arXiv subjects

Mirko Giacchini

Publications and source records attributed to Mirko Giacchini.

7 recordsLinked to original sources

arXiv2026

Sketching Intersection Profiles: A Simple Proof and Three Applications

In this work we settle the complexity of three sketching problems. (i) We show that sketching vertex neighborhood sizes in graphs requires $\Omega(n^2)$ bits, standing in sharp contrast to the $\tilde{O}(n)$ complexity of sketching edge cuts. (ii) We obtain tight lower and upper bounds of $\tilde{\Theta}(n^2)$ for sketching coverage functions with additive and multiplicative errors. (iii) We prove an $\Omega(n^2)$ lower bound for sketching Random Utility Models under the $\ell_\infty$-norm, improving upon the previous $\Omega(n \log n)$ bound and matching a known upper bound to within logarithmic factors. These bounds are obtained through a connection with the problem of sketching the intersection profile of a distribution $D$ on $2^{[n]}$. Specifically, we seek a succinct data structure that, for any query set $S \subseteq [n]$, approximates the quantity $\Pr_{T \sim D}[T \cap S \neq \varnothing]$ to within a small constant additive error. One can obtain lower bounds for this latter problem directly from known results about the itemset frequency estimation problem in databases for which tight bounds are known. As an additional contribution, we also provide an alternative proof for the intersection profile sketching lower bound, in the setting in which the accuracy parameter is constant. This proof relies solely on elementary probability avoiding the heavier machinery used in previous proofs.

cs.DS↗

arXiv2026

On the LSH Distortion of Ulam and Cayley Similarities

Locality-sensitive hashing (LSH) has found widespread use as a fundamental primitive, particularly to accelerate nearest neighbor search. An LSH scheme for a similarity function $S:\mathcal{X} \times \mathcal{X} \to [0,1]$ is a distribution over hash functions on $\mathcal{X}$ with the property that the probability of collision of any two elements $x,y\in \mathcal{X}$ is exactly equal to $S(x,y)$. However, not all similarity functions admit exact LSH schemes. The notion of LSH distortion measures how multiplicatively close a similarity function is to having an LSH scheme. In this work, we study the LSH distortion of the Ulam and Cayley similarities, which are popular similarity measures on permutations of $n$ elements. We show that the Ulam similarity admits a sublinear LSH distortion of $O(n / \sqrt{\log n})$; we also prove a lower bound of $\Omega(n^{0.12})$ on the best LSH distortion achievable. On the other hand, we show that the LSH distortion of the Cayley similarity is $\Theta(n)$.

cs.DS↗

arXiv2026

Learning Multinomial Logits in $O(n \log n)$ time

A Multinomial Logit (MNL) model is composed of a finite universe of items $[n]=\{1,..., n\}$, each assigned a positive weight. A query specifies an admissible subset -- called a slate -- and the model chooses one item from that slate with probability proportional to its weight. This query model is also known as the Plackett-Luce model or conditional sampling oracle in the literature. Although MNLs have been studied extensively, a basic computational question remains open: given query access to slates, how efficiently can we learn weights so that, for every slate, the induced choice distribution is within total variation distance $\varepsilon$ of the ground truth? This question is central to MNL learning and has direct implications for modern recommender system interfaces. We provide two algorithms for this task, one with adaptive queries and one with non-adaptive queries. Each algorithm outputs an MNL $M'$ that induces, for each slate $S$, a distribution $M'_S$ on $S$ that is within $\varepsilon$ total variation distance of the true distribution. Our adaptive algorithm makes $O\left(\frac{n}{\varepsilon^{3}}\log n\right)$ queries, while our non-adaptive algorithm makes $O\left(\frac{n^{2}}{\varepsilon^{3}}\log n \log\frac{n}{\varepsilon}\right)$ queries. Both algorithms query only slates of size two and run in time proportional to their query complexity. We complement these upper bounds with lower bounds of $\Omega\left(\frac{n}{\varepsilon^{2}}\log n\right)$ for adaptive queries and $\Omega\left(\frac{n^{2}}{\varepsilon^{2}}\log n\right)$ for non-adaptive queries, thus proving that our adaptive algorithm is optimal in its dependence on the support size $n$, while the non-adaptive one is tight within a $\log n$ factor.

cs.DS↗

arXiv2025

Man, these New York Times games are hard! A computational perspective

The New York Times (NYT) games have found widespread popularity in recent years and reportedly account for an increasing fraction of the newspaper's readership. In this paper, we bring the computational lens to the study of New York Times games and consider four of them not previously studied: Letter Boxed, Pips, Strands and Tiles. We show that these games can be just as hard as they are fun. In particular, we characterize the hardness of several variants of computational problems related to these popular puzzle games. For Letter Boxed, we show that deciding whether an instance is solvable is in general NP-Complete, while in some parameter settings it can be done in polynomial time. Similarly, for Pips we prove that deciding whether a puzzle has a solution is NP-Complete even in some restricted classes of instances. We then show that one natural computational problem arising from Strands is NP-Complete in most parameter settings. Finally, we demonstrate that deciding whether a Tiles puzzle is solvable with a single, uninterrupted combo requires polynomial time.

cs.CC↗

arXiv2025

A New Impossibility Result for Online Bipartite Matching Problems

Online Bipartite Matching with random user arrival is a fundamental problem in the online advertisement ecosystem. Over the last 30 years, many algorithms and impossibility results have been developed for this problem. In particular, the latest impossibility result was established by Manshadi, Oveis Gharan and Saberi in 2011. Since then, several algorithms have been published in an effort to narrow the gap between the upper and the lower bounds on the competitive ratio. In this paper we show that no algorithm can achieve a competitive ratio better than $1- \frac e{e^e} = 0.82062\ldots$, improving upon the $0.823$ upper bound presented in (Manshadi, Oveis Gharan and Saberi, SODA 2011). Our construction is simple to state, accompanied by a fully analytic proof, and yields a competitive ratio bound intriguingly similar to $1 - \frac1e$, the optimal competitive ratio for the fully adversarial Online Bipartite Matching problem. Although the tightness of our upper bound remains an open question, we show that our construction is extremal in a natural class of instances.

cs.DS↗

arXiv2024

Coordinating "7 Billion Humans" is hard

In the video game "7 Billion Humans", the player is requested to direct a group of workers to various destinations by writing a program that is executed simultaneously on each worker. While the game is quite rich and, indeed, it is considered one of the best games for beginners to learn the basics of programming, we show that even extremely simple versions are already NP-Hard or PSPACE-Hard.

cs.CC↗

arXiv2023

Approximating a RUM from Distributions on k-Slates

In this work we consider the problem of fitting Random Utility Models (RUMs) to user choices. Given the winner distributions of the subsets of size $k$ of a universe, we obtain a polynomial-time algorithm that finds the RUM that best approximates the given distribution on average. Our algorithm is based on a linear program that we solve using the ellipsoid method. Given that its corresponding separation oracle problem is NP-hard, we devise an approximate separation oracle that can be viewed as a generalization of the weighted feedback arc set problem to hypergraphs. Our theoretical result can also be made practical: we obtain a heuristic that is effective and scales to real-world datasets.

cs.LG↗