Polynomial Time

A graph is well-covered if every maximal independent set has the same cardinality. The recognition problem of well-covered graphs is known to be co-NP-complete. Let w be a weight function defined on the vertices of G. Then G is w-well-covered if all maximal independent sets of G are of the same weight. The set of weight functions w for which a graph is w-well-covered is a vector space. We prove that finding the vector space of weight functions under which an input graph is w-well-covered can be done in polynomial time, if the input graph does not contain cycles of length 4, 5, 6 and 7.

Simulation of quantum computers using classical computers is a hard problem with high memory and computational requirements. Parallelization can alleviate this problem, allowing the simulation of more qubits at the same time or the same number of qubits to be simulated in less time. A promising approach is to exploit the high performance computing capabilities provided by the latest graphical processing units. In this paper we present a parallel implementation of the QC-lib quantum computer simulator on the GPU using the CUDA programming model. The proposed scheme for partitioning the terms that describe the state of a quantum register takes advantage of the specific characteristics of the CUDA memory spaces and allows for an efficient parallelization of the general singe qubit operator. Experimental results indicate that very good speed-ups can be obtained in contrast with the sequential implementation.

Many hard algorithmic problems dealing with graphs, circuits, formulas and constraints admit polynomial-time upper bounds if the underlying graph has small treewidth. The same problems often encourage reducing the maximal degree of vertices to simplify theoretical arguments or address practical concerns. Such degree reduction can be performed through a sequence of splittings of vertices, resulting in an expansion of the

We consider the problem of computing the outer-radii of point sets. In this problem, we are given integers n, d, k where k ≤ d, and a set P of n points in Rd. The goal is to compute the outer k-radius of P, denoted by Rk(P), which is the minimum, over all (d−k)-dimensional flats F, of maxp∈P d(p, F), where d(p, F) is the Euclidean distance between the point p and flat F. Computing the radii of point sets is a fundamental problem in computational convexity with significantly many applications. The problem admits a polynomial time algorithm when the dimension d is constant [9]. Here we are interested in the general case when the dimension d is not fixed and can be as large as n, where the problem becomes NP-hard even for k = 1. It has been known that Rk(P) can be approximated in polynomial time by a factor of (1 + ε), for any ε> 0, when d − k is a fixed constant [15, 2]. A factor of O(√logn) approx-imation for R1(P), the width of the point set P, is implied from the results of Nemi...

We discuss in this paper a method of finding skyline or non-dominated points in a set P of n points with respect to a set S of m sites. A point pi∈ P is non-dominated if and only if for each pj∈ P, j ¹ ij\ not= i, there exists at least one point s∈ S that is closer to pi than p j. We reduce this problem of determining non-dominated points to the problem of finding sites that have non-empty cells in an additively weighted Voronoi diagram under convex distance function. The weights of the said Voronoi diagram are derived from the co-ordinates of the ...

Detecting polygons defined by a set of line segments in a plane is an important step in the analysis of vectorial drawings. This paper presents an approach that combines several algorithms to detect basic polygons from a set of arbitrary line segments. The resulting algorithm runs in polynomial time and space, with complexities of O((N +M)) and O((N +M)) respectively, where N is the number of line segments and M is the number of intersections between line segments. Our choice of algorithms was made to strike a good compromise between efficiency and ease of implementation. The result is a simple and efficient solution to detect polygons from lines.

A function F defined on all subsets of a finite ground set E is quasi-concave if F(X∪ Y)≥{F(X),F(Y)} for all X,Y⊂ E. Quasi-concave functions arise in many fields of mathematics and computer science such as social choice, theory of graph, data mining, clustering and other fields. The maximization of quasi-concave function takes, in general, exponential time. However, if a quasi-concave function is defined by associated monotone linkage function then it can be optimized by the greedy type algorithm in a polynomial time. Quasi-concave functions defined as minimum values of monotone linkage functions were considered on antimatroids, where the correspondence between quasi-concave and bottleneck functions was shown (Kempner & Levit, 2003). The goal of this paper is to analyze quasi-concave functions on different families of sets and to investigate their relationships with monotone linkage functions.

This paper deals with a problem of scheduling jobs on the identical parallel machines, where job values are given as a power function of the job completion times. Minimization of the total loss of job values is considered as a criterion. We establish the computational complexity of the problem – strong NP-hardness of its general version and NP-hardness of its

We explore the automatic generation of test data that respect constraints expressed in the Object-Role Modeling (ORM) language. ORM is a popular conceptual modeling language, primarily targeting database applications, with significant uses in practice. The general problem of even checking whether an ORM diagram is satisfiable is quite hard: restricted forms are easily NP-hard and the problem is undecidable for some expressive formulations of ORM. Brute-force mapping to input for constraint and SAT solvers does not scale: state-of-the-art solvers fail to find data to satisfy uniqueness and mandatory constraints in realistic time even for small examples. We instead define a restricted subset of ORM that allows efficient reasoning yet contains most constraints overwhelmingly used in practice. We show that the problem of deciding whether these constraints are consistent (i.e., whether we can generate appropriate test data) is solvable in polynomial time, and we produce a highly efficien...

This 1964 paper developed as an off-shoot to the foundational query: Do we discover the natural numbers (Platonically), or do we construct them linguistically? The paper also assumes computational significance in the light of Agrawal, Kayal and Saxena's August 2000 paper, "PRIMES is in P", since both the TRIM and Compact Number Generating algorithms - each of which generates all the primes - are deterministic algorithms that run in polynomial time and suggest that the Prime Difference, d(n), is O(Number of primes < (p(n)^1/2)).

In this article, we introduce and study a framework, called peer data exchange , for sharing and exchanging data between peers. This framework is a special case of a full-fledged peer data management system and a generalization of data exchange between a source schema and a target schema. The motivation behind peer data exchange is to model authority relationships between peers, where a source peer may contribute data to a target peer, specified using source-to-target constraints, and a target peer may use target-to-source constraints to restrict the data it is willing to receive, but cannot modify the data of the source peer.A fundamental algorithmic problem in this framework is that of deciding the existence of a solution: given a source instance and a target instance for a fixed peer data exchange setting, can the target instance be augmented in such a way that the source instance and the augmented target instance satisfy all constraints of the setting? We investigate the computa...

Some typical optimization problems like knapsack problem, activity selection problem, coin change problem and subset sum problem are taught in different courses of theoretical computer science, or in the different modules of the same course. A few of these problems can be solved in polynomial time but the r est are NP-complete problems. Even though, every proble m has its own nature and description; however after changing certain constraints, they become equivalen t to each other. These problems are discussed independen tly in most textbooks which makes it harder for the students to interlink them. Students get confused very easil y while transitioning between different problem domains eve n if one problem is described in the language of another problem. In this paper, we have carefully analyzed and investigated links between the aforementioned probl ems and have also diagnosed the major causes of failure in understanding. One of the primary reasons students face difficulty is that ...

Let a⊕b = max(a,b) and a⊗b = a + b for a, b ∈ ℝ. Max-algebra is an analogue of linear algebra developed on the pair of operations (⊕, ⊗) extended to matrices and vectors. The system of equations A ⊗ x = b and inequalities C ⊗ x ≤ d have each been studied in the literature. We consider a problem consisting of these two systems and present necessary and sufficient conditions for its solvability. We also develop a polynomial algorithm for solving max-linear program whose constraints are max-linear equations and inequalities.