# Approximation algorithms for np-hard problems

An approximation algorithm for solving an optimization problem. The steiner tree problem has been determined to be an npcomplete problem. Approximation algorithms will be the focus of this course. In this paper, we present polynomialtime approximation algorithms for the problem, for special cases of the dag c. In this lecture we consider such approximation algorithms, for several important problems. Associate professor venkatesan guruswami computer science and engineering most combinatorial optimization problems are nphard to solve optimally. Even though assuming p 6 np we cant hope for a polynomialtime algorithm that always gets the best solution, can we develop. Developing approximation algorithms for np hard problems is now a very active field in mathematical programming and theoretical computer science. Approximation algorithms for binpacking a survey, approximation algorithms for np hard problems. Natarajan meghanathan associate professor of computer science jackson state university. For many np complete problems there are approximation algorithms that are fast and give an answer that is close to the optimal e.

An approximation algorithm is any algorithm that gives a feasible solution to an optimization problem. Approximation algorithms are typically used when finding an optimal solution is intractable, but can also be used in some situations where a nearoptimal solution can be found quickly and an exact solution is not needed. Approximation algorithms for npcomplete problems on planar. Following are some np complete problems, for which no polynomial time algorithm is known. Approximation algorithms for np hard problems in this section, we discuss a different approach to handling difficult problems of combinatorial optimization, such as the traveling salesman problem and the knapsack problem. There are such algorithms for the easiest known nphard problems which are npcomplete problems. With chapters contributed by leading researchers in the field, this book introduces unifying techniques in the analysis of approximation algorithms. Approximation algorithms for nphard problems is intended for computer scientists and operations with chapters contributed by leading researchers in the field, this book introduces unifying techniques in the analysis of approximation algorithms. If the distance function is symmetric, the longest tour can be approximated within 43 by a deterministic algorithm and within by a randomized algorithm.

Typical performance of approximation algorithms for nphard. Pdf approximation algorithms for npproblems deepak. In this section, we discuss a different approach to handling difficult problems of combinatorial optimization, such as the traveling salesman problem and the knapsack problem. How well can you cluster nodes so as to cheaply separate a network into components around a few centers. Determining whether a graph has a hamiltonian cycle. Use a known exponential algorithm and stick to small problems. Ajesh babu we will continue the survey of approximation algorithms in this lecture. This is the first book to fully address the study of approximation algorithms as a tool for coping with intractable problems. Approximation algorithms have developed in response to the impossibility of solving a great variety of important optimization problems. Approximation algorithms for nphard problems book, 1997. Algorithms are at the heart of problem solving in scientific computing and computer science. Approximation algorithms for nphard problems lecture by dan suthers for university of hawaii information and computer sciences course 311 on algorithms. Approximation algorithms for npcomplete problems on.

So far we have dealt mostly with polynomially solvable problems. In the worstcase paradigm, algorithms for nphard problems are typically characterized by their approximation ratio, defined as the ratio between the worstcase cost of the algorithm and the cost of an allpowerful optimal algorithm. Determining whether a boolean formula is satisfiable, etc. Because it is unlikely that there exist efficient algorithms for nphard maximization problems, a typical approach to solving such a problem is to find a p approximation algorithm. Approximation algorithms for nphard problems errata. There is a vast and very technical field where people study approximation algorithms, which will give you guarantees for being close to the solution of an np complete problem. Any problem in np could be transformed to nphard problem in polynomial time. Lecture 20 polynomialtime approximation algorithms.

The main part of the course will emphasize recent methods and results. The traveling salesman problem tsp given a set ofcitiesalong with the cost of travel between them. Hochbaum the primaldual method for approximation algorithms. Faced with an np hard optimization problem, your options include. There are a number of approximation algorithms for the steiner tree problem. Linear programming has long been used to design and analyze approximation algorithms for nphard problems, particularly for problems which can be naturally formulated as integer programs. Hardness of approximation theory deals with studying the approximation threshold of such problems.

