58 O N O P T I M A Mathematical Programming Society Newsletter JUNE 1998 David Karger was awarded the 1997 Tucker Prize by the Randomization in Society for his thesis Random Graph Optimization Sampling in Graph Optimiza- tion Problems. The jury found Problems: A Survey his contribution, on the inter- face between computer sci- David Karger* ence and optimization, deep and elegant. To make David’s Abstract work even better known among the membership, I Randomization has become a pervasive technique in combinatorial invited him to write a feature optimization. We survey our thesis and subsequent work, which article on some of his very uses four common randomization techniques to attack numerous interesting techniques and optimization problems on undirected graphs. results. David Karger obtained an A.B. degree from Harvard 1 Introduction University, a Math Certificate Randomization has become a pervasive technique in combinatorial optimization. Randomization has been used from Cambridge University, to develop algorithms that are faster, simpler, and/or better-performing than previous deterministic algorithms. and a Ph.D. degree from This article surveys our thesis [Kar94], which presents randomized algorithms for numerous problems on Stanford University. Apart undirected graphs. Our work uses four important randomization techniques: Random Selection, which lets us from the Tucker Prize, David easily choose a “typical” element of a set, avoiding rare “bad” elements; Random Sampling, which provides a has been awarded numerous quick way to build a small, representative subproblem of a larger problem for quick analysis; Randomized other fellowships and prizes, such as the ACM Doctoral Rounding, which lets us transform fractional problem solutions into integral ones; and Monte Carlo Simula- Dissertation award, an Alfred tion, which lets us estimate the probabilities of interesting events. P. Sloane Foundation Fellow- We apply these techniques to numerous optimization problems on undirected graphs. The graph is one of ship, and a David and Lucille the most common structures in computer science and optimization, modeling among other things roads, com- Packard Foundation Fellow- munication and transportation networks, electrical circuits, relationships between individuals, hypertext collec- ship. His main research inter- tions, resource allocations, project plans, database and program dependencies, and parallel architectures. Among ests are analysis of algorithms the graph problems we address are finding a minimum spanning tree, finding a maximum flow, determining and information retrieval, on the connectivity (minimum cut) of a graph, network design, graph coloring, and estimating the reliability which he has written numer- (disconnection probability) of a network with random edge failures. A great deal of work has been done on all ous papers that appeared in of these problems. Due to space limitations, we are unable to discuss all of this related work. Such a discussion both conference proceedings can be found in our thesis. and journals. Since 1995, David has been assistant pro- It is extremely important to note that we are not carrying out “expected case” analysis of algorithms running fessor at the Laboratory for on random inputs. Rather, we consider worst case inputs and use random choices by the algorithm to solve them Computer Science at MIT. efficiently. If you want to find out more We begin this article with a survey of many of our applications of the four randomization techniques, sketch- about David’s research activi- ing the methods, the problems, and the resulting algorithms. Afterwards, we will present a more detailed dis- ties, his home page (see note, cussion of a few algorithms and proofs which will hopefully give some flavor of our work. below) is an excellent source! – Karen Aardal PAGE TWO þ *MIT Laboratory for Computer Science, Cambridge, MA 02138 R e-mail: [email protected] E URL: http://theory.lcs.mit.edu/~karger Work supported in part by ARPA contract N00014-95-1-1246 and NSF contract CCR-9624239, B and Fellowships from the Alfred P. Sloane and David and Lucille Packard Foundations. M 58 U updated MIPLIB 12 conference notes 16 reviews 19 gallimaufry 23 N OPTIMA58JUNE 1998 PAGE 2 Mathematical Programming has a new publisher. 