Matching is as Easy as Matrix Inversion Ketan Mulmuley ’ Computer Science Department University of California, Berkeley Umesh V. Vazirani 2 Aiken Computation Laboratory Harvard University Vijay V. Vazirani 3 Computer Science Department Cornell University Abstract only computationally non-trivial step required in its execution is the inversion of a A new algorithm for finding a maximum single integer matrix. Since this step can be matching in a general graph is presented; its parallel&d, we get a simple parallel (RNC2) special feature being that the only computa- algorithm. Because of this simplicity, the tionally non-trivial step required in its execu- sequential version of our algorithm has some tion is the inversion of a single integer merits over the conventional matching algo- matrix. Since this step can be parallelized, rithms as well. we get a simple parallel (RNC2) algorithm. This algorithm was obtained while At the heart of our algorithm lies a proba- solving the matching problem from the bilistic lemma, the isolating lemma. We viewpoint of parallel computation. The main show applications of this lemma to parallel difficulty here is that the graph may contain computation and randomized reductions. exponentially many maximum matchings; how do we coordinate the processors so they seek the same matching in parallel? The key to achieving this coordination is a probabilis- 1. Introduction tic lemma, the isolating lemma, which lies at the heart of our algorithm; it helps to single A new algorithm for finding a max- out one matching in the graph. imum matching in a general graph is In its general form, the isolating presented; its special feature being that the lemma holds for an arbitrary set system. This yields a relationship between the paral- Permission to copy without fee all or part of this lei complexity of an arbitrary search problem material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication ’ Miller Fellow, University of California, Berkeley. and its date appear, and notice is given that copying is s Work done during a post-doctoral position at MSRI, by permission of the Association for Computing Berkeley. Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 3 Supported by NSF Grant BCR 8503611 and an IBM Faculty Development Award. Currently at AT&T Bell Labs, Murray Hill, NJ 07974. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. 0 1987ACM O-89791-221-7/87/0006-0345 75c 345 and the corresponding weighted decision Tutte in 1947 ~Tu], based on the work of problem. As an application, we give an Pfaff on skew-symmetric matrices. It states RNC’ algorithm for the Exact Matching that a graph has a perfect matching iff a cer- problem in general graphs. It is interesting tain matrix of indeterminates, called the to note that this problem is not known to be Tutte matrix,, is non-singular. Motivated by in P. The isolating lemma also yields a sim- an a1gorithmi.c use of this theorem, :Edmonds ple proof for the result in IValVaz] showing [Ed21 studied the complexity of computing the NP-hardness, under randomized reduc- determinants.. He gave a modified Gaussian tions, of instances of SAT having a unique elimination procedure for computing the solution. determinant of an integer matrix in a polyno- Matching was first shown to be in mial number of bit operations, and stated the Random NC @NC3 ) by Karp, Upfal and open problem of efficiently deciding whcthcr Wigderson [KUW 11. Their algorithm also a matrix of indeterminates is non-singular. utilizes matrix operations, and in fact these The first algorithm based on Tuttc’s are some of the most widely used tools for theorem was given by Lovasz l&01]. Using obtaining fast parallel algorithms. Several the fundamlental insight that polynomial problems are known to be NC’ reducible to identities can be efficiently tested by ran- computing the determinant of an integer domization [SC], Lovasz reduced, the deci- matrix. Cook [Co] defines DET to be the sion problem, ‘Does the given graph have a class of all such problems. DET G NC2, and perfect matching? ’ to testing if a given it is not known whether this inclusion is integer matrix is non-singular. Since the proper. All problems known to be in NC2 latter problem is in NC2 [Cs], this yields a are either in AC1 or in DET [Co]. Our algo- (Monte Carlo) RNC2 algorithm for the rithm puts the matching problem in RDET. former