arXiv subjects

Mahsa Derakhshan

Publications and source records attributed to Mahsa Derakhshan.

At least 19 recordsLinked to original sources

arXiv2026

A Unified Framework for Analysis of Randomized Greedy Matching Algorithms

Randomized greedy algorithms form one of the simplest yet most effective approaches for computing approximate matchings in graphs. In this paper, we focus on the class of vertex-iterative (VI) randomized greedy matching algorithms, which process the vertices of a graph $G=(V,E)$ in some order $\pi$ and, for each vertex $v$, greedily match it to the first available neighbor according to a preference order $\sigma(v)$. Various VI algorithms have been studied, each corresponding to a different distribution over $\pi$ and $\sigma(v)$. We develop a unified framework for analyzing this family of algorithms and use it to obtain improved approximation ratios for Ranking and FRanking, the state-of-the-art randomized greedy algorithms for the random-order and adversarial-order settings, respectively. In Ranking, the decision order is drawn uniformly at random and used as the common preference order, whereas FRanking uses an adversarial decision order and a uniformly random preference order shared by all vertices. We obtain an approximation ratio of $0.560$ for Ranking, improving on the $0.5469$ bound of Derakhshan et al. [SODA 2026]. For FRanking, we obtain a ratio of $0.539$, improving on the $0.521$ bound of Huang et al. [JACM 2020]. These results also imply state-of-the-art approximation ratios for oblivious matching and fully online matching problems on general graphs. Our analysis framework also enables us to prove improved approximation ratios for graphs with no short odd cycles. Such graphs form an intermediate class between general graphs and bipartite graphs. In particular, we show that Ranking is at least $0.570$-competitive for graphs that are both triangle-free and pentagon-free. For graphs whose shortest odd cycle has length at least $129$, we prove that Ranking is at least $0.615$-competitive.

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arXiv2026

Multi-Agent Reinforcement Learning Counteracts Delayed CSI in Multi-Satellite Systems

The integration of satellite communication networks with next-generation (NG) technologies is a promising approach towards global connectivity. However, the quality of services is highly dependant on the availability of accurate channel state information (CSI). Channel estimation in satellite communications is challenging due to the high propagation delay between terrestrial users and satellites, which results in outdated CSI observations on the satellite side. In this paper, we study the downlink transmission of multiple satellites acting as distributed base stations (BS) to mobile terrestrial users. We propose a multi-agent reinforcement learning (MARL) algorithm which aims for maximising the sum-rate of the users, while coping with the outdated CSI. We design a novel bi-level optimisation, procedure themes as dual stage proximal policy optimisation (DS-PPO), for tackling the problem of large continuous action spaces as well as of independent and non-identically distributed (non-IID) environments in MARL. Specifically, the first stage of DS-PPO maximises the sum-rate for an individual satellite and the second stage maximises the sum-rate when all the satellites cooperate to form a distributed multi-antenna BS. Our numerical results demonstrate the robustness of DS-PPO to CSI imperfections as well as the sum-rate improvement attached by the use of DS-PPO. In addition, we provide the convergence analysis for the DS-PPO along with the computational complexity.

cs.IT↗

arXiv2026

Approximation Algorithms for Action-Reward Query-Commit Matching

Matching problems under uncertainty arise in applications such as kidney exchange, hiring, and online marketplaces. A decision-maker must sequentially explore potential matches under local exploration constraints, while committing irrevocably to successful matches as they are revealed. The query-commit matching problem captures these challenges by modeling edges that succeed independently with known probabilities and must be accepted upon success, subject to vertex patience (time-out) constraints limiting the number of incident queries. In this work, we introduce the action-reward query-commit matching problem, a strict generalization of query-commit matching in which each query selects an action from a known action space, determining both the success probability and the reward of the queried edge. If an edge is queried using a chosen action and succeeds, it is irrevocably added to the matching, and the corresponding reward is obtained; otherwise, the edge is permanently discarded. We study the design of approximation algorithms for this problem on bipartite graphs. This model captures a broad class of stochastic matching problems, including the sequential pricing problem introduced by Pollner, Roghani, Saberi, and Wajc (EC~2022). On the positive side, Pollner et al. designed a polynomial-time approximation algorithm achieving a ratio of $0.426$ in the one-sided patience setting, which degrades to $0.395$ when both sides have bounded patience. In this work, we design computationally efficient algorithms for the action-reward query-commit in one-sided and two-sided patience settings, achieving approximation ratios of $1-1/e \approx 0.63$ and $\frac{1}{27}\!\left(19-67/e^3\right) \approx 0.58$ respectively. These results improve the state of the art for the sequential pricing problem, surpassing the previous guarantees of $0.426$ and $0.395$.

