arXiv subjects

Alireza Farhadi

Publications and source records attributed to Alireza Farhadi.

16 recordsLinked to original sources

arXiv2025

CombiGraph-Vis: A Curated Multimodal Olympiad Benchmark for Discrete Mathematical Reasoning

State-of-the-art (SOTA) LLMs have progressed from struggling on proof-based Olympiad problems to solving most of the IMO 2025 problems, with leading systems reportedly handling 5 of 6 problems. Given this progress, we assess how well these models can grade proofs: detecting errors, judging their severity, and assigning fair scores beyond binary correctness. We study proof-analysis capabilities using a corpus of 90 Gemini 2.5 Pro-generated solutions that we grade on a 1-4 scale with detailed error annotations, and on MathArena solution sets for IMO/USAMO 2025 scored on a 0-7 scale. Our analysis shows that models can reliably flag incorrect (including subtly incorrect) solutions but exhibit calibration gaps in how partial credit is assigned. To address this, we introduce agentic workflows that extract and analyze reference solutions and automatically derive problem-specific rubrics for a multi-step grading process. We instantiate and compare different design choices for the grading workflows, and evaluate their trade-offs. Across our annotated corpus and MathArena, our proposed workflows achieve higher agreement with human grades and more consistent handling of partial credit across metrics. We release all code, data, and prompts/logs to facilitate future research.

cs.AI↗

arXiv2025

RefGrader: Automated Grading of Mathematical Competition Proofs using Agentic Workflows

cs.AI↗

arXiv2025

Brains vs. Bytes: Evaluating LLM Proficiency in Olympiad Mathematics

Recent advances in large language models (LLMs) have shown impressive progress in mathematical reasoning tasks. However, current evaluation benchmarks predominantly focus on the accuracy of final answers, often overlooking the crucial logical rigor for mathematical problem solving. The claim that state-of-the-art LLMs can solve Math Olympiad-level problems requires closer examination. To explore this, we conducted both qualitative and quantitative human evaluations of proofs generated by LLMs, and developed a schema for automatically assessing their reasoning capabilities. Our study reveals that current LLMs fall significantly short of solving challenging Olympiad-level problems and frequently fail to distinguish correct mathematical reasoning from clearly flawed solutions. Our analyses demonstrate that the occasional correct final answers provided by LLMs often result from pattern recognition or heuristic shortcuts rather than genuine mathematical reasoning. These findings underscore the substantial gap between LLM performance and human expertise in advanced mathematical reasoning and highlight the importance of developing benchmarks that prioritize the soundness of the reasoning used to arrive at an answer rather than the mere correctness of the final answers.

cs.AI↗

arXiv2025

Connected Components in Linear Work and Near-Optimal Time

Computing the connected components of a graph is a fundamental problem in algorithmic graph theory. A major question in this area is whether we can compute connected components in $o(\log n)$ parallel time. Recent works showed an affirmative answer in the Massively Parallel Computation (MPC) model for a wide class of graphs. Specifically, Behnezhad et al. (FOCS'19) showed that connected components can be computed in $O(\log d + \log \log n)$ rounds in the MPC model. More recently, Liu et al. (SPAA'20) showed that the same result can be achieved in the standard PRAM model but their result incurs $Θ((m+n) \cdot (\log d + \log \log n))$ work which is sub-optimal. In this paper, we show that for graphs that contain \emph{well-connected} components, we can compute connected components on a PRAM in sub-logarithmic parallel time with \emph{optimal}, i.e., $O(m+n)$ total work. Specifically, our algorithm achieves $O(\log(1/λ) + \log \log n)$ parallel time with high probability, where $λ$ is the minimum spectral gap of any connected component in the input graph. The algorithm requires no prior knowledge on $λ$. Additionally, based on the \textsc{2-Cycle} Conjecture we provide a time lower bound of $Ω(\log(1/λ))$ for solving connected components on a PRAM with $O(m+n)$ total memory when $λ\le (1/\log n)^c$, giving conditional optimality to the running time of our algorithm as a parameter of $λ$.

