Improved lower bound on the on-line chain partitioning of semi-orders with representation

TL;DR

This paper improves the lower bound on the number of chains needed by any online algorithm to partition semi-orders, especially for unit-interval representations, and completely solves the case for width 3.

Contribution

It establishes a new lower bound of rac{3}{2}w for online chain partitioning of semi-orders, advancing understanding of the problem's complexity.

Findings

Any online algorithm can be forced to use rac{3}{2}w chains for semi-orders.

The bound is tight for width 3, fully solving the problem in that case.

Abstract

An on-line chain partitioning algorithm receives a poset, one element at a time, and irrevocably assigns the element to one of the chains in the partition. The on-line chain partitioning problem involves finding the minimal number of chains needed by an optimal on-line algorithm. Chrobak and \'Slusarek considered variants of the on-line chain partitioning problem in which the elements are presented as intervals and intersecting intervals are incomparable. They constructed an on-line algorithm which uses at most $3w-2$ chains, where $w$ is the width of the interval order, and showed that this algorithm is optimal. They also considered the problem restricted to intervals of unit-length and while they showed that first-fit needs at most $2w-1$ chains, over $30$ years later, it remains unknown whether a more optimal algorithm exists. In this paper, we improve upon previously known bounds and show that any on-line algorithm can be forced to use $\lceil\frac{3}{2}w\rceil$ chains to partition a semi-order presented in the form of its unit-interval representation. As a consequence, we completely solve the problem for $w=3$.