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arXiv2025

A Simple Analysis of Ranking in General Graphs

We provide a simple combinatorial analysis of the Ranking algorithm, originally introduced in the seminal work by Karp, Vazirani, and Vazirani [KVV90], demonstrating that it achieves a $(1/2 + c)$-approximate matching for general graphs for $c \geq 0.005$.

cs.DS↗

arXiv2025

Improved Approximation for Ranking on General Graphs

In this paper, we study Ranking, a well-known randomized greedy matching algorithm, for general graphs. The algorithm was originally introduced by Karp, Vazirani, and Vazirani [STOC 1990] for the online bipartite matching problem with one-sided vertex arrivals, where it achieves a tight approximation ratio of 1 - 1/e. It was later extended to general graphs by Goel and Tripathi [FOCS 2012]. The Ranking algorithm for general graphs is as follows: a permutation $σ$ over the vertices is chosen uniformly at random. The vertices are then processed sequentially according to this order, with each vertex being matched to the first available neighbor (if any) according to the same permutation $σ$. While the algorithm is quite well-understood for bipartite graphs-with the approximation ratio lying between 0.696 and 0.727, its approximation ratio for general graphs remains less well characterized despite extensive efforts. Prior to this work, the best known lower bound for general graphs was 0.526 by Chan et al. [TALG 2018], improving on the approximation ratio of 0.523 by Chan et al. [SICOMP 2018]. The upper bound, however, remains the same as that for bipartite graphs. In this work, we improve the approximation ratio of \textsc{Ranking} for general graphs to 0.5469, up from 0.526. This also surpasses the best-known approximation ratio of $0.531$ by Tang et al. [JACM 2023] for the oblivious matching problem. Our approach builds on the standard primal-dual analysis. The novelty of our work lies in proving new structural properties of Ranking by introducing the notion of the backup for vertices matched by the algorithm. For a fixed permutation, a vertex's backup is its potential match if its current match is removed. This concept helps characterize the rank distribution of the match of each vertex, enabling us to eliminate certain bad events that constrained previous work.

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arXiv2025

One-way Communication Complexity of Minimum Vertex Cover in General Graphs

We study the communication complexity of the Minimum Vertex Cover (MVC) problem on general graphs within the $k$-party one-way communication model. Edges of an arbitrary $n$-vertex graph are distributed among $k$ parties. The objective is for the parties to collectively find a small vertex cover of the graph while adhering to a communication protocol where each party sequentially sends a message to the next until the last party outputs a valid vertex cover of the whole graph. We are particularly interested in the trade-off between the size of the messages sent and the approximation ratio of the output solution. It is straightforward to see that any constant approximation protocol for MVC requires communicating $Ω(n)$ bits. Additionally, there exists a trivial 2-approximation protocol where the parties collectively find a maximal matching of the graph greedily and return the subset of vertices matched. This raises a natural question: \textit{What is the best approximation ratio achievable using optimal communication of $O(n)$?} We design a protocol with an approximation ratio of $(2-2^{-k+1}+ε)$ and $O(n)$ communication for any desirably small constant $ε>0$, which is strictly better than 2 for any constant number of parties. Moreover, we show that achieving an approximation ratio smaller than $3/2$ for the two-party case requires $n^{1 + Ω(1/\lg\lg n)}$ communication, thereby establishing the tightness of our protocol for two parties.

cs.CC↗

arXiv2024

New Philosopher Inequalities for Online Bayesian Matching, via Pivotal Sampling

We study the polynomial-time approximability of the optimal online stochastic bipartite matching algorithm, initiated by Papadimitriou et al. (EC'21). Here, nodes on one side of the graph are given upfront, while at each time $t$, an online node and its edge weights are drawn from a time-dependent distribution. The optimal algorithm is $\textsf{PSPACE}$-hard to approximate within some universal constant. We refer to this optimal algorithm, which requires time to think (compute), as a philosopher, and refer to polynomial-time online approximations of the above as philosopher inequalities. The best known philosopher inequality for online matching yields a $0.652$-approximation. In contrast, the best possible prophet inequality, or approximation of the optimum offline solution, is $0.5$. Our main results are a $0.678$-approximate algorithm and a $0.685$-approximation for a vertex-weighted special case. Notably, both bounds exceed the $0.666$-approximation of the offline optimum obtained by Tang, Wu, and Wu (STOC'22) for the vertex-weighted problem. Building on our algorithms and the recent black-box reduction of Banihashem et al. (SODA'24), we provide polytime (pricing-based) truthful mechanisms which $0.678$-approximate the social welfare of the optimal online allocation for bipartite matching markets. Our online allocation algorithm relies on the classic pivotal sampling algorithm (Srinivasan FOCS'01, Gandhi et al. J.ACM'06), along with careful discarding to obtain negative correlations between offline nodes. Consequently, the analysis boils down to examining the distribution of a weighted sum $X$ of negatively correlated Bernoulli variables, specifically lower bounding its mass below a threshold, $\mathbb{E}[\min(1,X)]$, of possible independent interest. Interestingly, our bound relies on an imaginary invocation of pivotal sampling.