1.1 Notation We address only undirected graphs; directed graphs have so far not succumbed to the techniques we apply e are happy to announce that here. Throughout our discussion, we will consider a Programming via Springer’s LINK service. starting in 1999, Mathematical graph G with m edges and n vertices. Each graph edge Springer is one of the world’s best known Programming will be published by may have a weight (reflecting cost or capacity) associated W publishers in mathematics, computer science Springer-Verlag. This decision was the result with it. To simplify our presentation here, we often focus and operations research and will launch a ma- of an extensive search and negotiation carried on unweighted graphs (that is, graphs with all edge jor international promotional campaign for out by a committee made up of Stephen our journal. Increased subscriptions generated weights equal to one), though many of our algorithms Wright, Jan Karel Lenstra and myself. The de- by Springer will also increase visibility of our apply equally well to weighted graphs. Unless it has liberations of the committee were reviewed society and journal in related fields of re- parallel edges (multiple edges with the same endpoints) periodically by the Mathematical Program- search. Members of the society are urged to a graph has m £ (n). 2 ming Society Council, with the final decision support this effort by encouraging “their” The notation Õ(f ) denotes O(f logd n) for some subsequently approved by the Council. The libraries to subscribe to the journal. constant d. three principal reasons for the decision: From August onwards, society members 1.2 Overview of Results Springer’s willingness to accept a pricing will have the chance for a “sneak preview” for scheme in line with society expectations; the We show how randomization can be used in several papers already accepted for publication in worldwide editorial and marketing capabilities ways on problems of varying degrees of difficulty. For Volume 84 for 1999. To see how the elec- of Springer; and a modern, successful facility the “easy to solve” minimum spanning tree problem, tronic journal facility is working, please look for providing electronic access to the journal. at http://link.springer.de/. where a long line of research has resulted in ever closer The contractual details were settled just If you have any questions concerning the to linear-time algorithms, random sampling gives the before Christmas 1997. As before, each mem- change, the editors to contact at Springer- final small increment to a truly linear time algorithm. ber of the society will receive a free copy of Verlag are: Dr. Martin Peters, Senior Math- For harder problems it improves running times by a Mathematical Programming. Additional ematics Editor, Springer-Verlag, more significant factor. For example, we improve the benefits include: Tiergartenstrasse 17, D-69121 Heidelberg, time needed to find minimum cuts from Õ(mn) to • Each quarter, Springer-Verlag will offer a list Germany ([email protected]); Achi Õ(m), and give the first efficient parallel algorithm for of 50 books selected to be of particular inter- Dosanjh, Mathematics Editor, Springer- the problem. Addressing some hard NP-complete est to MPS. These books will be available at Verlag, 175 Fifth Avenue, New York, NY problems such as network design and graph coloring, a discount of 25%. The list will be published 10010, USA ([email protected]); where finding an exact solution in polynomial time is in “OPTIMA” and as part of Springer’s mar- Maria Taylor, Executive Editor, Physical Sci- thought to be hopeless, we use randomized rounding keting information on the Internet (http:// ences Journals, Springer-Verlag, 175 Fifth to give better approximation algorithms than were pre- www.springer.de). Avenue, New York, NY 10010, USA viously known. Finally, for the problem of determin- • Members of the society will have free access ([email protected]). ing the reliability of a network (a #P-complete prob- to the electronic version of Mathematical –BOB BIXBY, CHAIR OF THE PUBLICATIONS COMMITTEE lem which is thought to be “even harder” than NP- complete ones) we use Monte Carlo simulation to give the first efficient approximation algorithm. but has a low probability of giving an incorrect answer ing it several times and taking the best answer; we shall 1.3 Randomized Algorithms is called Monte Carlo (MC). If the running time of the see several examples of this below. In particular, we can Our work deals with randomized algorithms. Our algorithm is a random variable but the correct answer reduce the failure probability so far that other unavoid- typical model is that the algorithm has a source of “ran- is given with certainty, then the algorithm is said to be able events (such as a power failure) are more likely than dom bits”–variables that are mutually independent and Las Vegas (LV). Depending on the circumstances, one an incorrect answer. take on values 0 or 1 with probability 1/2 each. Extract- type of algorithm may be better than the other. How- ing one random bit from the source is assumed to take ever, a Las Vegas algorithm is “stronger” in the follow- 2 A Survey of Techniques and constant time. If our algorithms use more complex ing sense. Results A Las Vegas algorithm can be made Monte Carlo by operations, such as flipping biased coins or generating In this section, we provide a high level overview of the having it terminate with an arbitrary wrong answer if it samples from more complex distributions, we take into four randomization techniques and the various algo- exceeds the time bound f(n).