The approximation algorithms framework provides a guarantee on the quality of the solution obtained. Module 6 p, np, npcomplete problems and approximation algorithms. Such problems are commonly addressed with heuristics that provide a solution, but not information on the solutions quality. The upcoming special issue approximation algorithms for nphard problems aims to provide a comprehensive view of the most recent advances in the design and development of approximate solutions for computationally difficult problems. Several approximation algorithms from the seventies use. Often, when people refer to \ np\ hard problems, they are referring to decision problems, which are algorithms for which the output is yes or no. Approximation algorithms for multiprocessor scheduling. In this chapter we introduce the important concept of approximation algorithms. Most combinatorial optimization problems are nphard to solve optimally. Here approximation algorithms must be mentioned in the first place. Approximation algorithms for nphard clustering problems. Thus optimization problems can be cate gorized according to the best accuracy achievable by a polynomialtime approximation algorithm for each problem. A wide variety of np hard graph problems have been shown to be solvable in polynomial time, or even linear time, when constrained to partial ktrees 2,3, 11.

Pdf approximation algorithms for scheduling problems. Figure out whether you can restrict your problem to a special case for which polynomial solutions are known. In computer science and operations research, approximation algorithms are efficient algorithms that find approximate solutions to optimization problems in particular np hard problems with provable guarantees on the distance of the returned solution to the optimal one. Approximation algorithms for nphard clustering problems ramgopal r. Many usefulimportant problems are np hard or worse. Numerous practical problems are integer optimization problems that are intractable. The kmedian problem asks us to identify k cluster centers that minimize cost. Module 6 p, np, npcomplete problems and approximation algorithms dr. Instead, we can focus on design approximation algorithm.

We need a notion of \np\hardness that applies to optimization problems as well. Approximation algorithms for np hard problems is intended for computer scientists and operations with chapters contributed by leading researchers in the field, this book introduces unifying techniques in the analysis of approximation algorithms. Approximation algorithms for nphard optimization problems. Approximation algorithms for vertex cover problem node cover problem is np complete duration. Coffman and others published approximation algorithms for nphard problems find, read and cite all the research you need on. Nphard problems vary greatly in their approximability. Below we show we have a 2approximation algorithm for nphard vertex cover. Approximation algorithms for nphard problems by dorit. Mettu 103014 4 the problems we study the facility location problem asks us to identify a set of cluster centers that minimize associated penalties as well as cost.

Unfortunately many of the combinatorial problems that arise in a computational context are nphard, so that optimal solutions are unlikely to be found in. Give up on optimality, and find or design an approximation algorithm that gives good enough results. Thus optimization problems can be categorized according to the best accuracy achievable by a polynomialtime approximation algorithm for each. For some of these problems polynomial time so lutions are possible for graphs of treewidth ologn or olognlog logn 3,11. Approximation algorithms for npcomplete problems on planar graphs brenda s.

Paschos an overview on polynomial approximation of nphard problems exact optimal algorithms, that compute optimal solutions for the problems but run in exponential time. We can replace the nondeterministic guessing of stage 1 with the deterministic algorithm for the decision problem, and then in stage 2. Many computationallyhard problems admit efficient worstcase approximations 28, 32. Approximation algorithms for nphard problems acm sigact news. A natural approach to cope with this intractability is to design an. These are examples of nphard combinatorial optimization problems. The corresponding maximization problem of finding the longest travelling salesman tour is approximable within 6338. Approximation algorithms for np complete problems on planar graphs brenda s.

Approximation algorithms many of the npcomplete problems are. Naveen garg of computer science department at the iitdelhi. Mettu 103014 3 measuring cluster quality the cost of a set of cluster centers is the sum, over all points, of the weighted distance from each point to the. In this chapter, we discuss approximation algorithms for optimization problems. Buy approximation algorithms for nphard problems on. Klein department of computer science brown university neal e. This thesis describes efficient approximation algorithms for some np hard deterministic machine scheduling and related problems. Furthermore, for many natural np hard optimization problems, approximation algorithms have been developed whose accuracy nearly matches the best achievable according to the theory of np completeness. This paper describes a general technique that can be used to obtain approximation. In fact, many of those problems are nphard1, which means that no polynomialtime algorithm exists that solves the problem optimally unless pnp. This is a graduate level course on the design and analysis of combinatorial approximation algorithms for np hard optimization problems. How efficiently can you pack objects into a minimum number of boxes.

We give a few examples of approximation algorithms for nphard combinatorial optimization problems below. The first part of the book presents a set of classical np hard problems, set covering, bin packing, knapsack, etc. Typical performance of approximation algorithms for nphard problems. The problem of multiprocessor scheduling under uncertainty was introduced in 21 and was shown to be nphard even when all the jobs are independent. Our results include an optimal fixedparameter tractable algorithm, a polynomialtime approximation algorithm, as well as an effective heuristic that can also be used in an online setting of the problem. Approximation algorithms for optimization problems in graphs. Approximation algorithms for binpacking a survey, approximation algorithms for nphard problems 1997 by e g coffman, m r garey add to metacart. Approximation algorithms an overview sciencedirect topics. Approximation algorithms for np hard problems is intended for computer scientists and operations researchers interested in specific algorithm implementations, as well as design tools for algorithms.

