This paper bridges discrete and continuous optimization approaches for decomposable submodular function minimization, in both the standard and parametric settings. We provide improved running times for this problem by reducing it to a number of calls to a maximum flow oracle. When each function in the decomposition acts on $O(1)$ elements of the ground set $V$ and is polynomially bounded, our running time is up to polylogarithmic factors equal to that of solving maximum flow in a sparse graph with $O(\vert V \vert)$ vertices and polynomial integral capacities. We achieve this by providing a simple iterative method which can optimize to high precision any convex function defined on the submodular base polytope, provided we can efficiently minimize it on the base polytope corresponding to the cut function of a certain graph that we construct. We solve this minimization problem by lifting the solutions of a parametric cut problem, which we obtain via a new efficient combinatorial reduction to maximum flow. This reduction is of independent interest and implies some previously unknown bounds for the parametric minimum $s,t$-cut problem in multiple settings.
翻译:在标准设置和参数设置中,这种纸上连接离散的和连续的优化方法可以使分解的子模块功能最小化。 我们通过将这一问题降低到一个最大流点的多个调频点,为这一问题提供了更好的运行时间。 当对地面的$O(1)美元元元元元元的分解动作中的每一函数设定了$V$, 并且是多元的, 我们的运行时间将达到多元性因素, 等同于用美元( vert V\verd) $( verd) 的悬浮图解析最大流, 以及多元整体能力。 我们通过提供简单的迭接合方法来实现这一目标, 可以优化到高精度, 任何在子模块基数聚点上定义的同系函数, 只要我们能有效地在与我们构建的某个图形的切断函数相对的基数多功能上将其最小化为最小化。 我们通过取消对等分解问题的解决办法来解决这个最小化问题, 我们通过新的高效的调控波减少至最大流点来获得。 这种减少具有独立的利益, 并意味着在多个环境中对准最小值最小值的最小值问题有一些未知的界限。