cs.DS↗

arXiv2022

Differentially Private Densest Subgraph

Given a graph, the densest subgraph problem asks for a set of vertices such that the average degree among these vertices is maximized. Densest subgraph has numerous applications in learning, e.g., community detection in social networks, link spam detection, correlation mining, bioinformatics, and so on. Although there are efficient algorithms that output either exact or approximate solutions to the densest subgraph problem, existing algorithms may violate the privacy of the individuals in the network, e.g., leaking the existence/non-existence of edges. In this paper, we study the densest subgraph problem in the framework of the differential privacy, and we derive the first upper and lower bounds for this problem. We show that there exists a linear-time $ε$-differentially private algorithm that finds a $2$-approximation of the densest subgraph with an extra poly-logarithmic additive error. Our algorithm not only reports the approximate density of the densest subgraph, but also reports the vertices that form the dense subgraph. Our upper bound almost matches the famous $2$-approximation by Charikar both in performance and in approximation ratio, but we additionally achieve differential privacy. In comparison with Charikar's algorithm, our algorithm has an extra poly-logarithmic additive error. We partly justify the additive error with a new lower bound, showing that for any differentially private algorithm that provides a constant-factor approximation, a sub-logarithmic additive error is inherent. We also practically study our differentially private algorithm on real-world graphs, and we show that in practice the algorithm finds a solution which is very close to the optimal

cs.CR↗

arXiv2022

Beating $(1-1/e)$-Approximation for Weighted Stochastic Matching

In the stochastic weighted matching problem, the goal is to find a large-weight matching of a graph when we are uncertain about the existence of its edges. In particular, each edge $e$ has a known weight $w_e$ but is realized independently with some probability $p_e$. The algorithm may query an edge to see whether it is realized. We consider the well-studied query commit version of the problem, in which any queried edge that happens to be realized must be included in the solution. Gamlath, Kale, and Svensson showed that when the input graph is bipartite, the problem admits a $(1-1/e)$-approximation. In this paper, we give an algorithm that for an absolute constant $δ> 0.0014$ obtains a $(1-1/e+δ)$-approximation, therefore breaking this prevalent bound.

cs.DS↗

arXiv2022

Online Bipartite Matching with Reusable Resources

We study the classic online bipartite matching problem with a twist: offline vertices, called resources, are $\textit{reusable}$. In particular, when a resource is matched to an online vertex it is unavailable for a deterministic time duration $d$ after which it becomes available again for a re-match. Thus, a resource can be matched to many different online vertices over a period of time. While recent work on the problem have resolved the asymptotic case where we have large starting inventory (i.e., many copies) of every resource, we consider the (more general) case of $\textit{unit inventory}$ and give the first algorithms that are provably better than the naive greedy approach which has a competitive ratio of (exactly) 0.5. Our first algorithm, which achieves a competitive ratio of $0.589$, generalizes the classic RANKING algorithm for online bipartite matching of non-reusable resources (Karp et al., 1990), by $\textit{reranking}$ resources independently over time. While reranking resources frequently has the same worst case performance as greedy, we show that reranking intermittently on a periodic schedule succeeds in addressing reusability of resources and performs significantly better than greedy in the worst case. Our second algorithm, which achieves a competitive ratio of $0.505$, is a primal-dual randomized algorithm that works by suggesting up to two resources as candidate matches for every online vertex, and then breaking the tie to make the final matching selection in a randomized correlated fashion over time. As a key component of our algorithm, we suitably adapt and extend the powerful technique of online correlated selection (Fahrbach et al., 2020) to reusable resources, in order to induce negative correlation in our tie breaking step and to beat the competitive ratio of $0.5$. Both of our results also extend to the case where offline vertices have weights.

cs.DS↗

arXiv2022

Generalized Stochastic Matching

In this paper, we generalize the recently studied Stochastic Matching problem to more accurately model a significant medical process, kidney exchange, and several other applications. Up until now the Stochastic Matching problem that has been studied was as follows: given a graph G = (V, E), each edge is included in the realized sub-graph of G mutually independently with probability p_e, and the goal is to find a degree-bounded sub-graph Q of G that has an expected maximum matching that approximates the expected maximum matching of the realized sub-graph. This model does not account for possibilities of vertex dropouts, which can be found in several applications, e.g. in kidney exchange when donors or patients opt out of the exchange process as well as in online freelancing and online dating when online profiles are found to be faked. Thus, we will study a more generalized model of Stochastic Matching in which vertices and edges are both realized independently with some probabilities p_v, p_e, respectively, which more accurately fits important applications than the previously studied model. We will discuss the first algorithms and analysis for this generalization of the Stochastic Matching model and prove that they achieve good approximation ratios. In particular, we show that the approximation factor of a natural algorithm for this problem is at least $0.6568$ in unweighted graphs, and $1/2 + ε$ in weighted graphs for some constant $ε> 0$. We further improve our result for unweighted graphs to $2/3$ using edge degree constrained subgraphs (EDCS).

