arXiv subjects

MohammadTaghi Hajiaghayi

Publications and source records attributed to MohammadTaghi Hajiaghayi.

At least 91 records · Page 5Linked to original sources

arXiv2019

MapReduce Meets Fine-Grained Complexity: MapReduce Algorithms for APSP, Matrix Multiplication, 3-SUM, and Beyond

Distributed processing frameworks, such as MapReduce, Hadoop, and Spark are popular systems for processing large amounts of data. The design of efficient algorithms in these frameworks is a challenging problem, as the systems both require parallelism---since datasets are so large that multiple machines are necessary---and limit the degree of parallelism---since the number of machines grows sublinearly in the size of the data. Although MapReduce is over a dozen years old~\cite{dean2008mapreduce}, many fundamental problems, such as Matrix Multiplication, 3-SUM, and All Pairs Shortest Paths, lack efficient MapReduce algorithms. We study these problems in the MapReduce setting. Our main contribution is to exhibit smooth trade-offs between the memory available on each machine, and the total number of machines necessary for each problem. Overall, we take the memory available to each machine as a parameter, and aim to minimize the number of rounds and number of machines. In this paper, we build on the well-known MapReduce theoretical framework initiated by Karloff, Suri, and Vassilvitskii ~\cite{karloff2010model} and give algorithms for many of these problems. The key to efficient algorithms in this setting lies in defining a sublinear number of large (polynomially sized) subproblems, that can then be solved in parallel. We give strategies for MapReduce-friendly partitioning, that result in new algorithms for all of the above problems. Specifically, we give constant round algorithms for the Orthogonal Vector (OV) and 3-SUM problems, and $O(\log n)$-round algorithms for Matrix Multiplication, All Pairs Shortest Paths (APSP), and Fast Fourier Transform (FFT), among others. In all of these we exhibit trade-offs between the number of machines and memory per machine.

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arXiv2019

Online Pandora's Boxes and Bandits

We consider online variations of the Pandora's box problem (Weitzman. 1979), a standard model for understanding issues related to the cost of acquiring information for decision-making. Our problem generalizes both the classic Pandora's box problem and the prophet inequality framework. Boxes are presented online, each with a random value and cost drew jointly from some known distribution. Pandora chooses online whether to open each box given its cost, and then chooses irrevocably whether to keep the revealed prize or pass on it. We aim for approximation algorithms against adversaries that can choose the largest prize over any opened box, and use optimal offline policies to decide which boxes to open (without knowledge of the value inside). We consider variations where Pandora can collect multiple prizes subject to feasibility constraints, such as cardinality, matroid, or knapsack constraints. We also consider variations related to classic multi-armed bandit problems from reinforcement learning. Our results use a reduction-based framework where we separate the issues of the cost of acquiring information from the online decision process of which prizes to keep. Our work shows that in many scenarios, Pandora can achieve a good approximation to the best possible performance.

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arXiv2019

Optimal Strategies of Blotto Games: Beyond Convexity

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 both of these objectives and show how it is possible to obtain (almost) optimal solutions that have few strategies in their support. One of the main technical challenges in obtaining bounded support strategies for the Colonel Blotto game is that the solution space becomes non-convex. This prevents us from using convex programming techniques in finding optimal strategies which are essentially the main tools that are used in the literature. However, we show through a set of structural results that the solution space can, interestingly, be partitioned into polynomially many disjoint convex polytopes that can be considered independently. Coupled with a number of other combinatorial observations, this leads to polynomial time approximation schemes for both of the aforementioned objectives.

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arXiv2018

Fast Algorithms for Knapsack via Convolution and Prediction

The \Problem{knapsack} problem is a fundamental problem in combinatorial optimization. It has been studied extensively from theoretical as well as practical perspectives as it is one of the most well-known NP-hard problems. The goal is to pack a knapsack of size $t$ with the maximum value from a collection of $n$ items with given sizes and values. Recent evidence suggests that a classic $O(nt)$ dynamic-programming solution for the \Problem{knapsack} problem might be the fastest in the worst case. In fact, solving the \Problem{knapsack} problem was shown to be computationally equivalent to the \Problem{$(\min, +)$ convolution} problem, which is thought to be facing a quadratic-time barrier. This hardness is in contrast to the more famous \Problem{$(+, \cdot)$ convolution} (generally known as \Problem{polynomial multiplication}), that has an $O(n\log n)$-time solution via Fast Fourier Transform. Our main results are algorithms with near-linear running times (in terms of the size of the knapsack and the number of items) for the \Problem{knapsack} problem, if either the values or sizes of items are small integers. More specifically, if item sizes are integers bounded by $\smax$, the running time of our algorithm is $\tilde O((n+t)\smax)$. If the item values are integers bounded by $\vmax$, our algorithm runs in time $\tilde O(n+t\vmax)$. Best previously known running times were $O(nt)$, $O(n^2\smax)$ and $O(n\smax\vmax)$ (Pisinger, J. of Alg., 1999).

