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

Yin Tat Lee, Aaron Sidford, Sam Chiu-wai Wong
2015 arXiv   pre-print
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⊂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 ϵ using an expected O(n(nR/ϵ)) oracle evaluations and additional time O(n^3^O(1)(nR/ϵ)). This matches the oracle complexity and improves upon the O(n^ω+1(nR/ϵ)) additional time of the previous fastest algorithm
more » ... d over 25 years ago by Vaidya for the current matrix multiplication constant ω<2.373 when R/ϵ=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 nM·EO+n^3^O(1)nM) and O(n^3^2 n·EO+n^4^O(1)n), improving upon the previous best of O((n^4EO+n^5) M) and O(n^5EO+n^6). Matroid Intersection: Our runtimes are O(nrT_rank n (nM) +n^3^O(1)(nM)) and O(n^2 (nM) T_ind+n^3 ^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 nCU·EO+n^3^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 Õ(n(n^2+m^ω+S)), improving upon the previous best of Õ(n(n^ω+m^ω+S)) for the regime where the number of nonzeros S is small.
arXiv:1508.04874v2 fatcat:74i57p5lardbnmstsdml66ovta