Fast Interior Point Methods for Bipartite Matching

Abstract

In this paper we show that by using path following interior point methods with nonlogarithmic potential functions that vary inversely with the $\beta $th power of the distances from the hyperplane (with $\beta = O ( \log v )$), it is possible to obtain an approximate bipartite matching with the number of edges within a factor of $( 1 - \frac{1}{\rho } )$ of that in the optimal matching for arbitrarily specified $\rho $ in $O^* ( \rho )$ matrix inversions ($O^* ( X ) = O( X\log^k n )$, i.e., we ignore logarithmic factors of n in stating most bounds in this paper). At present the best-known logarithmic time parallel algorithm for finding an approximate matching is that for finding a maximal matching that contains at least half of the edges in the optimal matching by Karp and Wigderson [J. ACM, 32 (1985), pp. 762–773]. By combining the approximate matching algorithm discussed in this paper with an augmenting path algorithm it is possible to derive the optimal matching in $O^* ( v^{1/2} )$ time. The previous fastest parallel algorithms for general bipartite graphs are those by Vaidya [Pros. 22nd Ann. ACM Symp. Theory Computing, 1990, pp. 583–589], which runs in $O^* ( ( v e )^{1/4} )$ time and that by Goldberg, Plotkin, and Vaidya [Proc. 29th IEEE Symp. Foundations of Computer Science, 1990, pp. 175–185], which obtains solutions in $O^* ( v^{2/3} )$ time.