NP-hard problem

Just as there are universal computers that, given a particular input, can simulate any other com-puter, there are NP-complete problems that, with the appropriate input, are effectively equivalent to any NP-hard problem of a given size. For example, Boolean satisfiability -i.e. the problem of determining truth values of the variable s of a Boolean expression so that the expression is true -is known to be an NP-complete problem. See section 12.3.5.2... 

Problem-1 is a formidable challenge for mathematical programming. It is an NP-hard problem, and consequently all computational attempts to solve it cannot be guaranteed to provide a solution in polynomial time. It is not surprising then that all previous efforts have dealt with simplified versions of Problem-1. These simplifications have led to a variety of... 

Unger, R. and Moult, J. (1993) Finding the lowest free energy conformation of a protein is a NP-hard problem proof and implications, Bull. Math. Biol., 55, 1183-1198. 

The concept of biclustering has emerged in the context of microarray experiments since a gene can be involved in more than one biological process or could be co-expressed with other genes for only a subset of the conditions. Several existing approaches use heuristic methods, discretize the expression level, and/or solve a simplified model to address this NP-hard problem [6]. An excellent review of several biclustering methods can be found in [7]. 

The simulation of frustrated systems suffers from a similar tunneling problem as the simulation of first order phase transitions local minima in energy space are separated by barriers that grow with system size. While the multicanonical or optimized ensembles do not help with the NP-hard problems faced by spin glasses, they are efEcient in speeding up simulations of frustrated magnets without disorder. 

For some problems so-called polynomial time algorithms are known to exist. A polynomial time algorithm implies that the number of computational steps (which is proportional to the amount of computer time) needed to find a schedule which achieves the optimum value of the objective function is a polynomial function of the parameters of the problem (e.g., the number of jobs, n and/or the number of machines, m). A polynomial time algorithm may require, for example, a number of steps that is on the order of or /f. There are problems, however, for which no polynomial time algorithm is known to exist. These problems are the so-called NP-hard problems. The most efficient algorithms for these problems are exponential in the parameters of the problems. Such algorithms may require, for example, a number of steps that is on the order of 3" or 4". 

Ehschaige Elimination System permits NP-hard problems, 1722 NPSOL, 2564... 

The meta-heuristic algorithm, namely FA, which idealizes some of the flashing characteristics of fireflies, has been recently developed by Xin-She Yang [30]. Although the FA has many similarities with other swarm intelligence based algorithms, it is indeed much simpler both in concept and implementation [32]. There are already several applications of FA to different optimization problems [2,5,15,20,22,31-33]. The authors reported that the FA is powerful and very efficient novel population-based method and can outperform other meta-heuristics, such as GA, in solving many optimization problems and particularly NP-hard problems. 

Keywords Subset-based ant colony optimisation High dimensional NP-hard problems Tournament selection Roulette wheel selection Knapsack problem... 

It is unlikely that any NP-complete or NP-hard problem is polynomially solvable because of the relationship between these classes and the class of decision problems that can be solved in polynomial nondeterministic time. 

Let A and B be two problems. Problem A is polynomi-ally reducible to problem B (abbreviated A reduces to B, and written as A a B) if the existence of a deterministic polynomial time algorithm for B implies the existence of a deterministic polynomial time algorithm for A. Thus if A Q B and B is polynomially solvable, then so also is A. A problem A is NP-hard iff satisfiability a A. An NP-hard problem A is NP-complete if A e NP. Observe that the relation a is transitive (i.e., if A a B and B a C, then A a C). Consequently, if A a B and satisfiabilitya A then B is NP-hard. So, to show that any problem B is NP-hard, we need merely show that A a B, where A is any known NP-hard problem. Some of the known NP-hard problems are as follows ... 

The importance of showing that a problem A is NP-hard lies in the P=NP problem. Since we do not expect that P=NP, we do not expect. NP-hard problems to be solvable by algorithms with a worst-case complexity that is polynomial in the size of the problem instance. From Table n, it is apparent that, if a problem cannot be solved in polynomial time (in particular, low-order polynomial time), it is intractable, for all practical purposes. If A is NP-complete and if it does turn out that P =NP, then A will be polynomially solvable. However, if A is only NP-hard, it is possible for P to equal NP and for A not to be inP. 

An optimization problem is a problem in which one wishes to optimize (i.e., maximize or minimize) an optimization function f(x) subject to certain constraints C(x). For example, the NP-hard problem NP2 (0/1-knapsack) is an optimization problem. Here, we wish to optimize (in this... 

For most NP-hard problems, the problem of finding fc-absolute approximations is also NP-hard. As an example, consider problem NP2 (011-knapsack). From any instance (Pi, Wi,l feasible solutions as the old. However, the values of the feasible solutions to the new instance are multiples of k + l. Consequently, every k-absolute approximate solution to the new instance is an optimal solution for both the new and the old instance. Hence, -absolute approximate solutions to the 0/1-knapsack problem cannot be found any faster than optimal solutions. 

For several NP-hard problems, the e-approximation problem is also known to be NP-hard. For others fast e-approximation algorithms are known. As an example, we consider the optimization version of the bin-packing problem (NP8). This differs from NP8 in that the number of bins k is not part of the input. Instead, we are to find a packing of the n objects into bins of size C using the fewest number of bins. Some fast heuristics that are also -approximation algorithms are the following ... 

When BF is used, objects 1 and 2 get into bins 1 and 2, respectively. Object 3 gets into bin 2, since this provides a better fit than bin I. Object 4 now fits into bin I. The packing obtained uses only two bins and has objects 1 and 4 in bin 1 and objects 2 and 3 in bin 2. For FFD and BFD, the objects are packed in the order 2,4,1,3. In both cases, two-bin packing is obtained. Objects 2 and 3 are in bin 1 and objects 1 and 4 in bin 2. Approximation schemes (in particular fully polynomial time approximation schemes) are also known for several NP-hard problems. We will not provide any examples here. 

Clearly, this class of problems requires a triple optimisation, so-called integrated optimisation, at the same time allocating available resources to each production line, production line sequencing and production line scheduling. It is a multidimensional, precedence-constrained, knapsack problem. The knapsack problem is a classical NP-hard problem, and it has been thoroughly studied in the last few decades [2]. 

Multi-robot task allocation is a typically NP-hard problem. Its challenges become even more complicated when operations in uncertain environments, such as unexpected interference between robots, stochastic task requests, inconsistent information, and various component failures, are considered [4]. In such cases, it is not worth spending time and resources to secure an optimal solution, if the solutimi keeps changing as operations go on. Moreover, if time-window constraints are imposed, there may not be enough time to compute an exact and global solution. 

Budget constraints with integrality requirements for the choice of bids lead to a knapsack type constraints and lead a NP-hard problems. 

Although this is a small problem, the solution is not obvious. If we consider problems with several thousand or million variables, this is almost impossible to do without efiftcient and sophisticated algorithms. In order to find a solution, one has to consider all possible assignments for the variables. Assuming a problem with 50,000 variables, we have to consider assignments. Once a fulfilling solution has been found, the solution is easy to verify. These kinds of problems are called NP-hard problems. 

Hamilton graph (p. 879) travelling salesman problem (p. 879) NP-hard problem (p. 879)... 

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