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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.

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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.

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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.

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arXiv2018

Massively Parallel Dynamic Programming on Trees

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 solution of graph problems such as minimum bisection, $k$-spanning tree, maximum independent set, longest path, etc., when the input graph is a tree.

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arXiv2018

Semi-MapReduce Meets Congested Clique

Graph problems are troublesome when it comes to MapReduce. Typically, to be able to design algorithms that make use of the advantages of MapReduce, assumptions beyond what the model imposes, such as the density of the input graph, are required. In a recent shift, a simple and robust model of MapReduce for graph problems, where the space per machine is set to be O(|V|), has attracted considerable attention. We term this model semi-MapReduce, or in short, semiMPC, and focus on its computational power. We show through a set of simulation methods that semiMPC is, perhaps surprisingly, equivalent to the congested clique model of distributed computing. However, semiMPC, in addition to round complexity, incorporates another practically important dimension to optimize: the number of machines. Furthermore, we show that algorithms in other distributed computing models, such as CONGEST, can be simulated to run in the same number of rounds of semiMPC while also using an optimal number of machines. We later show the implications of these simulation methods by obtaining improved algorithms for these models using the recent algorithms that have been developed.

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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.

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arXiv2018

Prophet Secretary for Combinatorial Auctions and Matroids

The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} show that for adversarial arrival order of random variables the optimal prophet inequalities give a $1/2$-approximation. For many settings, however, it's conceivable that the arrival order is chosen uniformly at random, akin to the secretary problem. For such a random arrival model, we improve upon the $1/2$-approximation and obtain $(1-1/e)$-approximation prophet inequalities for both matroids and combinatorial auctions. This also gives improvements to the results of Yan \cite{yan2011mechanism} and Esfandiari et al. \cite{esfandiari2015prophet} who worked in the special cases where we can fully control the arrival order or when there is only a single item. Our techniques are threshold based. We convert our discrete problem into a continuous setting and then give a generic template on how to dynamically adjust these thresholds to lower bound the expected total welfare.

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arXiv2017

Fair Allocation of Indivisible Goods: Improvement and Generalization

We study the problem of fair allocation for indivisible goods. We use the the maxmin share paradigm introduced by Budish as a measure for fairness. Procaccia and Wang (EC'14) were first to investigate this fundamental problem in the additive setting. In contrast to what real-world experiments suggest, they show that a maxmin guarantee (1-MMS allocation) is not always possible even when the number of agents is limited to 3. While the existence of an approximation solution (e.g. a $1/2$-MMS allocation) is quite straightforward, improving the guarantee becomes subtler for larger constants. Procaccia provide a proof for existence of a $2/3$-MMS allocation and leave the question open for better guarantees. Our main contribution is an answer to the above question. We improve the result of [Procaccia and Wang] to a $3/4$ factor in the additive setting. The main idea for our $3/4$-MMS allocation method is clustering the agents. To this end, we introduce three notions and techniques, namely reducibility, matching allocation, and cycle-envy-freeness, and prove the approximation guarantee of our algorithm via non-trivial applications of these techniques. Our analysis involves coloring and double counting arguments that might be of independent interest. One major shortcoming of the current studies on fair allocation is the additivity assumption on the valuations. We alleviate this by extending our results to the case of submodular, fractionally subadditive, and subadditive settings. More precisely, we give constant approximation guarantees for submodular and XOS agents, and a logarithmic approximation for the case of subadditive agents. Furthermore, we complement our results by providing close upper bounds for each class of valuation functions. Finally, we present algorithms to find such allocations for additive, submodular, and XOS settings in polynomial time.

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arXiv2017

A Polynomial Time Algorithm for Spatio-Temporal Security Games

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 (M). 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 M, which can be very large.

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arXiv2017

Beating 1-1/e for Ordered Prophets

Hill and Kertz studied the prophet inequality on iid distributions [The Annals of Probability 1982]. They proved a theoretical bound of $1-\frac{1}{e}$ on the approximation factor of their algorithm. They conjectured that the best approximation factor for arbitrarily large n is $\frac{1}{1+1/e} \approx 0.731$. This conjecture remained open prior to this paper for over 30 years. In this paper we present a threshold-based algorithm for the prophet inequality with n iid distributions. Using a nontrivial and novel approach we show that our algorithm is a 0.738-approximation algorithm. By beating the bound of $\frac{1}{1+1/e}$, this refutes the conjecture of Hill and Kertz. Moreover, we generalize our results to non-iid distributions and discuss its applications in mechanism design.

