Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings

TL;DR

This work studies the online minimum-weight perfect matching problem, where a sequence of metric points arrive in pairs, and one has to maintain a perfect matching at all times, and studies the new stochastic online embedding.

Abstract

Low-distortional metric embeddings are a crucial component in the modern algorithmic toolkit. In an online metric embedding, points arrive sequentially and the goal is to embed them into a simple space irrevocably, while minimizing the distortion. Our first result is a deterministic online embedding of a general metric into Euclidean space with distortion $O(\log n)\cdot\min\{\sqrt{\logΦ},\sqrt{n}\}$ (or, $O(d)\cdot\min\{\sqrt{\logΦ},\sqrt{n}\}$ if the metric has doubling dimension $d$), solving a conjecture by Newman and Rabinovich (2020), and quadratically improving the dependence on the aspect ratio $Φ$ from Indyk et al.\ (2010). Our second result is a stochastic embedding of a metric space into trees with expected distortion $O(d\cdot \logΦ)$, generalizing previous results (Indyk et al.\ (2010), Bartal et al.\ (2020)). Next, we study the \emph{online minimum-weight perfect matching} problem, where a sequence of $2n$ metric points arrive in pairs, and one has to maintain a perfect matching at all times. We allow recourse (as otherwise the order of arrival determines the matching). The goal is to return a perfect matching that approximates the \emph{minimum-weight} perfect matching at all times, while minimizing the recourse. Our third result is a randomized algorithm with competitive ratio $O(d\cdot \log Φ)$ and recourse $O(\log Φ)$ against an oblivious adversary, this result is obtained via our new stochastic online embedding. Our fourth result is a deterministic algorithm against an adaptive adversary, using $O(\log^2 n)$ recourse, that maintains a matching of weight at most $O(\log n)$ times the weight of the MST, i.e., a matching of lightness $O(\log n)$. We complement our upper bounds with a strategy for an oblivious adversary that, with recourse $r$, establishes a lower bound of $Ω(\frac{\log n}{r \log r})$ for both competitive ratio and lightness.

Figures (8)

• Figure 1: $(a)$ Example where the weight of an online matching is arbitrarily far form optimum assuming irrevocable decisions (i.e., no recourse). The metric is the real line. We first receive the pairs $\{i,W+i\}_{i=1}^{n}$, and then the pairs $\{i+\varepsilon,W+i+\varepsilon\}_{i=1}^{n}$, for sufficiently small $\varepsilon$ and large $W$. The weight of the online perfect matching (specified in the illustration) is $2n\cdot W$, while the cost of the optimal perfect matching is $2\varepsilon\cdot n$. $(b)$ Example of the drastic non-monotonicity of minimum-weight perfect matching. The metric is the real line, where each point $\{1,2\dots,2n\}$ appears twice, while the points $\{0,2n+1\}$ appear once. Then the weight of a perfect matching is $2n+1$. After introducing the pair $\{0,2n+1\}$ (red in the figure), the weight of the perfect matching drops to $0$.
• Figure 2: A "duet" between online metric embeddings and minimum-weight perfect matchings: a Venn diagram of the relationship between the various results in this paper.
• Figure 3: Illustration of the partition of a single scale $\Delta=2^i$ and the paddedness of each point. The source metric here $(X,d_X)$ is induced by the Euclidean plane. The net points $N_i$, and cluster centers are $\{x_1,\dots,x_7\}$, where their respective clusters $P_1^i,\dots,P_7^i$ are colored orange, violet, grey, turquoise, purple, green, and blue. The rest of the metric points are represented by smaller unnamed dots. The paddedness is the distance to the boundary of the cluster each point belongs to. In the figure, the paddedness of each point is equal to the length of the dart originating from it.
• Figure 4: The Laakso Graph. On the left represented the Laakso graphs $G_1, G_2, G_3$. On the right is our version, where only a single edge is replaced.
• Figure 5: One iteration of repair$(M)$ replaces edges $a_1b_1\prec a_2b_2$ with $a_2a_1$ and $b_1b_2$.

Theorems & Definitions (76)

• Definition 1: Online Embedding
• Conjecture : NR20
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Theorem 4
• Theorem 5
• Theorem 6
• Definition 2: $k$-HST