On the Online Weighted Non-Crossing Matching Problem

Abstract

We introduce and study the weighted version of an online matching problem in the Euclidean plane with non-crossing constraints: points with non-negative weights arrive online, and an algorithm can match an arriving point to one of the unmatched previously arrived points. In the classic model, the decision on how to match (if at all) a newly arriving point is irrevocable. The goal is to maximize the total weight of matched points under the constraint that straight-line segments corresponding to the edges of the matching do not intersect. The unweighted version of the problem was introduced in the offline setting by Atallah in 1985, and this problem became a subject of study in the online setting with and without advice in several recent papers. We observe that deterministic online algorithms cannot guarantee a non-trivial competitive ratio for the weighted problem, but we give upper and lower bounds on the problem with bounded weights. In contrast to the deterministic case, we show that using randomization, a constant competitive ratio is possible for arbitrary weights. We also study other variants of the problem, including revocability and collinear points, both of which permit non-trivial online algorithms, and we give upper and lower bounds for the attainable competitive ratios. Finally, we prove an advice complexity bound for obtaining optimality, improving the best known bound.

Figures (5)

• Figure 1: An illustration of the adversary's strategy for Case 2 with $k=3$. The two arcs form the active region. The unnamed points have weight 1. Suppose in the first phase $\textsc{ALG}\xspace$ matched $x_{R_1}$ and $y_{R_1}$, which became responsible for region $R_1$. Note that the number of unmatched points (of weight 1) in $R_1$ is 6, which is less than $2^k-1 =7$. Thus, in the second phase, the adversary plans to give points $p_1, p_2,p_3$ of weights $a_1, a_2, a_3$ into $R_1$. Suppose $\textsc{ALG}\xspace$ matches $p_1$ of weight $a_1$; then the adversary gives $p_2$ and $p_3$ below the line segment between the matched pair (there are fewer unmatched points there). Similarly, after the point $p_2$ of weight $a_2$ is matched, the adversary gives $p_3$ to the side of the resulting segment with no unmatched points. This ensures that some point of weight $a_i$ (here $a_3$) stays unmatched and is mapped to the matched pairs.
• Figure 2: An illustration of the mapping used to analyze Wam. In this example, we have $k=2$ and $8$ points with weights in $\{1,\sqrt{U}, U\}$. Here, $[t, w]$ indicates the $t^{\text{th}}$ point in the input sequence having weight $w$. Note that points $1$ and $3$ are mapped to the segment corresponding to the imaginary points $(-\infty,0)$ and $(-\infty,1)$ of weight $U$.
• Figure 3: An illustration of the adversary's construction of the randomized input in the first few steps, guided by the random variables $L_i$ and $F_i$.
• Figure 4: On the left is the input and how Tgm divides and partitions the plane, and on the right is the tree it creates from the input. Based on this tree, it matches nodes with their parents randomly, such that every node upon its arrival gets matched with a probability of $1/3$.
• Figure 5: (1) The first two points arrive, partitioning the line into $(-\infty,p_1)$, $[\,p_1,p_2]\,$, and $(p_2, \infty)$. (2) Point $p_3$ arrives in $[\,p_1,p_2]\,$. Rrm revokes $\overline{p_1p_2}$, randomly matches $p_3$ to $p_1$, and partitions $[\,p_1,p_2]\,$ into $[\,p_1,p_3]\,$ and $(p_3,p_2]\,$. (3) Point $p_4$ arrives in $(p_2,\infty)$ and remains unmatched as the interval it arrived in is empty. (4) Point $p_5$ arrives in $(p_3,p_2]\,$ and is matched to $p_2$, splitting the interval into $(p_3,p_5)$ and $[\,p_5,p_2]\,$.

Theorems & Definitions (16)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5
• Lemma 6
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10