Parametrized complexity versus approximation algorithms. An approximation algorithm for an np hard optimization problem is a. Limits of approximation algorithms 2 feb, 2010 imsc. Garg clearly explains a very hard topic without the use of. We will show how standard algorithm design techniques such as greedy and local search methods have been used to. Johnson approximating covering and packing problems. These algorithms are extracted from a number of fundamental papers, which are of long, delicate presentations. Next 10 dynamic placement of virtual machines for managing sla violations. It is widely believed that p np so that it is impossible to solve the problems in poly momial time.

Introduction to approximation algorithms many important computational problems are di. Approximation algorithms for nphard problems acm sigact. Approximation algorithms for nphard problems is intended for computer scientists and operations researchers interested. The focus of this chapter is on the design of approximation algorithms for np hard optimization problems.

Approximation algorithms for np hard clustering problems ramgopal r. Special issue approximation algorithms for nphard problems. Mestre j a primaldual approximation algorithm for partial vertex cover proceedings of the 8th international workshop on approximation, randomization and combinatorial optimization problems, and proceedings of the 9th international conference on randamization and computation. Travelling salesman problem thomas sauerwald easter 2016.

Improved approximation algorithms for maximum cut and. In this section, we discuss a wellknown approximation algorithm developed by kou, markowsky, and berman in 1981 461, which we will refer to. Approximation algorithms for nphard problems guide books. No, there are only approximations algorithms for nphard problems. The class p consists of all polynomialtime solvable decision problems. In the remaining chapters we shall indicate some strategies to cope with np hard combinatorial optimization problems. In the realm of nphard problems, the best known approximation factor for various problems varies greatly, i. Polynomialtime approximation algorithms nphard problems are a vast family of problems that, to the best of our knowledge, cannot be solved in polynomial time.

There are lots of nphard problems that are fpt, however, there are many problems in np that are believed to not be fixed parameter tractable. This is a short lecture on the p versus np problem by prof. Approximation algorithms for nphard problems dorit. Can we just use the same polynomial time reduction, and solve a, to get a. For example, a decision version of the graph coloring problem is determining whether or not a graph is 3colorable. Module 6 p, np, npcomplete problems and approximation. Approximation algorithms for nphard problems by dorit hochbaum. Approximation algorithms for nphard problems part ii lecturer. Young department of computer science dartmouth college chapter 34, algorithms and theory of computation handbook. Citeseerx citation query approximation algorithms for.

Approximation algorithms for nphard p roblems 1479 algorithms that are e. Condensed matter disordered systems and neural networks. We have taken several particular perspectives in writing the book. An optimization problem is \ np\ hard if it can be used as a subroutine to solve an \ np\ hard decision problem in polynomial time, with the optimization problem used as a black box. This book is actually a collection of survey articles written by some of the foremost experts in this field. Approximation algorithms for nphard problems is intended for computer scientists and operations researchers interested in specific algorithm implementations, as well as design tools for algorithms.

Limits of approximation algorithms 28 jan, 2010 tifr lec. While i have a basic knowledge of computational complexity i more or less know what it means to be in p, np, nphard and so on, i tried to study some basic texts on approximability but im not sure i really get what this inapproximability is. Approximating nphard problems efficient algorithms and their limits. Are there any polynomialtime algorithms for nphard. Too frequently, when attempting to get a solution for a problem, one is confronted with the fact that the problem is nphard. Too frequently, when attempting to get a solution for a problem, one is confronted with the fact that the problem is np hard. An optimization problem consists in finding the best cheapest, heaviest, etc. Vazirami presented the problems and solutions in a unified framework. A natural approach to cope with this intractability is to design an approximation. P, npproblems class np nondeterministic polynomial is the class of decision problems that can be solved by nondeterministic polynomial algorithms.

Many problems that are nphard are also nonapproximable assuming p. Some of these algorithms are useable in the real world. In computer science and operations research, approximation algorithms are efficient algorithms. In the early 1990s, with the development of pcp theory, it became clear that many more approximation problems were hard to approximate, and that unless p np many known approximation algorithms achieved the best possible approximation ratio. For harder, or are considered to be, problems than npcomplete such as pspace\np.

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