introduced recently by Khan and Fong. These methods are surprising versatile, enabling us to develop algorithms for, and determine the complexity of, a number of different problems related to workflow satisfiability.Comment: arXiv admin note: text overlap with arXiv:1205.0852; to appear in Proceedings of SACMAT 201

Games for One, Games for Two: Computationally Complex Fun for Polynomial-Hierarchical Families

In the first half of this thesis, we explore the polynomial-time hierarchy, emphasizing an intuitive perspective that associates decision problems in the polynomial hierarchy to combinatorial games with fixed numbers of turns. Specifically, problems in are thought of as 0-turn games, as 1-turn “puzzle” games, and in general ₖ as -turn games, in which decision problems answer the binary question, “can the starting player guarantee a win?” We introduce the formalisms of the polynomial hierarchy through this perspective, alongside definitions of -turn CIRCUIT SATISFIABILITY games, whose ₖ-completeness is assumed from prior work (we briefly justify this assumption on intuitive grounds, but no proof is given). In the second half, we introduce and explore the properties of a novel family of games called the -turn GRAPH 3-COLORABILITY games. By embedding boolean circuits in proper graph 3-colorings, we construct reductions from -turn CIRCUIT SATISFIABILITY games to -turn 3-COLORABILITY games, thereby showing that -turn 3-COLORABILITY is ₖ-complete Finally, we conclude by discussing possible future generalizations of this work, vis-à-vis extending arbitrary -complete puzzles to interesting ₖ-complete games

Optimal Data Reduction for Graph Coloring Using Low-Degree Polynomials

The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring. First, we use a recent technique of finding redundant constraints by representing them as low-degree polynomials, to obtain a kernel of bitsize O(k^(q-1) log k) for q-Coloring parameterized by Vertex Cover for any q >= 3. This size bound is optimal up to k^o(1) factors assuming NP is not a subset of coNP/poly, and improves on the previous-best kernel of size O(k^q). Our second result shows that 3-Coloring does not admit non-trivial sparsification: assuming NP is not a subset of coNP/poly, the parameterization by the number of vertices n admits no (generalized) kernel of size O(n^(2-e)) for any e > 0. Previously, such a lower bound was only known for coloring with q >= 4 colors