The online $k$-taxi problem is a natural generalization of the $k$-server problem: $k$ taxis must serve a sequence of requests in a metric space. Each request is a pair of points $(r,s)$, and the algorithm serves the request by first moving a taxi to $r$ (to pick up a passenger) and then to $s$ (to drop off the passenger). The cost is the total distance traveled by the taxis without a passenger. Previous bounds for general $k$ all depended on the number of points $n$ or the aspect ratio $Δ$ of the metric space, both of which can be infinite on general metrics, and competitive algorithms were previously known on general metrics only for $k \le 3$. In this work, we give a $9^{k-1}$-competitive randomized algorithm against adaptive online adversaries and, as a corollary, an $81^{k-1}$-competitive deterministic algorithm for the online $k$-taxi problem.
We study metric distortion in one-sided matching. Given $n$ agents and $n$ items in an unknown metric space, our goal is to match agents to items so that the total distance between matched pairs is small. However, instead of direct access to the distances in the metric, we are only given each agent's ranking of the items by distance. What is the smallest worst-case approximation factor (called the distortion) that a mechanism can guarantee in expectation? Despite significant interest over the last decade, the best-known guarantee has been the $O(n)$ distortion of Random Serial Dictatorship since the beginning. We give a matching mechanism with distortion $O(n^{\log_4 3})$, the first sublinear upper bound. The key idea is to use a partial matching derived from a truncated version of either Random Serial Dictatorship or the Probabilistic Serial rule. We then complete this matching using a new, general-purpose cluster completion algorithm that ensures unmatched agents receive nearby items. Along the way, we pin down the optimal distortion of Random Serial Dictatorship and Probabilistic Serial up to logarithmic and constant factors, respectively, and give improved lower bounds for truthful mechanisms.
The online $k$-server problem is one of the most fundamental problems in online algorithms. In this problem, $k$ servers are located in a metric space and serve a sequence of requests at points in the metric space by moving a server there. The Harmonic algorithm is a very natural algorithm: it chooses a server with probability inversely proportional to its distance from the request. It is also highly efficient, being memoryless and answering each request in $O(k)$ time. However, its competitive ratio has remained unsettled between the $O(2^k \log k)$ upper bound (Bartal and Grove, JACM 2000) and the $k(k+1)/2$ lower bound (Raghavan and Snir, 1994, IBM J. Res. Dev.). We prove that Harmonic is $2k^4$-competitive on every metric space against adaptive online adversaries. Our analysis uses a potential combining corrected electrical energy with the distances of online servers to their assigned adversary locations.
We study simple auctions for one additive bidder whose values for $m$ items are independent. Write $\mathrm{REV}$ for the optimal revenue, $\mathrm{SREV}$ and $\mathrm{BREV}$ for the optimal revenues from separate selling and grand bundling, and $\mathrm{BSPA}_2$ for the grand-bundle second-price revenue with two independent bidders. For independent regular items, we prove $\mathrm{REV} \le \mathrm{BSPA}_2 \le 2\,\mathrm{BREV}$. Thus, one additional bidder lets a reserve-free second-price auction for the grand bundle match the optimal revenue from one bidder, regardless of the number of items. This gives a Bulow-Klemperer theorem for bundling and resolves an open question of Beyhaghi, Cai, Feng, Li, and Weinberg (EC 2025) on whether constant competition suffices for bundling. The factor-$2$ bundling approximation is also tight. For general independent items, we prove $\mathrm{REV} \le \mathrm{SREV} + \frac{π^2}{6}\mathrm{BREV} \le 2.645 \max\{\mathrm{SREV}, \mathrm{BREV}\}$. This improves the previous approximation factors of $6$ from Babaioff, Immorlica, Lucier, and Weinberg (FOCS 2014, JACM 2020) and $5.2$ from Ma and Simchi-Levi (AISTATS 2021). For the lower bound, we construct an instance in which pricing three disjoint bundles earns more than $2.11\max\{\mathrm{SREV},\mathrm{BREV}\}$, improving the previous lower bound of $2$ from Rubinstein (ITCS 2016). Together, these results narrow the tight approximation ratio to the range $[2.11, 2.645]$.