5 research outputs found
Galois correspondence for counting quantifiers
We introduce a new type of closure operator on the set of relations,
max-implementation, and its weaker analog max-quantification. Then we show that
approximation preserving reductions between counting constraint satisfaction
problems (#CSPs) are preserved by these two types of closure operators.
Together with some previous results this means that the approximation
complexity of counting CSPs is determined by partial clones of relations that
additionally closed under these new types of closure operators. Galois
correspondence of various kind have proved to be quite helpful in the study of
the complexity of the CSP. While we were unable to identify a Galois
correspondence for partial clones closed under max-implementation and
max-quantification, we obtain such results for slightly different type of
closure operators, k-existential quantification. This type of quantifiers are
known as counting quantifiers in model theory, and often used to enhance first
order logic languages. We characterize partial clones of relations closed under
k-existential quantification as sets of relations invariant under a set of
partial functions that satisfy the condition of k-subset surjectivity. Finally,
we give a description of Boolean max-co-clones, that is, sets of relations on
{0,1} closed under max-implementations.Comment: 28 pages, 2 figure
Complexity of approximating #CSPs
Constraint satisfactions is a framework to express combinatorial problems. #CSP is the problem of finding the number of solutions for a constraint satisfaction problem instance. In this work, we study complexity of approximately solving the #CSP. We provide several techniques for approximation preserving reductions among counting problems. Most of this work focuses on reduction to #BIS, the problem of finding the number of independent sets in a bipartite graph. We prove that approximately solving #CSP(Γ) over relations which we call monotone, is not harder than #BIS. We also prove that approximately solving #Hom(H) for reflexive oriented graphs is not easier than #BIS. Finally, we investigate monotone reflexive graphs
Touring a Sequence of Weighted Polygons
In this paper we will solve a generalization of the problem “Touring a Sequence of Polygons” where polygons can be concave and a weight is considered when visiting them. The main idea to solve the problem is triangulation of the polygons and adding some Steiner on the edges. We will also use a modified version of BUSHWHACK algorithm to find the shortest path among Steiner points. The running time of the algorithm will be O ( n
Abstract
Given a subdivision of plane into convex polygon regions, a sequence of polygons to meet, a start point s, and a target point t, we are interested in determining the shortest weighted path on this plane which starts at s, visits each of the polygons in the given order, and ends at t. The length of a path in weighted regions is defined as the sum of the lengths of the sub-paths within each region. We will present an approximation algorithm with maximum δ cost additive. Our algorithm is based on the shortest weighted path algorithm proposed by Mata and Mitchel [2]. The algorithm runs in O(((n 3 LW + RW) k δ)3) time, where n is the number of vertices of the region boundaries, L is the longest boundary, W is the maximum weight in the region, R is the sum of the perimeters of the regions, and k is the number of polygons. The main idea in the algorithm is to add Steiner points on the region boundaries and polygon edges. In addition, we will also present a solution to the query version of this problem. We will extend our result in unweighted version of the “Touring a Sequence of Polygons ” problem [3]. We will give an approximation algorithm to solve the general case of the problem (with non-convex intersecting polygons)