Any graph problem, which is NP-hard in general graphs, becomes polynomial-time solvable when restricted to graphs in special classes. When does a difficult problem become easy? To answer this question we study the notion of boundary classes. In the present paper we define this notion in its most general form, describe several approaches to identify boundary classes and apply them to various gra...

Metaheuristics is one of the most well-known field researches uses to find optimum solution for Non-deterministic polynomial hard problems (NP-Hard), that are difficult an optimal in a time. Over time many algorithms have been developed based on heuristics solve real-life problems, this paper will introduce Metaheuristic-based and its classifications, problems. It also compare performance two m...

This paper has been worked on a RAP with multi-state components and the performance rate of each component working state may increase by spending technical and organizational activities costs. Whereas RAP belongs to Np-Hard problems, we used Genetic algorithm (GA) and simulated annealing (SA) and for solving the presented problem and calculating system reliability universal generating function ...

We will continue the survey of approximation algorithms in this lecture. First, we will discuss a (1+ε)-approximation algorithm for Knapsack in time poly(n, 1/ε). We will then see applications of some heavy hammers such as linear programming (LP) and semi-definite programming (SDP) towards approximation algorithms. More specifically, we will see LPbased approximation for MAXSAT and MAXCUT. In t...

Aiming at developing a theoretical framework for the formal study of NP-hard optimization problems, which is built on precise mathematical foundations, we have focused on structural properties of optimization problems related to approximative issue. From the observation that, intuitively, there are many connections among categorical concepts and structural complexity notions, in this work we pr...

We present some of the recent results on computational complexity of approximating bounded degree combinatorial optimization problems. In particular, we present the best up to now known explicit nonapproximability bounds on the very small degree optimization problems which are of particular importance on the intermediate stages of proving approximation hardness of some other generic optimizatio...

We present a comprehensive review of past research into adiabatic quantum computation and then propose a scalable architecture for an adiabatic quantum computer that can treat NP-hard problems without requiring local coherent operations. Instead, computation can be performed entirely by adiabatically varying a magnetic field applied to all the qubits simultaneously. Local (incoherent) operation...

During recent years, it has been shown that a number of problems in matrix theory are NP-hard, including robust nonsingularity, robust stability, robust positive semide"niteness, robust bounded norm, state feedback stabilization with structural and norm constraints, etc. In this paper, we use standard bounds on empirical probabilities as well as recent results from statistical learning theory o...