problem (see also [BGH]). Rabin and Vazirani IRV] extended this approach, using a theorem of Frobenius, to give a simple ran- domizing algorithm which finds a perfect matching by sequentially inverting IV I /2 2. History matrices. The search problem, i.e. actually find- The maximum matching problem is a ing a perfect matching in parallel, is much natural problem, and its study has led to harder. The first parallel (RNC3) algorithm conceptual breakthroughs in the field of for this long-standing open problem was algorithms. In fact, the characterization of given by Karp, Upfal and Wigderson ‘tractable problems’ as ‘polynomial time solv- [KUWl]. Tbey use the Tutte matrix to able problems’ was first proposed by implement (in RNC2) a ‘rank’ function. Edmonds [Edl] in the context of the general Their algorithm probabilistically prunes out graph matching problem. Solving this prob- edges from the graph; the rank function lem from the viewpoint of parallel computa- guarantees that the remaining graph has a tion has also been quite fruitful. perfect matching, and a probabilistic lemma Whereas sequential algorithms for the ensures that a constant fraction of the edges maximum matching problem are based on are pruned at each stage. Hence, after finding ‘blossoms’ and ‘augmenting paths’ in O(Iog IVI) stages only a perfect matching graphs (see lEdl]), the known parallel algo- remains. This algorithm is also Monte Carlo rithms require a new approach; they use pro- in that it may fail to give a perfect matching. babilistic and algebraic methods. In fact, the Using the Gallai-Edmonds Structure matching problem emerged from algebra Theorem (see [LOO]) Karloff [Ka] gives a around the turn of this century in the works complementary Monte Carlo (RNC2) algo- of Petersen, Frobenius and Konig (for a rithm for bounding the size of a maximum detailed history see [LPI), A key ingredient matching from above, thus yielding a Las in the new approach is a theorem proved by Vegas extension. 346 Our algorithm is conceptually different Since S contains n elements, in that it directly finds a perfect matching in the graph. It is somewhat faster (RI@) and Pr [There exists a singular requires 0 (n3*‘m) processors. element] 5 ( ‘1 xn= l/2. 2n Thus, with probability at least l/2, no ele- ment is singular. The lemma follows from 3. The Isolating Lemma the observation that there is a unique minimum weight set iff no element is singu- lar. Definition: A set system (S, F) consists of a Notice that by the same argument, the finite set S of elements, maximum weight set will be unique with pro- s = {Xl, x2, . ..) x,), and a family F of sub- bability at least l/2 as well. An extension of sets of S, i.e. F = {S,, S2, * - - Sk}, SQ, the isolating lemma is required in Theorem for 1 5 j I k. 2(a). In this extension we are given integers Let us assign a weight wi to each ele- al,a2r ’ ’ ’ at, and the weight of set Si is ment Xi E S and let us define the weight of defined to be aj + C Wi, 15 j5 k. The the set Sj to be C Wi. x&sj proof given above works for this case as well. Lemma 1: Let (S, F) be a set system whose elements are assigned integer weights chosen uniformly and independently from 11, 2n], Then, Pr [There is a unique minimum weight set 4. The Matching Algorithm 1 in Fl2 2’ We will first consider the simpler case of a bipartite graph: Proof: Fix the weights of all elements except Xi. Define the threshold for element Xi, to be Input: A bipartite graph G(iJ,V,E), having the real number a; such that if wi I oi then a perfect matching. Xi is contained in some minimum weight sub- set, Si, and if wi > oi then Xi is in no Problem : Find a perfect matching in G. minimum weight subset. We will view the edges in E and the Clearly, if Wi < oi, then the element Xi set of perfect matchings in G as a set system. must be in every minimum weight subset. Let us assign random integer weights to the Thus ambiguity about element Xi occurs iff edges of the graph, chosen uniformly and Wi = (xi, since in this case there is a independently from [I, 2m1, where minimum weight subset that contains Xi and m = [El. Now by lemma 1, the minimum another which does not. In this case we shall weight perfect matching in G will be unique say that the element Xi is singular. with probability at least l/2. Our parallel We now make the crucial observation algorithm will pick out this perfect matching.