Optimal Geometric Flows via Dual Programs

Abstract

Considering potentials in the dual of a planar network has proved to be a powerful tool for computing planar maximum flows. In this paper we explore the use of potentials for giving algorithmic and combinatorial results on continuous flows in geometric domains -a (far going) generalization of discrete flows in unit-capacity planar networks.A continuous flow in a polygonal domain is a divergencefree vector field whose magnitude at every point is bounded by a given constant -the domain's permeability. The flow enters the domain through a source edge on the outer boundary of the domain, and leaves through a sink edge on the boundary. The value of the flow is the total flow amount that comes in through the source (the same amount comes out of the sink, due to the vanishing divergence). The cost of the flow is the total length of its streamlines. The flow is called monotone if its streamlines are x-monotone curves.Our main result is an algorithm to find (an arbitrarily close approximation to) the minimum-cost monotone flow in a polygonal domain, by formulating the problem as a convex program with an interesting choice of variables: one variable per hole in the domain. Our approach is based on the following flow of ideas: flow is the gradient of a potential function; a potential function can be extended from free space to holes so that it is constant over each hole; instead of the potential function (a value per point in the domain) we can thus speak of a potential vector (a value per hole); a potential uniquely (up to homotopic equivalence) defines "thick" paths in the domain; the paths define a flow. We show that the flow cost is convex in the potential; this allows us to employ the ellipsoid method to find the mincost flow, using holes potentials as variables. At each ellipsoid iteration the separation oracle is implemented by feasibility checks in the "critical graph" of the domain -as we prove, the graph defines linear constraints on the potentials.Using potentials and critical graphs we also prove MaxFlow/ MinCut theorems for geometric flows -both for monotone and for general ones. Formulating and proving the monotone Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Permissions@acm.org.

Most Vital Segment Barriers

“…Other modeling choices could have been made; e.g, another way to avoid complete blockage could be to introduce a "protected zone" around s à la in works on geographic mincut[24]. Also a more generic view, outside our scope, could be to combine the flow and path problems into considering minimum-cost flows[22,10] (the shortest path is the mincost flow of value 0) and explore how the barriers could influence both the capacity of the domain and the cost of the flow.…”