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arXiv2024

Settling the Competition Complexity of Additive Buyers over Independent Items

The competition complexity of an auction setting is the number of additional bidders needed such that the simple mechanism of selling items separately (with additional bidders) achieves greater revenue than the optimal but complex (randomized, prior-dependent, Bayesian-truthful) optimal mechanism without the additional bidders. Our main result settles the competition complexity of $n$ bidders with additive values over $m < n$ independent items at $Θ(\sqrt{nm})$. The $O(\sqrt{nm})$ upper bound is due to [BW19], and our main result improves the prior lower bound of $Ω(\ln n)$ to $Ω(\sqrt{nm})$. Our main result follows from an explicit construction of a Bayesian IC auction for $n$ bidders with additive values over $m m$: $\text{SRev}_n(\mathcal{ER}^m) = nm+O_m(\ln (n)) \leq \text{SRev}_{n+O_m(\ln (n))}(\mathcal{ER}^m)$, and therefore our result can only follow by considering all possible truncations.

cs.GT↗

arXiv2024

Learning and Collusion in Multi-unit Auctions

We consider repeated multi-unit auctions with uniform pricing, which are widely used in practice for allocating goods such as carbon licenses. In each round, $K$ identical units of a good are sold to a group of buyers that have valuations with diminishing marginal returns. The buyers submit bids for the units, and then a price $p$ is set per unit so that all the units are sold. We consider two variants of the auction, where the price is set to the $K$-th highest bid and $(K+1)$-st highest bid, respectively. We analyze the properties of this auction in both the offline and online settings. In the offline setting, we consider the problem that one player $i$ is facing: given access to a data set that contains the bids submitted by competitors in past auctions, find a bid vector that maximizes player $i$'s cumulative utility on the data set. We design a polynomial time algorithm for this problem, by showing it is equivalent to finding a maximum-weight path on a carefully constructed directed acyclic graph. In the online setting, the players run learning algorithms to update their bids as they participate in the auction over time. Based on our offline algorithm, we design efficient online learning algorithms for bidding. The algorithms have sublinear regret, under both full information and bandit feedback structures. We complement our online learning algorithms with regret lower bounds. Finally, we analyze the quality of the equilibria in the worst case through the lens of the core solution concept in the game among the bidders. We show that the $(K+1)$-st price format is susceptible to collusion among the bidders; meanwhile, the $K$-th price format does not have this issue.

cs.GT↗

arXiv2023

Query Efficient Weighted Stochastic Matching

In this paper, we study the weighted stochastic matching problem. Let $G=(V, E)$ be a given edge-weighted graph and let its realization $\mathcal{G}$ be a random subgraph of $G$ that includes each edge $e\in E$ independently with a known probability $p_e$. The goal in this problem is to pick a sparse subgraph $Q$ of $G$ without prior knowledge of $G$'s realization, such that the maximum weight matching among the realized edges of $Q$ (i.e. the subgraph $Q\cap \mathcal{G}$) in expectation approximates the maximum weight matching of the entire realization $\mathcal{G}$. Attaining any constant approximation ratio for this problem requires selecting a subgraph of max-degree $Ω(1/p)$ where $p=\min_{e\in E} p_e$. On the positive side, there exists a $(1-ε)$-approximation algorithm by Behnezhad and Derakhshan, albeit at the cost of max-degree having exponential dependence on $1/p$. Within the $\text{poly}(1/p)$ regime, however, the best-known algorithm achieves a $0.536$ approximation ratio due to Dughmi, Kalayci, and Patel improving over the $0.501$ approximation algorithm by Behnezhad, Farhadi, Hajiaghayi, and Reyhani. In this work, we present a 0.68 approximation algorithm with $O(1/p)$ queries per vertex, which is asymptotically tight. This is even an improvement over the best-known approximation ratio of $2/3$ for unweighted graphs within the $\text{poly}(1/p)$ regime due to Assadi and Bernstein. The $2/3$ approximation ratio is proven tight in the presence of a few correlated edges in $\mathcal{G}$, indicating that surpassing the $2/3$ barrier should rely on the independent realization of edges. Our analysis involves reducing the problem to designing a randomized matching algorithm on a given stochastic graph with some variance-bounding properties.

