arXiv subjects

Etienne Bamas

Publications and source records attributed to Etienne Bamas.

9 recordsLinked to original sources

arXiv2026

Blind-Spots-Bench: Evaluating Blind Spots in Multimodal Models

Modern AI models achieve strong performance on many established benchmarks, yet they still fail on tasks that humans find almost trivial, such as manipulating a string or drawing a dog with five legs. These examples suggest that existing benchmarks may under-measure persistent blind spots in current systems. We introduce $\texttt{blind-spots-bench}$, a benchmark designed to expose such blind spots through tasks that appear simple for humans but remain challenging for modern AI. We collect raw questions from students in an AI course, clean and annotate them with structured reference solutions, and propose a task taxonomy tailored to the resulting dataset of 235 samples. We further develop an automated grading pipeline to evaluate a wide range of models, including open-weight and closed-source language, vision-language, and image-generation models. Our analysis on $\texttt{blind-spots-bench}$ reveals that closed-source frontier models can substantially outperform open-weight models with even $\approx10\%$ gap, even when they attain comparable performance on existing benchmarks. A more fine-grained analysis shows that no single model dominates across all task types, and that some tasks remain challenging for all evaluated models. These results highlight the value of $\texttt{blind-spots-bench}$ as a diagnostic stress test for identifying concrete weaknesses in current modern models.

cs.AI↗

arXiv2026

$k$-server-bench: Automating Potential Discovery for the $k$-Server Conjecture

We introduce a code-based challenge for automated, open-ended mathematical discovery based on the $k$-server conjecture, a central open problem in competitive analysis. The task is to discover a potential function satisfying a large graph-structured system of simple linear inequalities. The resulting evaluation procedure is sound but incomplete: any violated inequality definitively refutes a candidate, whereas satisfying all inequalities does not by itself constitute a proof of the corresponding conjecture's special case. Nevertheless, a candidate that passes all constraints would be strong evidence toward a valid proof and, to the best of our knowledge, no currently known potential achieves this under our formulation in the open $k=4$ circle case. As such, a successful candidate would already be an interesting contribution to the $k$-server conjecture, and could become a substantial theoretical result when paired with a full proof. Experiments on the resolved $k=3$ regime show that current agentic methods can solve nontrivial instances, and in the open $k=4$ regime they reduce the number of violations relative to existing potentials without fully resolving the task. Taken together, these results suggest that the task is challenging but plausibly within reach of current methods. Beyond its relevance to the $k$-server community, where the developed tooling enables researchers to test new hypotheses and potentially improve on the current record, the task also serves as a useful \emph{benchmark} for developing code-based discovery agents. In particular, our $k=3$ results show that it mitigates important limitations of existing open-ended code-based benchmarks, including early saturation and the weak separation between naive random baselines and more sophisticated methods.

cs.MS↗

arXiv2025

Randomized Rounding over Dynamic Programs

We show that under mild assumptions for a problem whose solutions admit a dynamic programming-like recurrence relation, we can still find a solution under additional packing constraints, which need to be satisfied approximately. The number of additional constraints can be very large, for example, polynomial in the problem size. Technically, we reinterpret the dynamic programming subproblems and their solutions as a network design problem. Inspired by techniques from, for example, the Directed Steiner Tree problem, we construct a strong LP relaxation, on which we then apply randomized rounding. Our approximation guarantees on the packing constraints have roughly the form of a $(n^ε \mathrm{polylog}\ n)$-approximation in time $n^{O(1/ε)}$, for any $ε> 0$. By setting $ε=\log \log n/\log n$, we obtain a polylogarithmic approximation in quasi-polynomial time, or by setting $ε$ as a constant, an $n^ε$-approximation in polynomial time. While there are necessary assumptions on the form of the DP, it is general enough to capture many textbook dynamic programs from Shortest Path to Longest Common Subsequence. Our algorithm then implies that we can impose additional constraints on the solutions to these problems. This allows us to model various problems from the literature in approximation algorithms, many of which were not thought to be connected to dynamic programming. In fact, our result can even be applied indirectly to some problems that involve covering instead of packing constraints, for example, the Directed Steiner Tree problem, or those that do not directly follow a recurrence relation, for example, variants of the Matching problem.

