On the Heaviest Increasing or Decreasing Subsequence of a Permutation, and Paths and Matchings on Weighted Point Sets

Abstract

Let S = {s(1), …, s(n) } be a permutation of the integers {1,…, n}. A subsequence of S with elements {s(i₁), …, s(i _k )} is called an increasing subsequence if s(i₁) < ⋯ < s(i _k ); It is called a decreasing subsequence if s(i₁) > ⋯ > s(i _k ). The weight of a subsequence of S, is the sum of its elements. In this paper, we prove that any permutation of {1, …, n} contains an increasing or a decreasing subsequence of weight greater than $n\sqrt{2n/3}$.

Our motivation to study the previous problem arises from the following problem: Let P be a set of n points on the plane in general position, labeled with the integers {1, …,n} in such a way that the labels of different points are different. A non-crossing path Π with vertices in P is an increasing path if when we travel along it, starting at one of its end-points, the labels of its vertices always increase. The weight of an increasing path, is the sum of the labels of its vertices. Determining lower bounds on the weight of the heaviest increasing path a point set always has.

We also study the problem of finding a non-crossing matching of the elements of P of maximum weight, where the weight of an edge with endpoints i, j ∈ P is min{i,j}.