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Approximating the Minimum Equivalent Digraph

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Approximating the Minimum Equivalent Digraph Chapter 21 Approximating the Minimum Equivalent Digraph Samir Khuller * Balaji Raghavachari t Neal Young $ Abstract 1 Introduction The MEG (minimum equivalent graph) problem Connectivity is fundamental to the study of graphs is the following: “Given a directed graph, find and graph algorithms. Recently, many approxima- a smallest subset of the edges that maintains tion algorithms for finding subgraphs that meet all reachability relations between nodes.” The given connectivity requirements have been devel- MEG problem is NP-hard; this paper gives an oped [l, 9, 11, 15, 16, 241. These results provide approximation algorithm achieving a performance practical approximation algorithms for NP-hard guarantee of about 1.64 in polynomial time. We network-design problems via an increased under- give a modification that improves the performance standing of connectivity properties. guarantee to about 1.61. The algorithm achieves a Until now, the techniques developed have been performance guarantee of 1.75 in the time required applicable only to undirected graphs. We consider for transitive closure. a basic network-design problem in directed graphs The heart of the MEG problem is the min- [2, 12, 13, 181 w h ic h is as follows: given a digraph, imum SCSS (strongly connected spanning sub- find a smallest subset of the edges (forming a graph) problem - the MEG problem restricted minimum equivalent graph (MEG)) that maintains to strongly connected digraphs. For the mini- all reachability relations of the original graph. mum SCSS problem, the paper gives a practical, When the MEG problem is restricted to graphs nearly linear-time implementation achieving a per- which are strongly connected, we call it the mini- formance guarantee of 1.75. mum SCSS (strongly connected spanning subgraph) The algorithm and its analysis are based on problem. When the MEG problem is restricted to the simple idea of contracting long cycles. The acyclic graphs we call it the acyclic MEG prob- analysis applies directly to Z-EXCHANGE, a general Zem. The MEG problem reduces in linear time [5] “local improvement” algorithm, showing that its to a single acyclic problem given by the so-called performance guarantee is 1.75. “strong component graph”, together with one min- Keywords: directed graph, reachability, approx- imum SCSS problem for each strong component imation algorithm, strong connectivity, local im- (given by the subgraph induced by that com- provement . ponent). Furthermore, approximating the MEG problem is linear-time equivalent to approximating both restricted versions. *Computer Science Department and Institute for Advanced Moyles and Thompson [18] observe this de- Computer Studies, University of Maryland, College Park, MD 20742. Research supported by NSF Research Initiation composition and give exponential-time algorithms Award CCR-9307462. Email : samirQcs .umd. edu. for the restricted problems. Hsu [13] gives a tcomputer Science Department, The University of Texas at polynomial-time algorithm for the acyclic MEG Dallas, Box 830688, Richardson, TX 75083-0688. E-mail : problem and corrects some errors in the paper by rbkQutdallas . edu. Moyles and Thompson. *Computer Science Department, Princeton University, Prince- The related problem of finding a transitive re- ton, NJ 08544. E-mail : neyQcs .princeton.edu. This work was done while at University of Maryland Institute for Advanced Com- duction of a digraph - a smallest set of edges yield- puter Studies, College Park, MD 20742 and was supported by NSF ing the same reachability relations was studied by grants CCR-8906949 and CCR-9111348. 177 178 KHULLER ET AL. Aho, Garey and Ullman [a]. Transitive reduction called Z-EXCHANGE. %E,XcHANGE starts with the differs from the MEG problem in that the edges in given digraph and perfor:ms the following local im- the transitive reduction are not required to be in provement step as long as it is applicable: find two the original graph. However, the transitive reduc- edges in the current gra,ph that can be replaced tion problem decomposes just like the MEG prob- by one edge from the original graph, maintaining lem into acyclic and strongly connected instances. strong connectivity. Similar local-improvement al- For any strongly connected instance, a transitive gorithms are natural can&dates for many optimiza- reduction is given by any Hamilton cycle through tion problems but often elude analysis. We prove the vertices. For an acyclic instance, the transitive that the performance guarantee of Z-EXCHANGE is reduction is unique and, as Aho, Garey and Ullman 1.75. observe, is equivalent to the MEG problem: it con- A natural improvement to the cycle-contrac- sists of those edges (u,~) for which there is no al- tion algorithm is to modify the algorithm to solve