The Slotted Online One-Sided Crossing Minimization Problem on 2-Regular Graphs

TL;DR

This paper investigates an online variant of the crossing minimization problem on bipartite graphs, focusing on 2-regular graphs, and establishes bounds on the competitive ratio for online algorithms.

Contribution

It introduces the online slotted OSCM problem, analyzes its difficulty, and provides bounds on the competitive ratio specifically for 2-regular graphs.

Findings

Online slotted OSCM-k is not competitive for k ≥ 2.

Lower bound of 4/3 on the competitive ratio for 2-regular graphs.

Upper bound of 5 on the competitive ratio for 2-regular graphs.

Abstract

In the area of graph drawing, the One-Sided Crossing Minimization Problem (OSCM) is defined on a bipartite graph with both vertex sets aligned parallel to each other and all edges being drawn as straight lines. The task is to find a permutation of one of the node sets such that the total number of all edge-edge intersections, called crossings, is minimized. Usually, the degree of the nodes of one set is limited by some constant k, with the problem then abbreviated to OSCM-k. In this work, we study an online variant of this problem, in which one of the node sets is already given. The other node set and the incident edges are revealed iteratively and each node has to be inserted into placeholders, which we call slots. The goal is again to minimize the number of crossings in the final graph. Minimizing crossings in an online way is related to the more empirical field of dynamic graph drawing. Note the slotted OSCM problem makes instances harder to solve for an online algorithm but in the offline case it is equivalent to the version without slots. We show that the online slotted OSCM-k is not competitive for any k greater or equal 2 and subsequently limit the graph class to that of 2-regular graphs, for which we show a lower bound of 4/3 and an upper bound of 5 on the competitive ratio.