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 $1leq rleq O(log n)$, computes a $(log m)^{r^2}$-approximation for Minimum Balanced Cut on $G$, in time $Oleft ( m^{1+O(1/r)+o(1)}cdot (log m)^{O(r^2)}right )$. 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.
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