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Thanos Tolias

Publications and source records attributed to Thanos Tolias.

11 recordsLinked to original sources
arXiv2026

Hardness, Tractability and Density Thresholds of finite Pinwheel Scheduling Variants

The k-Visits problem is a recently introduced finite version of Pinwheel Scheduling [Kanellopoulos et al., SODA 2026]. Given the deadlines of n tasks, the problem asks whether there exists a schedule of length kn executing each task exactly k times, with no deadline expiring between consecutive visits (executions) of each task. In this work we prove that 2-Visits is strongly NP-complete even when the maximum multiplicity of the input is equal to 2, settling an open question from [Kanellopoulos et al., SODA 2026] and contrasting the tractability of 2-Visits for simple sets. On the other hand, we prove that 2-Visits is in RP when the number of distinct deadlines is constant, thus making progress on another open question regarding the parameterization of 2-Visits by the number of numbers. We then generalize all existing positive results for 2-Visits to a version of the problem where some tasks must be visited once and some other tasks twice, while providing evidence that some of these results are unlikely to transfer to 3-Visits. Lastly, we establish bounds for the density thresholds of k-Visits, analogous to the $(5/6)$-threshold of Pinwheel Scheduling [Kawamura, STOC 2024]; in particular, we show a $\sqrt{2}-1/2\approx 0.9142$ lower bound for the density threshold of 2-Visits and prove that the density threshold of k-Visits approaches $5/6\approx 0.8333$ for $k \to \infty$.

cs.DS↗
arXiv2026

A Nearly Tight Lower Bound for Matroid Intersection Prophet Inequalities

We study prophet inequalities under intersections of $q$ partition matroids, where an online algorithm irrevocably selects elements with independent nonnegative values drawn from known distributions and revealed in an adversarial order. We prove an $Ω(q/\log q)$ lower bound on the competitive ratio. Together with the known $O(q)$ upper bounds, this resolves, up to a logarithmic factor, the optimal dependence on $q$, an open question posed by Correa, Cristi, Fielbaum, Pollner, and Weinberg (IPCO 2022) and Saxena, Velusamy, and Weinberg (ITCS 2023). Our construction also yields an $Ω(d/\log d)$ lower bound for $d$-single-minded auctions, where buyers request fixed bundles of at most $d$ unit-capacity items. Our construction and analysis build on the "big-decisions-first" framework of Rubinstein and Singla (STOC 2026).

cs.GT↗
arXiv2026

The Knapsack Secretary Problem is Strictly Harder Than the Secretary Problem

The knapsack secretary problem is a generalization of the classical secretary problem where the accepted items must satisfy a knapsack constraint. A line of work has developed constant-competitive algorithms for this problem, with successive improvements culminating in the current best-known competitive ratio of $0.153$. A natural open question was whether the optimal $1/e$ competitive ratio for the classical secretary problem is also achievable for the knapsack secretary problem. We answer this question negatively by showing that no $(1/e - 0.0001)$-competitive algorithm exists for the knapsack secretary problem. The analysis of the family of hard instances we construct proceeds in three steps. First, we reduce the cardinal problem on these instances to an almost-ordinal problem. Second, we formulate a linear program that captures the performance of almost-ordinal algorithms on this instance family. Finally, we exhibit a feasible dual solution whose objective value is strictly below $1/e$. We also give an algorithm that improves the best-known competitive ratio from $0.153$ to $0.178$.