cs.DS↗

arXiv2024

Lift-and-Project Integrality Gaps for Santa Claus

This paper is devoted to the study of the MaxMinDegree Arborescence (MMDA) problem in layered directed graphs of depth $\ell\le O(\log n/\log \log n)$, which is an important special case of the Santa Claus problem. Obtaining a polylogarithmic approximation for MMDA in polynomial time is of high interest as it is a necessary condition to improve upon the well-known 2-approximation for makespan scheduling on unrelated machines by Lenstra, Shmoys, and Tardos [FOCS'87]. The only way we have to solve the MMDA problem within a polylogarithmic factor is via an elegant recursive rounding of the $(\ell-1)^{th}$ level of the Sherali-Adams hierarchy, which needs time $n^{O(\ell)}$ to solve. However, it remains plausible that one could obtain a polylogarithmic approximation in polynomial time by using the same rounding with only $1$ round of the Sherali-Adams hierarchy. As a main result, we rule out this possibility by constructing an MMDA instance of depth $3$ for which an integrality gap of $n^{Ω(1)}$ survives $1$ round of the Sherali-Adams hierarchy. This result is tight since it is known that after only $2$ rounds the gap is at most polylogarithmic on depth-3 graphs. Second, we show that our instance can be ``lifted'' via a simple trick to MMDA instances of any depth $\ell\in Ω(1)\cap o(\log n/\log \log n)$ (the whole range of interest), for which we conjecture that an integrality gap of $n^{Ω(1/\ell)}$ survives $Ω(\ell)$ rounds of Sherali-Adams. We show a number of intermediate results towards this conjecture, which also suggest that our construction is a significant challenge to the techniques used so far for Santa Claus.

cs.DS↗

arXiv2024

The Submodular Santa Claus Problem

We consider the problem of allocating indivisible resources to players so as to maximize the minimum total value any player receives. This problem is sometimes dubbed the Santa Claus problem and its different variants have been subject to extensive research towards approximation algorithms over the past two decades. In the case where each player has a potentially different additive valuation function, Chakrabarty, Chuzhoy, and Khanna [FOCS'09] gave an $O(n^ε)$-approximation algorithm with polynomial running time for any constant $ε> 0$ and a polylogarithmic approximation algorithm in quasi-polynomial time. We show that the same can be achieved for monotone submodular valuation functions, improving over the previously best algorithm due to Goemans, Harvey, Iwata, and Mirrokni [SODA'09], which has an approximation ratio of more than $\sqrt{n}$. Our result builds up on a sophisticated LP relaxation, which has a recursive block structure that allows us to solve it despite having exponentially many variables and constraints.

cs.DS↗

arXiv2023

An Analysis of $D^α$ seeding for $k$-means

One of the most popular clustering algorithms is the celebrated $D^α$ seeding algorithm (also know as $k$-means++ when $α=2$) by Arthur and Vassilvitskii (2007), who showed that it guarantees in expectation an $O(2^{2α}\cdot \log k)$-approximate solution to the ($k$,$α$)-means cost (where euclidean distances are raised to the power $α$) for any $α\ge 1$. More recently, Balcan, Dick, and White (2018) observed experimentally that using $D^α$ seeding with $α>2$ can lead to a better solution with respect to the standard $k$-means objective (i.e. the $(k,2)$-means cost). In this paper, we provide a rigorous understanding of this phenomenon. For any $α>2$, we show that $D^α$ seeding guarantees in expectation an approximation factor of $$ O_α\left((g_α)^{2/α}\cdot \left(\frac{σ_{\mathrm{max}}}{σ_{\mathrm{min}}}\right)^{2-4/α}\cdot (\min\{\ell,\log k\})^{2/α}\right)$$ with respect to the standard $k$-means cost of any underlying clustering; where $g_α$ is a parameter capturing the concentration of the points in each cluster, $σ_{\mathrm{max}}$ and $σ_{\mathrm{min}}$ are the maximum and minimum standard deviation of the clusters around their means, and $\ell$ is the number of distinct mixing weights in the underlying clustering (after rounding them to the nearest power of $2$). We complement these results by some lower bounds showing that the dependency on $g_α$ and $σ_{\mathrm{max}}/σ_{\mathrm{min}}$ is tight. Finally, we provide an experimental confirmation of the effects of the aforementioned parameters when using $D^α$ seeding. Further, we corroborate the observation that $α>2$ can indeed improve the $k$-means cost compared to $D^2$ seeding, and that this advantage remains even if we run Lloyd's algorithm after the seeding.