cs.DS↗

arXiv2020

Almost Envy-freeness, Envy-rank, and Nash Social Welfare Matchings

Envy-free up to one good (EF1) and envy-free up to any good (EFX) are two well-known extensions of envy-freeness for the case of indivisible items. It is shown that EF1 can always be guaranteed for agents with subadditive valuations. In sharp contrast, it is unknown whether or not an EFX allocation always exists, even for four agents and additive valuations. In addition, the best approximation guarantee for EFX is $(ϕ-1) \simeq 0.61$ by Amanitidis et al.. In order to find a middle ground to bridge this gap, in this paper we suggest another fairness criterion, namely envy-freeness up to a random good or EFR, which is weaker than EFX, yet stronger than EF1. For this notion, we provide a polynomial-time $0.73$-approximation allocation algorithm. For our algorithm, we introduce Nash Social Welfare Matching which makes a connection between Nash Social Welfare and envy freeness. We believe Nash Social Welfare Matching will find its applications in future work.

cs.GT↗

arXiv2020

Asymmetric Streaming Algorithms for Edit Distance and LCS

The edit distance (ED) and longest common subsequence (LCS) are two fundamental problems which quantify how similar two strings are to one another. In this paper, we consider these problems in the asymmetric streaming model introduced by Andoni et al. (FOCS'10) and Saks and Seshadhri (SODA'13). In this model we have random access to one string and streaming access the other string. Our main contribution is a constant factor approximation algorithm for ED with the memory of $\tilde O(n^δ)$ for any constant $δ> 0$. In addition to this, we present an upper bound of $\tilde O_ε(\sqrt{n})$ on the memory needed to approximate ED or LCS within a factor $1+ε$. All our algorithms are deterministic and run in a single pass. For approximating ED within a constant factor, we discover yet another application of triangle inequality, this time in the context of streaming algorithms. Triangle inequality has been previously used to obtain subquadratic time approximation algorithms for ED. Our technique is novel and elegantly utilizes triangle inequality to save memory at the expense of an exponential increase in the runtime.

cs.DS↗

arXiv2019

Approximate Maximum Matching in Random Streams

In this paper, we study the problem of finding a maximum matching in the semi-streaming model when edges arrive in a random order. In the semi-streaming model, an algorithm receives a stream of edges and it is allowed to have a memory of $\tilde{O}(n)$ where $n$ is the number of vertices in the graph. A recent inspiring work by Assadi et al. shows that there exists a streaming algorithm with the approximation ratio of $\frac{2}{3}$ that uses $\tilde{O}(n^{1.5})$ memory. However, the memory of their algorithm is much larger than the memory constraint of the semi-streaming algorithms. In this work, we further investigate this problem in the semi-streaming model, and we present simple algorithms for approximating maximum matching in the semi-streaming model. Our main results are as follows. We show that there exists a single-pass deterministic semi-streaming algorithm that finds a $\frac{3}{5} (= 0.6)$ approximation of the maximum matching in bipartite graphs using $\tilde{O}(n)$ memory. This result significantly outperforms the state-of-the-art result of Konrad that finds a $0.539$ approximation of the maximum matching using $\tilde{O}(n)$ memory. By giving a black-box reduction from finding a matching in general graphs to finding a matching in bipartite graphs, we show there exists a single-pass deterministic semi-streaming algorithm that finds a $\frac{6}{11} (\approx 0.545)$ approximation of the maximum matching in general graphs, improving upon the state-of-art result $0.506$ approximation by Gamlath et al.

cs.DS↗

arXiv2018

Polynomial-time Approximation Scheme for Minimum k-cut in Planar and Minor-free Graphs

The $k$-cut problem asks, given a connected graph $G$ and a positive integer $k$, to find a minimum-weight set of edges whose removal splits $G$ into $k$ connected components. We give the first polynomial-time algorithm with approximation factor $2-ε$ (with constant $ε> 0$) for the $k$-cut problem in planar and minor-free graphs. Applying more complex techniques, we further improve our method and give a polynomial-time approximation scheme for the $k$-cut problem in both planar and minor-free graphs. Despite persistent effort, to the best of our knowledge, this is the first improvement for the $k$-cut problem over standard approximation factor of $2$ in any major class of graphs.

