Experimental Analysis of Fixed Parameter Algorithms for Odd Cycle Transversals and Feedback Vertex Sets
Total Page:16
File Type:pdf, Size:1020Kb
www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT): 2394-6962 Volume-5, Issue-5, October-2015 International Journal of Engineering and Management Research Page Number: 656-667 Experimental Analysis of Fixed Parameter Algorithms for Odd Cycle Transversals and Feedback Vertex Sets S. Shivaprasad1, Dr. M Sadanandam2, Anand Agrawal3 1,3Assistant Professor, Vignan’s Univeristy, INDIA 2Assistant Professor, Kakatiya University, INDIA ABSTRACT natural way to cope with problems that are As implementing approximation considered intractable according to the classical P algorithms for NP-hard problems seems to be versus NP dichotomy. In practice, large instances impractical, a fixed-parameter algorithm is one that of NP-complete problems are solved exactly every provides an optimal solution to a discrete day. The reason for why this is possible is that combinatorial problem. The fundamental idea is to problem instances that occur naturally often have a re- strict the corresponding, seemingly unavoidable, combinatorial explosion that causes the exponential hidden structure. The basic idea of Parameterized growth in the running time of certain problem- Algorithms and Complexity is to extract and specific parameters.An odd cycle transversal (OCT) is harness the power of this structure. In a subset of the vertices of a graph G that hits all the Parameterized Algorithms and Complexity every odd cycles in G. Clearly the deletion of such a vertex problem instance comes with a relevant secondary set leaves a bipartite graph. Thus the problem of measurement k, called a parameter . The parameter finding an odd cycle transversal of min- imum measures how difficult the instance is, the larger the cardinality is just the classical graph bipartization parameter, the harder the instance. The hope is that problem. A feedback vertex set F (briefly, FVS) for G is at least for small values of the parameter, the a set of vertices in G such that every directed cycle in G contains at least one vertex in F, or equivalently, problem instances can be solved efficiently. This that the removal of F from the graph G leaves a naturally leads to the definition of fixed parameter directed acyclic graph (i.e., a DAG). Odd Cycle tractability. We say that a problem is fixed parameter Transversal and Feedback Vertex Sets are the NP- tractable (FPT) if problem instances of size n can be Hard problems which can be practically implemented O(1) by fixed parameter algorithms. This re- port is an solved in f (k)n time for some function f study of algorithms for OCT and FVS.The study shows independent of n. the influence of various factors, such as number of nodes, optimum solution size, edge density, and Over the last two decades, Parameterized parameter size, on the performance of the FPT Algorithms and Complexity has established itself as algorithm. an important subfield of Theoretical Computer Science. Parameterized Algorithms has its roots in Keywords---- Graph, Odd Cycle Transversals, the Graph Minors project of Robertson and Feedback vertex Sets, Bipartization. Seymour [1], but in fact, such algorithms have been made since the early seventies [8, 9]. In the late eighties several FPT algorithms were discovered[10, 11] but despite considerable efforts no fixed I. INTRODUCTION parameter tractable algorithm was found for problems like Independent Set and Dominating A. Parameterised Algorithms Set. In the early nineties, Downey and Fellows There has been a long history of [13, 12] developed a framework for showing in- exponential-time algorithms for finding opti- mal feasibility of FPT algorithms for parameterized solutions to NP-hard problems. Exponential problems under certain com- plexity theoretic running time at first glance seems to be assumptions, and showed that the existence of FPT impractical. This conception has been challenged algorithms for Independent Set and Dominating Set by the view of parameterized complexity. is unlikely. The possibility of showing lower as well Parameterized Algorithms and Complexity is a as upper bounds sparked interest in the field, and 656 Copyright © 2011-15. Vandana Publications. All Rights Reserved. www.ijemr.net ISSN (ONLINE): 2250-0758, ISSN (PRINT): 2394-6962 considerable progress has been made in machine as the underlying machine model developing methods to show tractability and in- throughout this thesis. In the random-access tractability of parameterized problems. Notable machine any simple oper- ation (arithmetic, if- methods that have been developed include statements, memory-access etc.) takes unit length