Approximations for the Maximum Acyclic Subgraph Problem

Given a directed graph G V A the maximum acyclic subgraph problem is to compute a subset A of arcs of maximum size or total weight so that G V A is acyclic We discuss several approximation algorithms for this problem Our main result is an O jAj d max algorithm that produces a solution with at least a fraction p dmax of the number of arcs in an optimal solution Here dmax is the maximum vertex degree in G Introduction Given a directed graph G V A V f ng with arc weights wij i j A the maximum acyclic subgraph problem is to nd a subset A A such that G V A is acyclic and w A P i j A wij is maximized An alternative statement of this problem the minimum feedback arc set problem requires to nd a minimum weight subset A A such that every directed cycle of G contains at least one arc in A The problem is NP hard Kp It belongs to the class of edge deletion problems Y M It has been shown to be complete for the class of permutation optimization problems MAX SNP de ned in PY that can be approximated within a xed error ratio The problem is polynomially solvable when G is planar L Kv Fr and Ch in GLS The best complexity of these algorithms is O n Ga and O n log nW where W is the largest magnitude of an arc weight and the weights are assumed to be integral Ga The problem is also polynomially solvable for the more general class of K free graphs PN and the classes of reducible ow graphs Ram and weakly acyclic graphs GJR A variation of the problem in which the objective is to minimize the greatest outdegree of a vertex in the subgraph V A can be solved in linear time LS The problem has a variety of applications such as ordering alternatives by group voting deter mining of a hierarchy of the sectors of an economy determining ancestry relationships analysis of systems with feedback and certain scheduling problems Fl J Ram Flood Fl used the relation of the problem to quadratic assignment for developing an e cient branch and bound algorithm J unger J studied the acyclic subgraph polytope The maximum acyclic subgraph and minimum feedback arc set problems are equivalent with respect to their optimal solution However bounded error polynomial approximations are known only for the maximum acyclic subgraph version The simplest algorithm is the following K Let A f i j A j i jg A f i j A j i jg Clearly both V A and V A are acyclic and since A A A maxfw A w A g w A Therefore it is a approximation for the problem Note that the algorithm has linear complexity Korte and Hausmann KH proved that the greedy algorithm i e construct a solution by repeatedly selecting the arc of maximum weight that does not form a directed cycle with the already chosen arcs does not guarantee any xed error ratio A more sophisticated algorithm for the unweighted problem was proposed by Berger and Shor BS They note that without loss of generality we can assume that G has no cycles of length since any bound that can be achieved under this assumption can also be achieved without it Information Processing Letters Department of Statistics and Operations Research School of Mathematical Sciences Tel Aviv University Tel Aviv Israel email fhassin shlomg math tau ac il by a simple modi cation of the algorithm They then develop an algorithm producing an acyclic subgraph of at least p dmax jAj arcs where dmax is the maximum vertex degree of G Even when cycles of length two exist the solution contains at least a fraction p dmax of the number of arcs in an optimal solution The running time of the algorithm is O jAjjV j In this paper we examine a variety of algorithms for the problem Our main contribution is an algorithm for the unweighted problem that guarantees a bound similar to that achieved by Berger and Shore but with time complexity O jAj d max which is better than O jAjjV j in certain cases Inducing a solution from a permutation Call an acyclic subgraph of G maximal if it is not strictly contained in another acyclic subgraph of G Since we assume that the arc weights are positive the maximum acyclic subgraph is also maximal A permutation of f ng induces an acyclic subgraph G V A where A f i j A j i j g Note that G may not be maximal if it is not connected However every maximal acyclic subgraph of G is induced by some permutation One can always renumber the vertices of an acyclic graph so that each arc i j in it satis es i j and if the graph is maximal then it is the one induced by this permutation Therefore the maximum acyclic subgraph problem is exactly the problem of computing a permutation whose induced subgraph is of maximum weight We rst describe a prototype algorithm that generates a permutation such that w A w A For i V and S V let win i S P j S wji wout i S P j S wij Algorithm Set S V u n Choose i S Set S Snfig If w i S w i S set i If w i S w i S set i u u u If u go to Step Else stop and output The weight of the arcs preserved by the algorithm is at least one half of the total weight of A since this property holds in every iteration with respect to the arcs incident with vertex i in the subgraph induced by S