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

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 {\em deterministic, almost-linear time} approximation algorithm for this problem. Specifically, our algorithm, given an $n$-vertex $m$-edge graph $G$ and any parameter $1\leq r\leq O(\log n)$, computes a $(\log m)^{{r2}$-approximation} for Minimum Balanced Cut on $G$, in time $O\left ( m^{{1+O(1/r)+o(1)}\cdot} (\log m)^{{O(r2)}\right} )$. In particular, we obtain a $(\log m)^{{1/\epsilon}$-approximation} in time $m^{{1+O(1/\sqrt{\epsilon})}$} for any constant $\epsilon$, and a $(\log m)^{{f(m)}$-approximation} in time $m^{1+o(1)}$, for any slowly growing function $m$. 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 $n^{o(1)}$, 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.