Maximum flows and minimum cuts in the plane

Abstract

Maximum flow, Minimum cut, Capacity constraint, Cheeger,

Approximation of maximal Cheeger sets by projection

“…With weights f and g as above, we will again refer to h(Ω) as the Cheeger constant and to solutions of (1.3) as Cheeger sets. In this case, the minimization problem (1.2) and the value h(Ω) are related to the so-called maximal flow problem, see Strang [24,25] and Section 2.2 below.…”

“…We first spend some time in Section 2 to motivate the interest of problem (1.2) with general weights f and g. More precisely, we describe two different applied settings where (1.2) and the Cheeger constant h(Ω) naturally arise: landslide modelling [13,[19][20][21] and the continuous maximal flow problem (see Strang [24,25]). In Section 3, we recall the results of [6] on the selection of the maximal Cheeger set.…”

“…The duality between (1.1) and the continuous maximal flow problem is essentially due to Strang [24]. We refer to the papers of Strang [24,25] for a detailed exposition of the problem, extensions and related questions. This continuous duality is used by Appleton and Talbot [4] to solve computer vision problems such as image segmentation.…”

Maximum area with Minkowski measures of perimeter

“…This application of duality is remarkable for the fact that the minimum cut ∂S may be easily determined from (1.1), while the flow vector v(x, y) that fills the cut to capacity has only recently been approximated [37]. No analytic expression for v has been discovered [50,52].…”

“…We refer to our survey [52] for the essential role of the coarea formula in converting the dual to maximum flow into the constrained isoperimetric problem that directly identifies the minimum cut ∂S:…”