We provide faster running times for exactly solving discounted Markov Decision Processes (DMDPs) in strongly polynomial time. We obtain our results by efficiently reducing computing optimal values and policies in DMDPs to the easier tasks of policy evaluation and computing approximately optimal values. We provide both a straightforward deterministic reduction and a more efficient randomized variant that, together with advances in approximately solving DMDPs, yield our results.

We give a combinatorial algorithm for computing exact maximum flows in directed graphs with n vertices and edge capacities from {1, …, U} in $\tilde O\left({{n^2}\log U}\right)$ time, which is near-optimal on dense graphs. This shaves an no(1)factor from the recent result of [Bernstein–Blikstad–Saranurak–Tu FOCS’24] and, more importantly, greatly simplifies their algorithm. We believe that ours is by a significant margin the simplest of all algorithms that go beyond $\tilde O(m\sqrt n )$ time in general graphs. To highlight this relative simplicity, we provide a full implementation of the algorithm in C++.The only randomized component of our work is the cut-matching game. Via existing tools, we show how to derandomize it for vertex-capacitated max flow and obtain a deterministic $\tilde O\left({{n^2}}\right)$ time algorithm. This marks the first deterministic near-linear time algorithm for this problem (or even for the special case of bipartite matching) in any density regime.

We provide an algorithm that maintains, against an adaptive adversary, a (1 — ε )-approximate maximum matching in n-node m-edge general (not necessarily bipartite) undirected graph undergoing edge deletions with high probability with (amortized) O (poly(ε-1, log n )) time per update. We also obtain the same update time for maintaining a fractional approximate weighted matching (and hence an approximation to the value of the maximum weight matching) and an integral approximate weighted matching in dense graphs.1 Our unweighted result improves upon the prior state-of-the-art which includes a poly(log n ) · 2O (1/ɛ2) update time [Assadi-Bernstein-Dudeja 2022] and an update time [Gupta-Peng 2013], and our weighted result improves upon the log n ) update time due to [Gupta-Peng 2013].

We consider the foundational problem of maintaining a (1 — ε )-approximate maximum weight matching (MWM) in an n-node dynamic graph undergoing edge insertions and deletions. We provide a general reduction that reduces the problem on graphs with a weight range of poly(n ) to poly(1/ε ) at the cost of just an additive poly(1/ε ) in update time. This improves upon the prior reduction of Gupta-Peng (FOCS 2013) which reduces the problem to a weight range of ε-O (1/ε) with a multiplicative cost of O (log n ).

We present a combinatorial algorithm for computing exact maximum flows in directed graphs with$n$vertices and edge capacities from$\{1, \ldots, U\}$in$n^{2+o(1)}\log U$time, which is almost optimal in dense graphs. Our algorithm is a novel implementation of the classical augmenting-path framework; we list augmenting paths more efficiently using a new variant of the push-relabel algorithm that uses additional edge weights to guide the algorithm, and we derive the edge weights by constructing a directed expander hierarchy. Even in unit-capacity graphs, this breaks the long-standing$O(m \cdot\min\{\sqrt{m},n^{2/3}\})$time bound of the previous combinatorial algorithms by Karzanov (1973) and Even and Tarjan (1975) when the graph has$m=\omega(n^{4/3})$edges. Notably, our approach does not rely on continuous optimization nor heavy dynamic graph data structures, both of which are crucial in the recent developments that led to the almost-linear time algorithm by Chen et al. (FOCS 2022). Our running time also matches the$n^{2+o(1)}$time bound of the independent combinatorial algorithm by Chuzhoy and Khanna (STOC 2024) for computing the maximum bipartite matching, a special case of maximum flow.

We initiate the study of matroid problems in a new oracle model called dynamic oracle. Our algorithms in this model lead to new bounds for some classic problems, and a “unified” algorithm whose performance matches previous results developed in various papers for various problems. We also show a lower bound that answers some open problems from a few decades ago. Concretely, our results are as follows.