Papers by Mahsa Derakhshan
Proceedings of the AAAI Conference on Artificial Intelligence
In the Colonel Blotto game, which was initially introduced by Borel in 1921, two colonels simulta... more In the Colonel Blotto game, which was initially introduced by Borel in 1921, two colonels simultaneously distribute their troops across different battlefields.The winner of each battlefield is determined independently by a winner-take-all rule. The ultimate payoff of each colonel is the number of battlefields he wins. This game is commonly used for analyzing a wide range of applications such as the U.S presidential election, innovative technology competitions, advertisements, etc. There have been persistent efforts for finding the optimal strategies for the Colonel Blotto game. After almost a century Ahmadinejad, Dehghani, Hajiaghayi, Lucier, Mahini, and Seddighin provided a poly-time algorithm for finding the optimal strategies. They first model the problem by a Linear Program (LP) with exponential number of constraints and use Ellipsoid method to solve it. However, despite the theoretical importance of their algorithm, it ishighly impractical. In general, even Simplex method (despite it...
Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing
Suppose that we are given an arbitrary graph G = (V, E) and know that each edge in E is going to ... more 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 [10, 4] to 0.5001 [5] to 0.6568 [7] and eventually to 2/3 [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 [4, 5] towards achieving (1−ε) approximation based on existence of dense Ruzsa-Szemerédi graphs.
arXiv: Data Structures and Algorithms, Sep 11, 2018
Dynamic programming is a powerful technique that is, unfortunately, often inherently sequential. ... more Dynamic programming is a powerful technique that is, unfortunately, often inherently sequential. That is, there exists no unified method to parallelize algorithms that use dynamic programming. In this paper, we attempt to address this issue in the Massively Parallel Computations (MPC) model which is a popular abstraction of MapReduce-like paradigms. Our main result is an algorithmic framework to adapt a large family of dynamic programs defined over trees. We introduce two classes of graph problems that admit dynamic programming solutions on trees. We refer to them as "(poly log)-expressible" and "linear-expressible" problems. We show that both classes can be parallelized in O(log n) rounds using a sublinear number of machines and a sublinear memory per machine. To achieve this result, we introduce a series of techniques that can be plugged together. To illustrate the generality of our framework, we implement in O(log n) rounds of MPC, the dynamic programming solution of graph problems such as minimum bisection, k-spanning tree, maximum independent set, longest path, etc., when the input graph is a tree.
arXiv: Distributed, Parallel, and Cluster Computing, Jul 17, 2018
The success of modern parallel paradigms such as MapReduce, Hadoop, or Spark, has attracted a sig... more The success of modern parallel paradigms such as MapReduce, Hadoop, or Spark, has attracted a significant attention to the Massively Parallel Computation (MPC) model over the past few years, especially on graph problems. In this work, we consider symmetry breaking problems of maximal independent set (MIS) and maximal matching (MM), which are among the most intensively studied problems in distributed/parallel computing, in MPC.
Operations Research, 2022
The Colonel Blotto game (initially introduced by Borel in 1921) is commonly used for analyzing a ... more The Colonel Blotto game (initially introduced by Borel in 1921) is commonly used for analyzing a wide range of applications from the U.S.Ppresidential election to innovative technology competitions to advertising, sports, and politics. After around a century Ahmadinejad et al. provided the first polynomial-time algorithm for computing the Nash equilibria in Colonel Blotto games. However, their algorithm consists of an exponential-size LP solved by the ellipsoid method, which is highly impractical. In “Fast and Simple Solutions of Blotto Games,” Behnezhad, Dehghani, Derakhshan, Hajighayi, and Seddighin provide the first polynomial-size LP formulation of the optimal strategies for the Colonel Blotto game using linear extension techniques. They use this polynomial-size LP to provide a simpler and significantly faster algorithm for finding optimal strategies of the Colonel Blotto game. They further show this representation is asymptotically tight, which means there exists no other linea...
Proceedings of the 2017 ACM Conference on Economics and Computation, 2017
An ever-important issue is protecting infrastructure and other valuable targets from a range of t... more An ever-important issue is protecting infrastructure and other valuable targets from a range of threats from vandalism to theft to piracy to terrorism. The "defender" can rarely afford the needed resources for a 100% protection. Thus, the key question is, how to provide the best protection using the limited available resources. We study a practically important class of security games that is played out in space and time, with targets and "patrols" moving on a real line. A central open question here is whether the Nash equilibrium (i.e., the minimax strategy of the defender) can be computed in polynomial time. We resolve this question in the affirmative. Our algorithm runs in time polynomial in the input size, and only polylogarithmic in the number of possible patrol locations (). Further, we provide a continuous extension in which patrol locations can take arbitrary real values. Prior work obtained polynomial-time algorithms only under a substantial assumption, e.g., a constant number of rounds. Further, all these algorithms have running times polynomial in , which can be very large.
