A Faster Cutting Plane Method and its Implications for Combinatorial and Convex Optimization

We improve upon the running time for finding a point in a convex set given a separation oracle. In particular, given a separation oracle for a convex set $K\subset \mathbb{R}^n$ contained in a box of radius $R$, we show how to either find a point in $K$ or prove that $K$ does not contain a ball of radius $\epsilon$ using an expected $O(n\log(nR/\epsilon))$ oracle evaluations and additional time $O(n^3\log^{O(1)}(nR/\epsilon))$. This matches the oracle complexity and improves upon the $O(n^{\omega+1}\log(nR/\epsilon))$ additional time of the previous fastest algorithm achieved over 25 years ago by Vaidya for the current matrix multiplication constant $\omega<2.373$ when $R/\epsilon=n^{O(1)}$. Using a mix of standard reductions and new techniques, our algorithm yields improved runtimes for solving classic problems in continuous and combinatorial optimization: Submodular Minimization: Our weakly and strongly polynomial time algorithms have runtimes of $O(n^2\log nM\cdot\text{EO}+n^3\log^{O(1)}nM)$ and $O(n^3\log^2 n\cdot\text{EO}+n^4\log^{O(1)}n)$, improving upon the previous best of $O((n^4\text{EO}+n^5)\log M)$ and $O(n^5\text{EO}+n^6)$. Matroid Intersection: Our runtimes are $O(nrT_{\text{rank}}\log n\log (nM) +n^3\log^{O(1)}(nM))$ and $O(n^2\log (nM) T_{\text{ind}}+n^3 \log^{O(1)} (nM))$, achieving the first quadratic bound on the query complexity for the independence and rank oracles. In the unweighted case, this is the first improvement since 1986 for independence oracle. Submodular Flow: Our runtime is $O(n^2\log nCU\cdot\text{EO}+n^3\log^{O(1)}nCU)$, improving upon the previous bests from 15 years ago roughly by a factor of $O(n^4)$. Semidefinite Programming: Our runtime is $\tilde{O}(n(n^2+m^{\omega}+S))$, improving upon the previous best of $\tilde{O}(n(n^{\omega}+m^{\omega}+S))$ for the regime where the number of nonzeros $S$ is small.Comment: 111 pages, FOCS 201

Fair Integral Network Flows

A strongly polynomial algorithm is developed for finding an integer-valued feasible $st$-flow of given flow-amount which is decreasingly minimal on a specified subset $F$ of edges in the sense that the largest flow-value on $F$ is as small as possible, within this, the second largest flow-value on $F$ is as small as possible, within this, the third largest flow-value on $F$ is as small as possible, and so on. A characterization of the set of these $st$-flows gives rise to an algorithm to compute a cheapest $F$-decreasingly minimal integer-valued feasible $st$-flow of given flow-amount. Decreasing minimality is a possible formal way to capture the intuitive notion of fairness.Comment: 37 page