Online Matching on $3$-Uniform Hypergraphs

TL;DR

This work provides an optimal primal-dual fractional algorithm, which achieves the optimal competitive ratio of 1/2 on 3-uniform hypergraphs when every online node has degree at most 2, and presents a carefully constructed adversarial instance which shows this ratio is in fact optimal.

Abstract

The online matching problem was introduced by Karp, Vazirani and Vazirani (STOC 1990) on bipartite graphs with vertex arrivals. It is well-known that the optimal competitive ratio is $1-1/e$ for both integral and fractional versions of the problem. Since then, there has been considerable effort to find optimal competitive ratios for other related settings. In this work, we go beyond the graph case and study the online matching problem on $k$-uniform hypergraphs. For $k=3$, we provide an optimal primal-dual fractional algorithm, which achieves a competitive ratio of $(e-1)/(e+1)\approx 0.4621$. As our main technical contribution, we present a carefully constructed adversarial instance, which shows that this ratio is in fact optimal. It combines ideas from known hard instances for bipartite graphs under the edge-arrival and vertex-arrival models. For $k\geq 3$, we give a simple integral algorithm which performs better than greedy when the online nodes have bounded degree. As a corollary, it achieves the optimal competitive ratio of 1/2 on 3-uniform hypergraphs when every online node has degree at most 2. This is because the special case where every online node has degree 1 is equivalent to the edge-arrival model on graphs, for which an upper bound of 1/2 is known.

Figures (6)

• Figure 1: An illustration of the region $R = \{(a, b) \in [0,1]^2 : f(a) + f(b) \leq 1\}$. The symmetric point at the boundary of the region has both coordinates $\ln((e+1)/2) \approx 0.62$. When an online node $w$ arrives, a threshold respecting algorithm ensures that the fractional matching $x$ satisfies $(x(\delta(u)), x(\delta(v))) \in R$ for every hyperedge $h = \{u,v,w\} \in \delta(w)$ with $x_h>0$ at the end of that iteration.
• Figure 2: The partial matchings $\mathcal{M}_i^{(t)}$ if the fractional algorithm ensures that every edge reaches the threshold at the end of every phase, meaning that $f(\ell_u^{(t)}) + f(\ell_v^{(t)}) \geq 1$ for every $(u,v) \in \mathcal{M}_i^{(t)}.$ The maximum matching at the end of phase $5$ has size five and consists of $\mathcal{M}_i^{(5)}$.
• Figure 3: In this example, the algorithm does not increase the edge $(1,3)$, and thus by symmetry the edge $(3,1)$, up to the threshold during phase $t = 3$. Hence, $f(\ell_1^{(3)}) + f(\ell_3^{(3)}) < 1$ and nodes $1$ and $3$ become inactive from that point on. The maximum matching at the end of phase $5$ still has size five and consists of $\mathcal{M}_i^{(5)}$, in addition to the two edges $(1,3)$ and $(3,1)$ that are below the threshold.
• Figure 4: Plot of the $\ell(t,i)$ process for two different values of $t$ if the algorithm exactly matches the threshold at every phase. Observe that, since $t$ is odd, $(\sigma_t(q_t), \sigma_t(q_t)) \in \mathcal{M}_{i}^{(t)}$ with the load of node $\sigma_t(q_t)$ staying at $\ln((e+1)/2) \approx 0.62$.
• Figure 5: Plot of $\psi(t,y)$ for two different values of $t$, the horizontal axis represents $y \in \mathbb{Z}/2$.

Theorems & Definitions (37)

• Theorem 1.1
• Theorem 1.2
• Theorem 3.1
• proof : Proof of \ref{['thm:optimal_algo']}
• Definition 4.1
• Remark
• Theorem 4.1
• Claim 4.1
• proof
• Lemma 4.1