The Colonel Blotto game, first introduced by Borel in 1921, is a well-studied game theory classic... more The Colonel Blotto game, first introduced by Borel in 1921, is a well-studied game theory classic. Two colonels each have a pool of troops that they divide simultaneously among a set of battlefields. The winner of each battlefield is the colonel who puts more troops in it and the overall utility of each colonel is the sum of weights of the battlefields that s/he wins. Over the past century, the Colonel Blotto game has found applications in many different forms of competition from advertisements to politics to sports. Two main objectives have been proposed for this game in the literature: (i) maximizing the guaranteed expected payoff, and (ii) maximizing the probability of obtaining a minimum payoff u. The former corresponds to the conventional utility maximization and the latter concerns scenarios such as elections where the candidates' goal is to maximize the probability of getting at least half of the votes (rather than the expected number of votes). In this paper, we consider...
Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2022
We study the minimum vertex cover problem in the following stochastic setting. Let G be an arbitr... more We study the minimum vertex cover problem in the following stochastic setting. Let G be an arbitrary given graph, p ∈ (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 nontrivial bound was known for MVC prior to our work. In this work, we present a: • (2 + ε)-approximation for general graphs which queries O(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 ∼ 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].
Proceedings of the 2018 ACM Conference on Economics and Computation, 2018
Protecting valuable targets from an adversary is an ever-important international concern with far... more Protecting valuable targets from an adversary is an ever-important international concern with far-reaching applications in wildlife protection, border protection, counter-terrorism, protection of ships from piracy, etc. As a successful recent approach, security games cast these issues as two-player games between a defender and an attacker. The defender decides on how to allocate the available resources to protect targets against the attacker who strives to in ict damage on them. The main question of interest here is equilibrium computation. Our focus in this paper is on spatio-temporal security games. However, inspired by the paper of Xu [EC'16], we start with a general model of security games and show that any approximation (of any factor) for the defender's best response (DBR) problem leads to an approximation of the same factor for the actual game. In most applications of security games, the targets are mobile. This leads to a well-studied class of succinct games, namely spatio-temporal security games, that is played in space and time. In such games, the defender has to specify a time-dependent patrolling strategy over a spatial domain to protect a set of moving targets. We give a generalized model of prior spatio-temporal security games that is played on a base graph G. That is, the patrols can be placed on the vertices of G and move along its edges over time. This uni es and generalizes prior spatio-temporal models that only consider speci c spatial domains such as lines or grids. Graphs can further model many other domains of practical interest such as roads, internal maps of buildings, etc. Finding an optimal defender strategy becomes NP-hard on general graphs. To overcome this, we give an LP relaxation of the DBR problem and devise a rounding technique to obtain an almost optimal integral solution. More precisely, we show that one can achieve a (1 − ϵ)-approximation in polynomial time if we allow the defender to use ln(1/ϵ) times more patrols. We later show that this result is in some sense the best possible polynomial time algorithm (unless P=NP). Furthermore, we show that by using a novel dependent rounding technique, the same LP relaxation gives an optimal solution for speci c domains of interest, such as one-dimensional spaces. This result simpli es and improves upon the prior algorithm of Behnezhad et al. [EC'17] on several aspects and can be generalized to other graphs of interest such as cycles. Lastly, we note that most prior algorithms for security games assume that the attacker attacks only once and become intractable for a super-constant number of attacks. Our algorithms are fully polynomial in the input size and work for any given number of attacks. CCS Concepts: • Theory of computation → Algorithmic game theory and mechanism design;
2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), 2020
Let G = (V, E) be a given edge-weighted graph and let its realization G be a random subgraph of G... more Let G = (V, E) be a given edge-weighted graph and let its realization G be a random subgraph of G that includes each edge e ∈ E independently with probability p. In the stochastic matching problem, the goal is to pick a sparse subgraph Q of G without knowing the realization G, such that the maximum weight matching among the realized edges of Q (i.e. graph Q ∩ G) in expectation approximates the maximum weight matching of the whole realization G. In this paper, we prove that for any desirably small ε ∈ (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. * Supported by a Google PhD Fellowship.
Proceedings of the 21st ACM Conference on Economics and Computation, 2020
On online platforms, consumers face an abundance of options that are displayed in the form of a p... more On online platforms, consumers face an abundance of options that are displayed in the form of a position ranking. Only products placed in the first few positions are readily accessible to the consumer, and she needs to exert effort to access more options. For such platforms, we develop a two-stage sequential search model where in the first stage, the consumer sequentially screens positions to observe the preference weight of the products placed in them and forms a consideration set. In the second stage, she observes the additional idiosyncratic utility that she can derive from each product and chooses the highest-utility product within her consideration set. For this model, we first characterize the optimal sequential search policy of a welfare-maximizing consumer. We then study how platforms with different objectives should rank products. We focus on two objectives: (i) maximizing the platform's market share and (ii) maximizing the consumer's welfare. Somewhat surprisingly, we show that ranking products in decreasing order of their preference weights does not necessarily maximize market share or consumer welfare. Such a ranking may shorten the consumer's consideration set due to the externality effect of high-positioned products on low-positioned ones, leading to insufficient screening. We then show that both problems---maximizing market share and maximizing consumer welfare---are NP-complete. We develop novel near-optimal polynomial-time ranking algorithms for each objective. Further, we show that even though ranking products in decreasing order of their preference weights is suboptimal, such a ranking enjoys strong performance guarantees for both objectives. We complement our theoretical developments with numerical studies using synthetic data in which we show (1) that heuristic versions of our algorithms that do not rely on model primitives perform well and (2) that our model can be effectively estimated using a maximum likelihood estimator.