cs.DS↗

arXiv2022

A Simple LP-Based Approximation Algorithm for the Matching Augmentation Problem

The Matching Augmentation Problem (MAP) has recently received significant attention as an important step towards better approximation algorithms for finding cheap $2$-edge connected subgraphs. This has culminated in a $\frac{5}{3}$-approximation algorithm. However, the algorithm and its analysis are fairly involved and do not compare against the problem's well-known LP relaxation called the cut LP. In this paper, we propose a simple algorithm that, guided by an optimal solution to the cut LP, first selects a DFS tree and then finds a solution to MAP by computing an optimum augmentation of this tree. Using properties of extreme point solutions, we show that our algorithm always returns (in polynomial time) a better than $2$-approximation when compared to the cut LP. We thereby also obtain an improved upper bound on the integrality gap of this natural relaxation.

cs.DS↗

arXiv2020

The Combinatorial Santa Claus Problem or: How to Find Good Matchings in Non-Uniform Hypergraphs

We consider hypergraphs on vertices $P\cup R$ where each hyperedge contains exactly one vertex in $P$. Our goal is to select a matching that covers all of $P$, but we allow each selected hyperedge to drop all but an $(1/α)$-fraction of its intersection with $R$ (thus relaxing the matching constraint). Here $α$ is to be minimized. We dub this problem the Combinatorial Santa Claus problem, since we show in this paper that this problem and the Santa Claus problem are almost equivalent in terms of their approximability. The non-trivial observation that any uniform regular hypergraph admits a relaxed matching for $α= O(1)$ was a major step in obtaining a constant approximation rate for a special case of the Santa Claus problem, which received great attention in literature. It is natural to ask if the uniformity condition can be omitted. Our main result is that every (non-uniform) regular hypergraph admits a relaxed matching for $α= O(\log\log(|R|))$, when all hyperedges are sufficiently large (a condition that is necessary). In particular, this implies an $O(\log\log(|R|))$-approximation algorithm for the Combinatorial Santa Claus problem with large hyperedges.

cs.DS↗

arXiv2020

The Submodular Santa Claus Problem in the Restricted Assignment Case

The submodular Santa Claus problem was introduced in a seminal work by Goemans, Harvey, Iwata, and Mirrokni (SODA'09) as an application of their structural result. In the mentioned problem $n$ unsplittable resources have to be assigned to $m$ players, each with a monotone submodular utility function $f_i$. The goal is to maximize $\min_i f_i(S_i)$ where $S_1,\dotsc,S_m$ is a partition of the resources. The result by Goemans et al. implies a polynomial time $O(n^{1/2 +\varepsilon})$-approximation algorithm. Since then progress on this problem was limited to the linear case, that is, all $f_i$ are linear functions. In particular, a line of research has shown that there is a polynomial time constant approximation algorithm for linear valuation functions in the restricted assignment case. This is the special case where each player is given a set of desired resources $Γ_i$ and the individual valuation functions are defined as $f_i(S) = f(S \cap Γ_i)$ for a global linear function $f$. This can also be interpreted as maximizing $\min_i f(S_i)$ with additional assignment restrictions, i.e., resources can only be assigned to certain players. In this paper we make comparable progress for the submodular variant. Namely, if $f$ is a monotone submodular function, we can in polynomial time compute an $O(\log\log(n))$-approximate solution.

cs.DS↗