cs.DS↗
arXiv2026

Online Sorting with Our Eyes Wide Shut

In Online Sorting, we are given an array $A$ of $n$ initially empty cells. At each time step $t\in[n]$, an element $x_t\in[0,1]$ arrives and must be placed irrevocably into an empty cell, without knowledge of future arrivals. The objective is to minimize the sum of absolute differences between elements assigned to adjacent cells. The problem has been studied under both adversarial and stochastic input models. For adversarial sequences, Aamand, Abrahamsen, Beretta, and Kleist (SODA'23) gave a tight $O\sqrt n)$-competitive algorithm, fully resolving the worst-case setting. For stochastic sequences, in which the elements are drawn i.i.d.\ from $U[0,1]$, Hu (SODA'26) gave an $\log n\cdot 2^{O(\log^* n)}$-competitive algorithm in expectation and proved an $Ω(\log n)$ lower bound, while Kalavas, Platanos, and Tolias (STACS'26) gave an $O(\log^2 n)$-competitive algorithm with high probability. Very recently, Hermansen (ESA'26) closed the remaining gap by designing an $O(\log n)$-competitive algorithm in expectation. In this work, we study Random-Order Online Sorting, a model interpolating between the adversarial and stochastic settings, that was posed as a challenging open question by Hermansen (ESA'26). Here, the input is a multiset chosen adversarially, but its elements arrive in uniformly random order. We take a different point of view by solving the problem in rank space, and prove an $O(\log^2 n)$-competitive algorithm with high probability, matching the state-of-the-art high probability guarantee for the stochastic setting in this more general model. We also study a multidimensional generalization, which we call Random-Order Online TSP, and obtain an $O(\log^3 n)$-competitive algorithm with high probability.

cs.DS↗
arXiv2026

Repeated Descent: A Framework for Online Budget-Feasible Auctions

We study budget feasible procurement auctions, in which $n$ agents, each with a privately held service cost, offer their services to an employer. The employer seeks to maximize a public submodular valuation function over the set of hired agents, while facing a hard budget constraint. We consider an online posted-price setting, in which agents arrive in a uniformly random order (a.k.a. \emph{secretary arrivals}) and the employer must make irrevocable take-it-or-leave-it offers upon their arrival. The employer does not get any feedback about the agent service costs other than whether they accept the offer or not. We introduce Repeated Descent (a.k.a. \RED), a deterministic framework based on adaptive linear posted pricing. \RED enforces budget feasibility by adaptively adjusting its pricing and balancing each pricing level with the number of agents considered in it. Using \RED as the main building block, we obtain a $1046$-competitive posted-price mechanism for online budget feasible auctions with secretary agent arrivals and submodular valuations, thus improving on the previously best known ratio of (Charalampopoulos et al., EC 2025) by several orders of magnitude. Combining \RED with random subsampling, we obtain the first constant-competitive posted-price budget feasible mechanism for non-monotone submodular valuations. On the negative side, we show that every online budget feasible mechanism with XOS valuations has a competitive ratio of $Ω\!\left(\tfrac{\log n}{(\log\log n)^2}\right)$.

cs.GT↗
arXiv2026

Online Resource Allocation via Static Bundle Pricing

Online Resource Allocation addresses the problem of efficiently allocating limited resources to buyers with incomplete knowledge of future requests. In our setting, buyers arrive sequentially requesting a set of items, each with a value drawn from a known distribution. We study the efficiency of static and anonymous bundle pricing in environments where the buyers' valuations exhibit strong complementarities. In such settings, standard item pricing fails to leverage item multiplicities, while static bundle pricing mechanisms are only known for very restricted domains and their analysis relies on domain-specific arguments. We develop a unified bundle pricing framework for online resource allocation in three well-studied domains with complementarities: (i) single-minded combinatorial auctions with maximum bundle size $d$; (ii) general single-minded combinatorial auctions; and (iii) network routing, where each buyer aims to route a unit of flow from a source node $s$ to a target node $t$ in a capacitated network. Our approach yields static and anonymous bundle pricing mechanisms whose performance improves exponentially with item multiplicity. For the $d$-single-minded setting with minimum item multiplicity $B$, we obtain an $O(d^{1/B})$-competitive mechanism. For general single-minded combinatorial auctions and online network routing, we obtain $O(m^{1/(B+1)})$-competitive mechanisms, where $m$ is the number of items. We complement these results with information-theoretic lower bounds. We show that no online algorithm can achieve a competitive ratio better than $ \widetildeΩ(m^{1/(B+2)})$ for single-minded combinatorial auctions and $ \widetildeΩ(d^{1/(B+1)})$ for the $d$-single-minded setting. Our constructions exploit a deep connection to the extremal combinatorics problem of determining the maximum number of qualitatively independent partitions of a ground set.