Graph clustering is a fundamental task in many data-mining and machine-learning pipelines. In par... more Graph clustering is a fundamental task in many data-mining and machine-learning pipelines. In particular, identifying a good hierarchical structure is at the same time a fundamental and challenging problem for several applications. The amount of data to analyze is increasing at an astonishing rate each day. Hence there is a need for new solutions to efficiently compute effective hierarchical clusterings on such huge data. The main focus of this paper is on minimum spanning tree (MST) based clusterings. In particular, we propose affinity, a novel hierarchical clustering based on Boruvka's MST algorithm. We prove certain theoretical guarantees for affinity (as well as some other classic algorithms) and show that in practice it is superior to several other state-of-the-art clustering algorithms. Furthermore, we present two MapReduce implementations for affinity. The first one works for the case where the input graph is dense and takes constant rounds. It is based on a Massively Par...
We assume some sort of structure on the users’ demand which is called Leontief preferences, and s... more We assume some sort of structure on the users’ demand which is called Leontief preferences, and states that the utility of an agent is the fraction of its dominant resource that it can actually use, given its proportional demands and its allocation of the various resources. For example, an agent that requires twice as much CPU as RAM to run a task prefers to be allocated 4 CPU units and 2 RAM units to 2 CPU units and 1 RAM unit, but is indifferent between the former allocation and 5 CPU units and 2 RAM units. Let wi denote the units of resource i that an agent needs to run a task, and let xi denote the units of resource i that is allocated to this agent. The utility of the agent in this allocation is as follows:
arXiv: Computer Science and Game Theory, 2019
We study the problem of computing data-driven personalized reserve prices in eager second price a... more 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 NP-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.
We study the problem of finding personalized reserve prices for unit-demand buyers in multi-unit ... more 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 poly...
A valid edge-coloring of a graph is an assignment of “colors” to its edges such that no two incid... more A valid edge-coloring of a graph is an assignment of “colors” to its edges such that no two incident edges receive the same color. The goal is to find a proper coloring that uses few colors. (Note that the maximum degree, ∆, is a trivial lower bound.) In this paper, we revisit this fundamental problem in two models of computation specific to massive graphs, the Massively Parallel Computations (MPC) model and the Graph Streaming model: Massively Parallel Computation. We give a randomized MPC algorithm that with high probability returns a ∆ + Õ(∆3/4) edge coloring in O(1) rounds using O(n) space per machine and O(m) total space. The space per machine can also be further improved to n1−Ω(1) if ∆ = nΩ(1). Our algorithm improves upon a previous result of Harvey et al. [SPAA 2018]. Graph Streaming. Since the output of edge-coloring is as large as its input, we consider a standard variant of the streaming model where the output is also reported in a streaming fashion. The main challenge is...
Management Science, 2021
We study the problem of computing data-driven personalized reserve prices in eager second price a... more 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 r that maximize the revenue earned on these auctions by running eager second price auctions with reserve r. For this problem, which is known to be NP complete, we present a novel linear program (LP) formulation and a rounding procedure, which achieves a 0.684 approximation. This improves over the [Formula: see text]-approximation algorithm from 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. This paper was accepted by Chung Piaw Teo, Management Science Spec...
ArXiv, 2020
Let G = (V,E) be a given edge-weighted graph and let its realization G be a random subgraph of G ... more Let G = (V,E) be a given edge-weighted graph and let its realization G be a random subgraph of G that includes each edge e ∈ E independently with probability p. We study a stochastic matching problem where the goal is to non-adaptively pick a sparse subgraph Q of G (without knowing the realization G), such that the maximum weight matching among the realized edges of Q (i.e. graph Q ∩ G) in expectation approximates the maximum weight matching of the whole realization G. In this paper, we prove that for any ε ∈ (0, 1), every graph G has a subgraph Q that has maximum degree only Oε,p(1) and guarantees a (1− ε)-approximation. 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 grap...
ArXiv, 2018
Dynamic programming is a powerful technique that is, unfortunately, often inherently sequential. ... more Dynamic programming is a powerful technique that is, unfortunately, often inherently sequential. That is, there exists no unified method to parallelize algorithms that use dynamic programming. In this paper, we attempt to address this issue in the Massively Parallel Computations (MPC) model which is a popular abstraction of MapReduce-like paradigms. Our main result is an algorithmic framework to adapt a large family of dynamic programs defined over trees. We introduce two classes of graph problems that admit dynamic programming solutions on trees. We refer to them as "(polylog)-expressible" and "linear-expressible" problems. We show that both classes can be parallelized in $O(\log n)$ rounds using a sublinear number of machines and a sublinear memory per machine. To achieve this result, we introduce a series of techniques that can be plugged together. To illustrate the generality of our framework, we implement in $O(\log n)$ rounds of MPC, the dynamic programming...
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Papers by Mahsa Derakhshan