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arXiv2023

Stochastic Minimum Vertex Cover in General Graphs: a $3/2$-Approximation

Our main result is designing an algorithm that returns a vertex cover of $\mathcal{G}^\star$ with size at most $(3/2+ε)$ times the expected size of the minimum vertex cover, using only $O(n/εp)$ non-adaptive queries. This improves over the best-known 2-approximation algorithm by Behnezhad, Blum, and Derakhshan [SODA'22], who also show that $Ω(n/p)$ queries are necessary to achieve any constant approximation. Our guarantees also extend to instances where edge realizations are not fully independent. We complement this upper bound with a tight $3/2$-approximation lower bound for stochastic graphs whose edges realizations demonstrate mild correlations.

cs.DS↗

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

Max-Weight Online Stochastic Matching: Improved Approximations Against the Online Benchmark

In this paper, we study max-weight stochastic matchings on online bipartite graphs under both vertex and edge arrivals. We focus on designing polynomial time approximation algorithms with respect to the online benchmark, which was first considered by Papadimitriou, Pollner, Saberi, and Wajc [EC'21]. In the vertex arrival version of the problem, the goal is to find an approximate max-weight matching of a given bipartite graph when the vertices in one part of the graph arrive online in a fixed order with independent chances of failure. Whenever a vertex arrives we should decide, irrevocably, whether to match it with one of its unmatched neighbors or leave it unmatched forever. There has been a long line of work designing approximation algorithms for different variants of this problem with respect to the offline benchmark (prophet). Papadimitriou et al., however, propose the alternative online benchmark and show that considering this new benchmark allows them to improve the 0.5 approximation ratio, which is the best ratio achievable with respect to the offline benchmark. They provide a 0.51-approximation algorithm which was later improved to 0.526 by Saberi and Wajc [ICALP'21]. The main contribution of this paper is designing a simple algorithm with a significantly improved approximation ratio of (1-1/e) for this problem. We also consider the edge arrival version in which, instead of vertices, edges of the graph arrive in an online fashion with independent chances of failure. Designing approximation algorithms for this problem has also been studied extensively with the best approximation ratio being 0.337 with respect to the offline benchmark. This paper, however, is the first to consider the online benchmark for the edge arrival version of the problem. For this problem, we provide a simple algorithm with an approximation ratio of 0.5 with respect to the online benchmark.

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arXiv2021

Stochastic Vertex Cover with Few Queries

We study the minimum vertex cover problem in the following stochastic setting. Let $G$ be an arbitrary given graph, $p \in (0, 1]$ a parameter of the problem, and let $G_p$ be a random subgraph that includes each edge of $G$ independently with probability $p$. We are unaware of the realization $G_p$, but can learn if an edge $e$ exists in $G_p$ by querying it. The goal is to find an approximate minimum vertex cover (MVC) of $G_p$ by querying few edges of $G$ non-adaptively. This stochastic setting has been studied extensively for various problems such as minimum spanning trees, matroids, shortest paths, and matchings. To our knowledge, however, no non-trivial bound was known for MVC prior to our work. In this work, we present a: * $(2+ε)$-approximation for general graphs which queries $O(\frac{1}{ε^3 p})$ edges per vertex, and a * $1.367$-approximation for bipartite graphs which queries $poly(1/p)$ edges per vertex. Additionally, we show that at the expense of a triple-exponential dependence on $p^{-1}$ in the number of queries, the approximation ratio can be improved down to $(1+ε)$ for bipartite graphs. Our techniques also lead to improved bounds for bipartite stochastic matching. We obtain a $0.731$-approximation with nearly-linear in $1/p$ per-vertex queries. This is the first result to break the prevalent $(2/3 \sim 0.66)$-approximation barrier in the $poly(1/p)$ query regime, improving algorithms of [Behnezhad et al; SODA'19] and [Assadi and Bernstein; SOSA'19].