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arXiv2017

Online Weighted Degree-Bounded Steiner Networks via Novel Online Mixed Packing/Covering

We design the first online algorithm with poly-logarithmic competitive ratio for the edge-weighted degree-bounded Steiner forest(EW-DB-SF) problem and its generalized variant. We obtain our result by demonstrating a new generic approach for solving mixed packing/covering integer programs in the online paradigm. In EW-DB-SF we are given an edge-weighted graph with a degree bound for every vertex. Given a root vertex in advance we receive a sequence of terminal vertices in an online manner. Upon the arrival of a terminal we need to augment our solution subgraph to connect the new terminal to the root. The goal is to minimize the total weight of the solution while respecting the degree bounds on the vertices. In the offline setting edge-weighted degree-bounded Steiner tree (EW-DB-ST) and its many variations have been extensively studied since early eighties. Unfortunately the recent advancements in the online network design problems are inherently difficult to adapt for degree-bounded problems. In contrast in this paper we obtain our result by using structural properties of the optimal solution, and reducing the EW-DB-SF problem to an exponential-size mixed packing/covering integer program in which every variable appears only once in covering constraints. We then design a generic integral algorithm for solving this restricted family of IPs. We demonstrate a new technique for solving mixed packing/covering integer programs. Define the covering frequency k of a program as the maximum number of covering constraints in which a variable can participate. Let m denote the number of packing constraints. We design an online deterministic integral algorithm with competitive ratio of O(k log m) for the mixed packing/covering integer programs. We believe this technique can be used as an interesting alternative for the standard primal-dual techniques in solving online problems.

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arXiv2017

Online Degree-Bounded Steiner Network Design

We initiate the study of degree-bounded network design problems in the online setting. The degree-bounded Steiner tree problem { which asks for a subgraph with minimum degree that connects a given set of vertices { is perhaps one of the most representative problems in this class. This paper deals with its well-studied generalization called the degree-bounded Steiner forest problem where the connectivity demands are represented by vertex pairs that need to be individually connected. In the classical online model, the input graph is given online but the demand pairs arrive sequentially in online steps. The selected subgraph starts off as the empty subgraph, but has to be augmented to satisfy the new connectivity constraint in each online step. The goal is to be competitive against an adversary that knows the input in advance. We design a simple greedy-like algorithm that achieves a competitive ratio of O(log n) where n is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio o(log n); thus our result is asymptotically tight. We further show strong hardness results for the group Steiner tree and the edge-weighted variants of degree-bounded connectivity problems. Fourer and Raghavachari resolved the online variant of degree-bounded Steiner forest in their paper in SODA'92. Since then, the natural family of degree-bounded network design problems has been extensively studied in the literature resulting in the development of many interesting tools and numerous papers on the topic. We hope that our approach in this paper, paves the way for solving the online variants of the classical problems in this family of network design problems.

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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.

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arXiv2017

From Duels to Battefields: Computing Equilibria of Blotto and Other Games

We study the problem of computing Nash equilibria of zero-sum games. Many natural zero-sum games have exponentially many strategies, but highly structured payoffs. For example, in the well-studied Colonel Blotto game (introduced by Borel in 1921), players must divide a pool of troops among a set of battlefields with the goal of winning (i.e., having more troops in) a majority. The Colonel Blotto game is commonly used for analyzing a wide range of applications from the U.S presidential election, to innovative technology competitions, to advertisement, to sports. However, because of the size of the strategy space, standard methods for computing equilibria of zero-sum games fail to be computationally feasible. Indeed, despite its importance, only a few solutions for special variants of the problem are known. In this paper we show how to compute equilibria of Colonel Blotto games. Moreover, our approach takes the form of a general reduction: to find a Nash equilibrium of a zero-sum game, it suffices to design a separation oracle for the strategy polytope of any bilinear game that is payoff-equivalent. We then apply this technique to obtain the first polytime algorithms for a variety of games. In addition to Colonel Blotto, we also show how to compute equilibria in an infinite-strategy variant called the General Lotto game; this involves showing how to prune the strategy space to a finite subset before applying our reduction. We also consider the class of dueling games, first introduced by Immorlica et al. (2011). We show that our approach provably extends the class of dueling games for which equilibria can be computed: we introduce a new dueling game, the matching duel, on which prior methods fail to be computationally feasible but upon which our reduction can be applied.

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