arXiv subjects

Tze-Yang Poon

Publications and source records attributed to Tze-Yang Poon.

3 recordsLinked to original sources

arXiv2025

Shortest Paths, Convexity, and Treewidth in Regular Hyperbolic Tilings

Hyperbolic tilings are natural infinite planar graphs where each vertex has degree $q$ and each face has $p$ edges for some $\frac1p+\frac1q<\frac12$. We study the structure of shortest paths in such graphs. We show that given a set of $n$ terminals, we can compute a so-called isometric closure (closely related to the geodesic convex hull) of the terminals in near-linear time, using a classic geometric convex hull algorithm as a black box. We show that the size of the convex hull is $O(N)$ where $N$ is the total length of the paths to the terminals from a fixed origin. Furthermore, we prove that the geodesic convex hull of a set of $n$ terminals has treewidth only $\max(12,O(\log\frac{n}{p + q}))$, a bound independent of the distance of the points involved. As a consequence, we obtain algorithms for subset TSP and Steiner tree with running time $O(N \log N) + \mathrm{poly}(\frac{n}{p + q}) \cdot N$.

cs.CG↗

arXiv2025

Online 3-Taxi on General Metrics

The online $k$-taxi problem, introduced in 1990 by Fiat, Rabani and Ravid, is a generalization of the $k$-server problem where $k$ taxis must serve a sequence of requests in a metric space. Each request is a pair of two points, representing the pick-up and drop-off location of a passenger. In the interesting ''hard'' version of the problem, the cost is the total distance that the taxis travel without a passenger. The problem is known to be substantially harder than the $k$-server problem, and prior to this work even for $k=3$ taxis it has been unknown whether a finite competitive ratio is achievable on general metric spaces. We present an $O(1)$-competitive algorithm for the $3$-taxi problem.

cs.DS↗

arXiv2025

Approximate Proportionality in Online Fair Division

We study the online fair division problem, where indivisible goods arrive sequentially and must be allocated immediately and irrevocably. Prior work establishes strong impossibility results for approximating classic notions such as envy-freeness up to one good (EF1) and maximin share (MMS) in this setting, but the approximability of proportionality up to one good (PROP1) has remained unresolved. We resolve this gap in two steps. First, we show that three natural greedy allocation rules (standard baselines in fair division) fail to guarantee any multiplicative approximation to PROP1 against an adaptive adversary. These limitations motivate two relaxations: (i) restricting attention to a non-adaptive adversary, and (ii) incorporating coarse predictions in the spirit of learning-augmented algorithms. Under a non-adaptive adversary, we show that the uniform random allocation achieves a meaningful PROP1 approximation with high probability, and this guarantee is essentially tight for this approach; moreover, when item values are sufficiently small, the allocation is near-PROP1 with high probability. Finally, given maximum item value (MIV) predictions, we design an online algorithm that achieves robust approximation guarantees for PROP1, and degrades gracefully under one-sided prediction error. In contrast, we show that EF1, MMS, and PROPX remain inapproximable even with perfect MIV predictions.

cs.GT↗