cs.DS↗

arXiv2018

Stochastic Matching with Few Queries: New Algorithms and Tools

We consider the following stochastic matching problem on both weighted and unweighted graphs: A graph $G(V, E)$ along with a parameter $p \in (0, 1)$ is given in the input. Each edge of $G$ is realized independently with probability $p$. The goal is to select a degree bounded (dependent only on $p$) subgraph $H$ of $G$ such that the expected size/weight of maximum realized matching of $H$ is close to that of $G$. This model of stochastic matching has attracted significant attention over the recent years due to its various applications. The most fundamental open question is the best approximation factor achievable for such algorithms that, in the literature, are referred to as non-adaptive algorithms. Prior work has identified breaking (near) half-approximation as a barrier for both weighted and unweighted graphs. Our main results are as follows: -- We analyze a simple and clean algorithm and show that for unweighted graphs, it finds an (almost) $4\sqrt{2}-5$ ($\approx 0.6568$) approximation by querying $O(\frac{\log (1/p)}{p})$ edges per vertex. This improves over the state-of-the-art $0.5001$ approximate algorithm of Assadi et al. [EC'17]. -- We show that the same algorithm achieves a $0.501$ approximation for weighted graphs by querying $O(\frac{\log (1/p)}{p})$ edges per vertex. This is the first algorithm to break $0.5$ approximation barrier for weighted graphs. It also improves the per-vertex queries of the state-of-the-art by Yamaguchi and Maehara [SODA'18] and Behnezhad and Reyhani [EC'18]. Our algorithms are fundamentally different from prior works, yet are very simple and natural. For the analysis, we introduce a number of procedures that construct heavy fractional matchings. We consider the new algorithms and our analytical tools to be the main contributions of this paper.

cs.DS↗

arXiv2018

Lower Bounds for External Memory Integer Sorting via Network Coding

Sorting extremely large datasets is a frequently occuring task in practice. These datasets are usually much larger than the computer's main memory; thus external memory sorting algorithms, first introduced by Aggarwal and Vitter (1988), are often used. The complexity of comparison based external memory sorting has been understood for decades by now, however the situation remains elusive if we assume the keys to be sorted are integers. In internal memory, one can sort a set of $n$ integer keys of $Θ(\lg n)$ bits each in $O(n)$ time using the classic Radix Sort algorithm, however in external memory, there are no faster integer sorting algorithms known than the simple comparison based ones. In this paper, we present a tight conditional lower bound on the complexity of external memory sorting of integers. Our lower bound is based on a famous conjecture in network coding by Li and Li, who conjectured that network coding cannot help anything beyond the standard multicommodity flow rate in undirected graphs. The only previous work connecting the Li and Li conjecture to lower bounds for algorithms is due to Adler et al. Adler et al. indeed obtain relatively simple lower bounds for oblivious algorithms (the memory access pattern is fixed and independent of the input data). Unfortunately obliviousness is a strong limitations, especially for integer sorting: we show that the Li and Li conjecture implies an $Ω(n \log n)$ lower bound for internal memory oblivious sorting when the keys are $Θ(\lg n)$ bits. This is in sharp contrast to the classic (non-oblivious) Radix Sort algorithm. Indeed going beyond obliviousness is highly non-trivial; we need to introduce several new methods and involved techniques, which are of their own interest, to obtain our tight lower bound for external memory integer sorting.

cs.DS↗

arXiv2018

On the Complexity of Chore Division

We study the proportional chore division problem where a protocol wants to divide an undesirable object, called chore, among $n$ different players. The goal is to find an allocation such that the cost of the chore assigned to each player be at most $1/n$ of the total cost. This problem is the dual variant of the cake cutting problem in which we want to allocate a desirable object. Edmonds and Pruhs showed that any protocol for the proportional cake cutting must use at least $Ω(n \log n)$ queries in the worst case, however, finding a lower bound for the proportional chore division remained an interesting open problem. We show that chore division and cake cutting problems are closely related to each other and provide an $Ω(n \log n)$ lower bound for chore division.

cs.GT↗

arXiv2017

Fair Allocation of Indivisible Goods to Asymmetric Agents

We study fair allocation of indivisible goods to agents with unequal entitlements. Fair allocation has been the subject of many studies in both divisible and indivisible settings. Our emphasis is on the case where the goods are indivisible and agents have unequal entitlements. This problem is a generalization of the work by Procaccia and Wang wherein the agents are assumed to be symmetric with respect to their entitlements. Although Procaccia and Wang show an almost fair (constant approximation) allocation exists in their setting, our main result is in sharp contrast to their observation. We show that, in some cases with $n$ agents, no allocation can guarantee better than $1/n$ approximation of a fair allocation when the entitlements are not necessarily equal. Furthermore, we devise a simple algorithm that ensures a $1/n$ approximation guarantee. Our second result is for a restricted version of the problem where the valuation of every agent for each good is bounded by the total value he wishes to receive in a fair allocation. Although this assumption might seem w.l.o.g, we show it enables us to find a $1/2$ approximation fair allocation via a greedy algorithm. Finally, we run some experiments on real-world data and show that, in practice, a fair allocation is likely to exist. We also support our experiments by showing positive results for two stochastic variants of the problem, namely stochastic agents and stochastic items.

cs.GT↗