cs.GT↗
arXiv2026

A Polylogarithmic Competitive Algorithm for Stochastic Online Sorting and TSP

In \emph{Online Sorting}, an array of $n$ initially empty cells is given. At each time step $t$, an element $x_t \in [0,1]$ arrives and must be placed irrevocably into an empty cell without any knowledge of future arrivals. We aim to minimize the sum of absolute differences between pairs of elements placed in consecutive array cells, seeking an online placement strategy that results in a final array close to a sorted one. An interesting multidimensional generalization, a.k.a. the \emph{Online Travelling Salesperson Problem}, arises when the request sequence consists of points in the $d$-dimensional unit cube and the objective is to minimize the sum of euclidean distances between points in consecutive cells. Motivated by the recent work of (Abrahamsen, Bercea, Beretta, Klausen and Kozma; ESA 2024), we consider the \emph{stochastic version} of Online Sorting (\textit{resp.} Online TSP), where each element (\textit{resp.} point) $x_t$ is an i.i.d. sample from the uniform distribution on $[0, 1]$ (\textit{resp.} $[0,1]^d$). By carefully decomposing the request sequence into a hierarchy of balls-into-bins instances, where the balls to bins ratio is large enough so that bin occupancy is sharply concentrated around its mean and small enough so that we can efficiently deal with the elements placed in the same bin, we obtain an online algorithm that approximates the optimal cost within a factor of $O(\log^2 n)$ with high probability. Our result comprises an exponential improvement on the previously best known competitive ratio of $\tilde{O}(n^{1/4})$ for Stochastic Online Sorting due to (Abrahamsen et al.; ESA 2024) and $O(\sqrt{n})$ for (adversarial) Online TSP due to (Bertram, ESA 2025).

cs.DS↗
arXiv2025

A Polylogarithmic Competitive Algorithm for Stochastic Online Sorting and TSP

In \emph{Online Sorting}, an array of $n$ initially empty cells is given. At each time step $t$, an element $x_t \in [0,1]$ arrives and must be placed irrevocably into an empty cell without any knowledge of future arrivals. We aim to minimize the sum of absolute differences between pairs of elements placed in consecutive array cells, seeking an online placement strategy that results in a final array close to a sorted one. An interesting multidimensional generalization, a.k.a. the \emph{Online Travelling Salesperson Problem}, arises when the request sequence consists of points in the $d$-dimensional unit cube and the objective is to minimize the sum of euclidean distances between points in consecutive cells. Motivated by the recent work of (Abrahamsen, Bercea, Beretta, Klausen and Kozma; ESA 2024), we consider the \emph{stochastic version} of Online Sorting (\textit{resp.} Online TSP), where each element (\textit{resp.} point) $x_t$ is an i.i.d. sample from the uniform distribution on $[0, 1]$ (\textit{resp.} $[0,1]^d$). By carefully decomposing the request sequence into a hierarchy of balls-into-bins instances, where the balls to bins ratio is large enough so that bin occupancy is sharply concentrated around its mean and small enough so that we can efficiently deal with the elements placed in the same bin, we obtain an online algorithm that approximates the optimal cost within a factor of $O(\log^2 n)$ with high probability. Our result comprises an exponential improvement on the previously best known competitive ratio of $\tilde{O}(n^{1/4})$ for Stochastic Online Sorting due to (Abrahamsen et al.; ESA 2024) and $O(\sqrt{n})$ for (adversarial) Online TSP due to (Bertram, ESA 2025).

