Faster Algorithms for Next Breakpoint and Max Value for Parametric Global Minimum Cuts

Abstract

The parametric global minimum cut problem concerns a graph $G = (V, E)$ where the cost of each edge is an affine function of a parameter $\mu \in \mathbb {R}^d$ for some fixed dimension d. We consider the problems of finding the next breakpoint in a given direction, and finding a parameter value with maximum minimum cut value. We develop strongly polynomial algorithms for these problems that are faster than a naive application of Megiddo’s parametric search technique. Our results indicate that the next breakpoint problem is easier than the max value problem.

H. Aissi—This research benefited from the support of the FMJH Program PGMO and from the support of EDF, Thales, Orange et Criteo.

5 Appendix

1.1 5.1 Geometric tools

A classical problem in computational geometry called point location in arrangements (PLA) is useful to our algorithm. PLA has been widely used in various contexts such as linear programming [1, 18] or parametric optimization [9, 30]. For more details, see [22, Chapter 5].

Given a simplex P, an arrangement $A(\mathcal {H})$ formed by a set $\mathcal {H}$ of hyperplanes in $\mathbb {R}^d$, let $A(\mathcal {H})\cap P$ denote the restriction of the arrangement $A(\mathcal {H})$ to P. The goal of PLA is to construct a data structure in order to quickly locate a cell of $A(\mathcal {H})\cap P$ containing an unknown target value $\bar{\mu }$. Solving PLA requires the explicit construction of the arrangement $A(\mathcal {H})$ which can be done in an excessive $O(|\mathcal {H}|^d)$ running time [22, Theorem 6.1.2]. For our purposes, it is sufficient to solve the following simpler form of PLA.

• $P_\mathrm{{reg}}(\mathcal {H},P,\bar{\mu })$ Given a simplex P, a set $\mathcal {H}$ of hyperplanes in $\mathbb {R}^d$, and a target value $\bar{\mu }$, locate a d-dimensional simplex $R\subseteq A(\mathcal {H})\cap P$ containing a target and unknown value $\bar{\mu }$.

Cohen and Megiddo [9] consider the problem Max(f) of maximizing a concave function $f: \mathbb {R}^d \rightarrow \mathbb {R}$ with fixed dimension d and give, under some conditions, a polynomial time algorithm. This algorithm also uses problem $P_\mathrm{{reg}}(\mathcal {H},P,\bar{\mu })$ as a subroutine, where in this context the target value $\bar{\mu }$ is the optimal value of Max(f). Let T(d) denote the time required to solve Max(f) with d parameters and T(0) denote the running time of evaluating f at any value in $\mathbb {R}^d$. The authors solve $P_\mathrm{{reg}}(\mathcal {H},P,\bar{\mu })$ recursively using multidimensional parametric search technique. See also [7, 8, 30].

Lemma 5

Given a simplex P, a set $\mathcal {H}$ of hyperplanes in $\mathbb {R}^d$, and a target and unknown value $\bar{\mu }$, $P_\mathrm{{reg}}(\mathcal {H},P,\bar{\mu })$ can be solved in $O(\log (|\mathcal {H}|)T(d-1) + |\mathcal {H}|)$ time.