Optimal Cuts and Bisections on the Real Line in Polynomial Time

TL;DR

This paper presents a polynomial-time algorithm for solving the longstanding open problem of optimal cuts and bisections on the real line, using a novel technique involving metric equalities and dynamic programming.

Contribution

It introduces the first exact polynomial-time algorithm for one-dimensional geometric cuts and bisections, resolving a major open problem.

Findings

The problem is solvable in polynomial time on the real line.

A new technique connecting metric equalities with dynamic programming was developed.

The method may be applicable to other geometric optimization problems.

Abstract

The exact complexity of geometric cuts and bisections is the longstanding open problem including even the dimension one. In this paper, we resolve this problem for dimension one (the real line) by designing an exact polynomial time algorithm. Our results depend on a new technique of dealing with metric equalities and their connection to dynamic programming. The method of our solution could be also of independent interest.