Bounded Search Trees, Kernelization, Color Coding of time, and the word size is sufficiently large to hold and Iterative Compression[14]. numbers that are singly exponential in the size of B. Basic Definitions: the input. Basic Notions : We assume that the reader is Gaph : A graph G, can be defined as a tuple, G = familiar with basic notions like sets, functions, (V,E), where V is a finite set and E is a collection of polynomials, relations, integers etc. unordered pairs of distinct elements of V. The Problems: A problem or language is a set L of elements of V are called vertices of G and the elements of E are called edges of G. Let |S| denote strings over a finite alphabet Σ, that is L (Σ) . A the cardinality of set S. We denote | V | by n and | E | string s L is a yes instance of L and a string sinLis∗ a by m. We consider all graphs, we deal with, to be ⊆ no instance of L. A parameterized problem is a subset Π finite, undirected and unweighted, without ∈ of (Σ) xN, that is, every instance of Π comes with a multiple edges. Odd Cycles :In graph theory, the term cycle may natural∗ number called the parameter . In an optimization problem we are given as input an refer to a closed path. If repeated vertices are instance s, defining a set F(s) of feasible solutions for allowed, it is more often called a closed walk. If a problem specific function F . The objective is to the path is a simple path, with no repeated vertices find an element x F(s) that minimizes or or edges other than the starting and ending maximizes a certain problem specific objective vertices, it may also be called a simplecycle, circuit, function g(x). If the ∈ objective is to minimize circle, or polygon A cycle graph or circulargraph is a (maximize)g(x) the problem is referred to as a graph that consists of a single cycle, or in other minimization (maximization) problem. The words, some number of vertices connected in a minimum (maximum) value of the objective closed chain. A cycle with an odd number of function over all the feasible solutions for an vertices is called an odd cycle instance s is called the optimalvalue of the objective Graph bipartization : Given an undirected graph G function. = (V, E) and a nonnegative integer k . Does G have Function Growth : We employ big − Oh notation an odd cycle cover C of size at most k, that is, is which suppresses constant factors and lower order there a subset C V of vertices with |C| ≤ k such that terms. That is, for two functions f : N → N and g : each odd cycle in G contains at least one vertex N → N we say f (n) = O(g(n)) if there are constants from C? Note that⊆ the removal of all vertices in C c0 , c1 > 0 such that for all x c0 , c.f (x) c1 g(x). from G results in a bipartite graph. We say that a function f is singly exponential if f = C. Odd Cycle Transversal n 2 An odd cycle transversal (or cover ) is a O(2 c ) for some constant c. If f = O(2 n ) for some subset of the vertices of a graph G that hits all constant c we say f is doubly exponential. We will the odd cycles in G. Clearlyc the deletion of such sometimes misuse notation and say that a vertex set leaves a bipartite graph. Thus the f (n) is at least O(g(n)), meaning that g(n) = O( f (n)). problem of finding an odd cycle transversal of NP-hard problem : NP-hard (non-deterministic minimum cardinality is just the classical graph polynomial-time hard), in computational bipartization problem. The objective is to find a set complexity theory, is a class of problems that are, S of at most k vertices whose deletion makes the infor- mally, ”at least as hard as the hardest graph bipartite , and a set S such that G \ S is bipartite problems in NP”. A problem H is NP-hard if and is called an odd cycle transversal of G . only if there is an NP-complete problem L that is Input: A graph G and an integer k. polyno- mial time Turing-reducible to H (i.e., LTH). Output : A vertex set X V(G) with |X| ≤ k such that In other words, L can be solved in polynomial time G-X is bipartite. by an oracle machine with an oracle for H. D. Feed Back Vertex Sets Informally, we can think of an algorithm that can Feed Back Sets problems deal with call such an oracle machine as a subroutine for destroying cycles in graphs using a minimum solving H, and solves L in polynomial time, if the number of vertex deletions or edge deletions. subrou- tine call takes only one step to compute. Feedback Arc Set (FAS) asks for a minimum NP-hard problems may be of any type: decision number of arcs to be deleted in order to obtain a problems, search problems, or optimization cycle-free directed graph Feedback Vertex Set problems. (FVS) asks for a minimum number of vertex Algorithms : An algorithm for the problem L is an deletions.