cs.DS↗

arXiv2020

LP-based Approximation for Personalized Reserve Prices

We study the problem of computing data-driven personalized reserve prices in eager second price auctions without having any assumption on valuation distributions. Here, the input is a data-set that contains the submitted bids of $n$ buyers in a set of auctions and the problem is to return personalized reserve prices $\textbf r$ that maximize the revenue earned on these auctions by running eager second price auctions with reserve $\textbf r$. For this problem, which is known to be APX-hard, we present a novel LP formulation and a rounding procedure which achieves a $(1+2(\sqrt{2}-1)e^{\sqrt{2}-2})^{-1} \approx 0.684$-approximation. This improves over the $\frac{1}{2}$-approximation algorithm due to Roughgarden and Wang. We show that our analysis is tight for this rounding procedure. We also bound the integrality gap of the LP, which shows that it is impossible to design an algorithm that yields an approximation factor larger than $0.828$ with respect to this LP.

cs.GT↗

arXiv2020

Beating Greedy For Approximating Reserve Prices in Multi-Unit VCG Auctions

We study the problem of finding personalized reserve prices for unit-demand buyers in multi-unit eager VCG auctions with correlated buyers. The input to this problem is a dataset of submitted bids of $n$ buyers in a set of auctions. The goal is to find a vector of reserve prices, one for each buyer, that maximizes the total revenue across all auctions. Roughgarden and Wang (2016) showed that this problem is APX-hard but admits a greedy $\frac{1}{2}$-approximation algorithm. Later, Derakhshan, Golrezai, and Paes Leme (2019) gave an LP-based algorithm achieving a $0.68$-approximation for the (important) special case of the problem with a single-item, thereby beating greedy. We show in this paper that the algorithm of Derakhshan et al. in fact does not beat greedy for the general multi-item problem. This raises the question of whether or not the general problem admits a better-than-$\frac{1}{2}$ approximation. In this paper, we answer this question in the affirmative and provide a polynomial-time algorithm with a significantly better approximation-factor of $0.63$. Our solution is based on a novel linear programming formulation, for which we propose two different rounding schemes. We prove that the best of these two and the no-reserve case (all-zero vector) is a $0.63$-approximation.

cs.GT↗

arXiv2020

Stochastic Weighted Matching: $(1-ε)$ Approximation

Let $G=(V, E)$ be a given edge-weighted graph and let its {\em realization} $\mathcal{G}$ be a random subgraph of $G$ that includes each edge $e \in E$ independently with probability $p$. In the {\em stochastic matching} problem, the goal is to pick a sparse subgraph $Q$ of $G$ without knowing the realization $\mathcal{G}$, such that the maximum weight matching among the realized edges of $Q$ (i.e. graph $Q \cap \mathcal{G}$) in expectation approximates the maximum weight matching of the whole realization $\mathcal{G}$. In this paper, we prove that for any desirably small $ε\in (0, 1)$, every graph $G$ has a subgraph $Q$ that guarantees a $(1-ε)$-approximation and has maximum degree only $O_{ε, p}(1)$. That is, the maximum degree of $Q$ depends only on $ε$ and $p$ (both of which are known to be necessary) and not for example on the number of nodes in $G$, the edge-weights, etc. The stochastic matching problem has been studied extensively on both weighted and unweighted graphs. Previously, only existence of (close to) half-approximate subgraphs was known for weighted graphs [Yamaguchi and Maehara, SODA'18; Behnezhad et al., SODA'19]. Our result substantially improves over these works, matches the state-of-the-art for unweighted graphs [Behnezhad et al., STOC'20], and essentially settles the approximation factor.

cs.DS↗

arXiv2020

Stochastic Matching with Few Queries: $(1-\varepsilon)$ Approximation

Suppose that we are given an arbitrary graph $G=(V, E)$ and know that each edge in $E$ is going to be realized independently with some probability $p$. The goal in the stochastic matching problem is to pick a sparse subgraph $Q$ of $G$ such that the realized edges in $Q$, in expectation, include a matching that is approximately as large as the maximum matching among the realized edges of $G$. The maximum degree of $Q$ can depend on $p$, but not on the size of $G$. This problem has been subject to extensive studies over the years and the approximation factor has been improved from $0.5$ to $0.5001$ to $0.6568$ and eventually to $2/3$. In this work, we analyze a natural sampling-based algorithm and show that it can obtain all the way up to $(1-ε)$ approximation, for any constant $ε> 0$. A key and of possible independent interest component of our analysis is an algorithm that constructs a matching on a stochastic graph, which among some other important properties, guarantees that each vertex is matched independently from the vertices that are sufficiently far. This allows us to bypass a previously known barrier towards achieving $(1-ε)$ approximation based on existence of dense Ruzsa-Szemerédi graphs.

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