We survey a collective achievement of a group of researchers: the PCP Theorems. They give new definitions of the class NP, and imply that computing approximate solutions to many NP-hard problems is itself NP-hard. Techniques developed to prove them have had many other consequences. 2000 Mathematics Subject Classification: 68Q10, 68Q15, 68Q17, 68Q25.

We investigate the computational complexity of a general “compression task” centrally occurring in the recently developed technique of iterative compression for exactly solving NP-hard minimization problems. The core issue (particularly but not only motivated by iterative compression) is to determine the computational complexity of the following task: given an already inclusion-minimal solution...

A popular myth is that, for NP -hard problems, there are no algorithms with worst-case running time better than that of brute-force search. Reality is more nuanced, and for many natural NP -hard problems, there are algorithms with (worst-case) running time much better than the naive brute-force algorithm (albeit still exponential). This lecture proves this point by revisiting three problems stu...

To date, thousands of natural optimization problems have been shown to be NP-hard [6, 13]. Designing approximation algorithms [4, 17, 21] has become a standard path to attack these problems. For some problem, however, it is even NP-hard to approximate the optimal solution to within a certain ratio. The TRAVELING SALESMAN PROBLEM (TSP), for instance, has no approximation algorithm, since finding...

bilevel programming, a tool for modeling decentralized decision problems, consists of the objective of the leader at its first level and that of the follower at the second level. bilevel programming has been proved to be an np-hard problem. numerous algorithms have been developed for solving bilevel programming problems. these algorithms lack the required efficiency for solving a real problem. ...

For an optimization problem known to be NP-Hard, the dichotomy study investigates reduction instances determine line separating polynomial-time solvable vs NP-Hard (easy hard instances). In this paper, we investigate well-studied Hamiltonian cycle (HCYCLE), and present interesting result on split graphs. T. Akiyama et al. (1980) have shown that HCYCLE is NP-complete planar bipartite graphs with...

Hundreds of interesting and important combinatorial optimization problems are NP-hard, and so it is unlikely that any of them can be solved by a wost-case efficient exact algorithm. Short of proving P = NP, when one deals with an NP-hard problem one must accept a relaxation of one or more of the requirements of having a an optimal algorithm that runs in polynomial time on all inputs. Possible w...

A formulation of a graph problem for scheduling parallel computations of multibody dynamic analysis is presented. The complexity of scheduling parallel computations for a multibody dynamic analysis is studied. The problem of finding a shortest critical branch spanning tree is described and transformed to a minimum radius spanning tree, which is solved by an algorithm of polynomial complexity. T...

The fixed-parameter approach is an algorithm design technique for solving combinatorially hard (mostly NP-hard) problems. For some of these problems, it can lead to algorithms that are both efficient and yet at the same time guaranteed to find optimal solutions. Focusing on their application to solving NP-hard problems in practice, we survey three main techniques to develop fixed-parameter algo...