ternate path from u to o. In fact, Aho, Garey and the problem optimally once the contracted graph Ullman show that the transitive reduction prob- has no cycles longer than a given length c. For lem is equivalent to the transitive closure problem. instance, for c = 3, this modification improves the Thus, the acyclic MEG problem reduces to transi- performance guarantee to a2/6- l/36 M 1.617. We tive closure. use SCSS, to denote the minimum SCSS problem The acyclic MEG problem can be solved in restricted to digraphs with no cycle longer than polynomial time, whereas the minimum SCSS c. The minimum SCSS2 problem is trivial. The problem is NP-hard [8]. Consequently, this pa- minimum SCSSa problem is at least as hard as per focuses on approximation algorithms for the bipartite matching; in fact we can show that it has minimum SCSS problem. By the observations of a polynomial-time algorithm. However, potential the preceding paragraphs, the performance guar- improvement in this direction is limited: we show antees obtained for the minimum SCSS problem that the minimum SCSS:, problem is NP-hard. In carry over to the general MEG problem with the fact, we show that the minimum SCSSrr problem overhead of solving a single instance of transitive is SNP-hard. This precludes the possibility of a closure. polynomial-time approximation scheme, assuming PfNP [4]. 1.1 Our Results. Given a strongly con- nected graph, our basic algorithm finds as long a 1.2 Other Related Work. The union of cycle as it can, contracts the cycle, and recurses. any incoming branching and any outgoing branch- The contracted graph remains strongly connected. ing from the same root yields an SCSS with at most When the graph finally collapses into a single ver- 2n - 2 edges (where n is the number of vertices in tex, the algorithm returns the set of edges con- the graph). This is a special case of the algorithm tracted during the course of the algorithm as the given by Frederickson and JBJQ [6] that uses min- desired SCSS. imum weight branchings to achieve a performance The algorithm achieves a performance guaran- guarantee of 2 for weighted graphs. Since any SCSS tee of any constant greater than 7r2/6 w 1.645 in has at least n edges, this yields a performance guar- polynomial time. We give a nearly linear-time ver- antee of 2 for the SCSS problem. sion that achieves a performance guarantee of 1.75. Any minimal SCSS ( one from which no edge We give examples showing lower bounds on the per- can be deleted) has at most 2n - 2 edges and also formance guarantees of the algorithm. For the gen- yields a performance guarantee of 2. A linear- eral algorithm, the lower bounds are slightly above time algorithm finding a minimal SCSS is given 1.5. For the nearly linear-time version, the lower by Simon [21]. A parallel algorithm is given bound is 1.75, matching the upper bound. by Gibbons, Karp, Ramachandran, Soroker and The performance guarantee analysis extends Tarjan [lo]. directly to a simple “local improvement” algorithm A related problem in undirected graphs is to APPROXIMATING THE MINIMUM EQUIVALENT DIGRAPH 179 find a smallest subset of the edges forming a Proof. Starting with a minimum-size subset biconnected (respectively bridge-connected (i.e., 2- that strongly connects the graph, repeatedly con- edge-connected)) spanning subgraph of a given tract cycles in the subset until no cycles are left. graph. These problems are NP-hard. Khuller Observe that the maximum cycle length does not and Vishkin [15] g ive a DFS-based algorithm that increase under contractions. Consequently, for achieves a factor of $ for biconnectivity and 5 for each cycle contracted, the ratio of the number of bridge-connectivity. Garg, Santosh and Singla [9] edges deleted to the decrease in the number of ver- subsequently improve the approximation factors, tices is at least A. Since the total decrease in using a similar approach, to $ and z, respectively. the number of vertices is n - 1, at least &(n - 1) None of these methods appear to extend to the edges are deleted. 0 minimum SCSS problem. Note that the above lemma gives a lower bound Undirected graphs having bounded cycle length which is existentially tight. For all values of C, have bounded tree width. Arnborg, Lagergren there exist graphs for which the bound given by and Seese [3] have shown that many NP-hard the lemma is equal to OPI(G). Also note that C problems, including the minimum biconnected- has a trivial upper bound of n and, using this, we spanning-subgraph problem, have polynomial-time get a lower bound of n for UP?+(G), which is the algorithms when restricted to such graphs. known trivial lower bound. LEMMA 3.2. (CONTRACTION LEMMA) For 2 Preliminaries any directed graph G and set of edges S, To contract a pair of vertices U,Z) of a digraph is OPI(G) 2 OPI(G/S). to replace u and w (and each occurrence of u or 2, in any edge) by a single new vertex, and to delete Proof.
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