cs.DS↗
arXiv2025

An algorithm for accurate and simple-looking metaphorical maps

"Metaphorical maps" or "contact representations" are visual representations of vertex-weighted graphs that rely on the geographic map metaphor. The vertices are represented by countries, the weights by the areas of the countries, and the edges by contacts/ boundaries among them. The accuracy with which the weights are mapped to areas and the simplicity of the polygons representing the countries are the two classical optimization goals for metaphorical maps. Mchedlidze and Schnorr [Metaphoric Maps for Dynamic Vertex-weighted Graphs, EuroVis 2022] presented a force-based algorithm that creates metaphorical maps that balance between these two optimization goals. Their maps look visually simple, but the accuracy of the maps is far from optimal - the countries' areas can vary up to 30% compared to required. In this paper, we provide a multi-fold extension of the algorithm in [Metaphoric Maps for Dynamic Vertex-weighted Graphs, EuroVis 2022]. More specifically: 1. Towards improving accuracy: We introduce the notion of region stiffness and suggest a technique for varying the stiffness based on the current pressure of map regions. 2. Towards maintaining simplicity: We introduce a weight coefficient to the pressure force exerted on each polygon point based on whether the corresponding point appears along a narrow passage. 3. Towards generality: We cover, in contrast to [Metaphoric Maps for Dynamic Vertex-weighted Graphs, EuroVis 2022], non-triangulated graphs. This is done by either generating points where more than three regions meet or by introducing holes in the metaphorical map. We perform an extended experimental evaluation that, among other results, reveals that our algorithm is able to construct metaphorical maps with nearly perfect area accuracy with a little sacrifice in their simplicity.

cs.DM↗
arXiv2025

A Competitive Posted-Price Mechanism for Online Budget-Feasible Auctions

We consider online procurement auctions, where the agents arrive sequentially, in random order, and have private costs for their services. The buyer aims to maximize a monotone submodular value function for the subset of agents whose services are procured, subject to a budget constraint on their payments. We consider a posted-price setting where upon each agent's arrival, the buyer decides on a payment offered to them. The agent accepts or rejects the offer, depending on whether the payment exceeds their cost, without revealing any other information about their private costs whatsoever. We present a randomized online posted-price mechanism with constant competitive ratio, thus resolving the main open question of (Badanidiyuru, Kleinberg and Singer, EC 2012). Posted-price mechanisms for online procurement typically operate by learning an estimation of the optimal value, denoted as OPT, and using it to determine the payments offered to the agents. The main challenge is to learn OPT within a constant factor from the agents' accept / reject responses to the payments offered. Our approach is based on an online test of whether our estimation is too low compared against OPT and a carefully designed adaptive search that gradually refines our estimation.

cs.GT↗
arXiv2024

Improved Bounds for Online Facility Location with Predictions

We consider Online Facility Location in the framework of learning-augmented online algorithms. In Online Facility Location (OFL), demands arrive one-by-one in a metric space and must be (irrevocably) assigned to an open facility upon arrival, without any knowledge about future demands. We focus on uniform facility opening costs and present an online algorithm for OFL that exploits potentially imperfect predictions on the locations of the optimal facilities. We prove that the competitive ratio decreases from sublogarithmic in the number of demands $n$ to constant as the so-called $η_1$ error, i.e., the sum of distances of the predicted locations to the optimal facility locations, decreases. E.g., our analysis implies that if for some $\varepsilon > 0$, $η_1 = \mathrm{OPT} / n^\varepsilon$, where $\mathrm{OPT}$ is the cost of the optimal solution, the competitive ratio becomes $O(1/\varepsilon)$. We complement our analysis with a matching lower bound establishing that the dependence of the algorithm's competitive ratio on the $η_1$ error is optimal, up to constant factors. Finally, we evaluate our algorithm on real world data and compare the performance of our learning-augmented approach against the performance of the best known algorithm for OFL without predictions.

cs.DS↗