A Deterministic Algorithm for Balanced Cut with Applications to Dynamic Connectivity, Flows, and Beyond

Abstract

We consider the classical Minimum Balanced Cut problem: given a graph G, compute a partition of its vertices into two subsets of roughly equal volume, while minimizing the number of edges connecting the subsets. We present the first deterministic, almost-linear time approximation algorithm for this problem. Specifically, our algorithm, given an n-vertex m-edge graph G and any parameter 1 ≤ r ≤ O(logn), computes a (logm) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r2</sup> -approximation for Minimum Balanced Cut in G, in time O(m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+O(1/r)+o(1)</sup> ·(logm) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(r2</sup> )). In particular, we obtain a (logm) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1/ε</sup> -approximation in time m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+O(√{ε})</sup> for any constant , and a (logm) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">f(m)</sup> -approximation in time m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1+o(1)</sup> , for any slowly growing function f(m). We obtain deterministic algorithms with similar guarantees for the Sparsest Cut and the Lowest-Conductance Cut problems. Our algorithm for the Minimum Balanced Cut problem in fact provides a stronger guarantee: it either returns a balanced cut whose value is close to a given target value, or it certifies that such a cut does not exist by exhibiting a large subgraph of G that has high conductance. We use this algorithm to obtain deterministic algorithms for dynamic connectivity and minimum spanning forest, whose worst-case update time on an n-vertex graph is n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">o(1)</sup> , thus resolving a major open problem in the area of dynamic graph algorithms. Our work also implies deterministic algorithms for a host of additional problems, whose time complexities match, up to subpolynomial in n factors, those of known randomized algorithms. The implications include almost-linear time deterministic algorithms for solving Laplacian systems and for approximating maximum flows in undirected graphs.