MohammadTaghi Hajiaghayi - MIT

MohammadTaghi Hajiaghayi
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MohammadTaghi Hajiaghayi

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Computer Science - Data Structures and Algorithms (26)
Computer Science - Computer Science and Game Theory (15)
Computer Science - Discrete Mathematics (5)
Computer Science - Computational Complexity (4)
Computer Science - Artificial Intelligence (2)
Mathematics - Probability (1)
Computer Science - Multiagent Systems (1)
Mathematics - Combinatorics (1)

Publications Authored By MohammadTaghi Hajiaghayi

In this paper, we study a stochastic variant of the celebrated k-server problem. In the k-server problem, we are required to minimize the total movement of k servers that are serving an online sequence of t requests in a metric. In the stochastic setting we are given t independent distributions Read More

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. Read More

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. Read More

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. Read More

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. Procacciafirst (EC'14) were first to investigate this fundamental problem in the additive setting. Read More

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. Read 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. Read More

In this paper we consider two special cases of the "cover-by-pairs" optimization problem that arise when we need to place facilities so that each customer is served by two facilities that reach it by disjoint shortest paths. These problems arise in a network traffic monitoring scheme proposed by Breslau et al. and have potential applications to content distribution. Read More

We study competition in a general framework introduced by Immorlica et al. and answer their main open question. Immorlica et al. Read More

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. Read More

Optimal stopping theory is a powerful tool for analyzing scenarios such as online auctions in which we generally require optimizing an objective function over the space of stopping rules for an allocation process under uncertainty. Perhaps the most classic problems of stopping theory are the prophet inequality problem and the secretary problem. The classical prophet inequality states that by choosing the same threshold OPT/2 for every step, one can achieve the tight competitive ratio of 0. Read More

An instance of the Connected Maximum Cut problem consists of an undirected graph G = (V, E) and the goal is to find a subset of vertices S $\subseteq$ V that maximizes the number of edges in the cut \delta(S) such that the induced graph G[S] is connected. We present the first non-trivial \Omega(1/log n) approximation algorithm for the connected maximum cut problem in general graphs using novel techniques. We then extend our algorithm to an edge weighted case and obtain a poly-logarithmic approximation algorithm. Read More

Online advertising is the main source of revenue for many Internet firms. A central component of online advertising is the underlying mechanism that selects and prices the winning ads for a given ad slot. In this paper we study designing a mechanism for the Combinatorial Auction with Identical Items (CAII) in which we are interested in selling $k$ identical items to a group of bidders each demanding a certain number of items between $1$ and $k$. Read More

Recently [Bhattacharya et al., STOC 2015] provide the first non-trivial algorithm for the densest subgraph problem in the streaming model with additions and deletions to its edges, i.e. Read More

Given an edge-weighted directed graph $G=(V,E)$ on $n$ vertices and a set $T=\{t_1, t_2, \ldots, t_p\}$ of $p$ terminals, the objective of the \scss ($p$-SCSS) problem is to find an edge set $H\subseteq E$ of minimum weight such that $G[H]$ contains an $t_{i}\rightarrow t_j$ path for each $1\leq i\neq j\leq p$. In this paper, we investigate the computational complexity of a variant of $2$-SCSS where we have demands for the number of paths between each terminal pair. Formally, the \sharinggeneral problem is defined as follows: given an edge-weighted directed graph $G=(V,E)$ with weight function $\omega: E\rightarrow \mathbb{R}^{\geq 0}$, two terminal vertices $s, t$, and integers $k_1, k_2$ ; the objective is to find a set of $k_1$ paths $F_1, F_2, \ldots, F_{k_1}$ from $s\leadsto t$ and $k_2$ paths $B_1, B_2, \ldots, B_{k_2}$ from $t\leadsto s$ such that $\sum_{e\in E} \omega(e)\cdot \phi(e)$ is minimized, where $\phi(e)= \max \Big\{|\{i\in [k_1] : e\in F_i\}|\ ,\ |\{j\in [k_2] : e\in B_j\}|\Big\}$. Read More

In this paper we present a simple but powerful subgraph sampling primitive that is applicable in a variety of computational models including dynamic graph streams (where the input graph is defined by a sequence of edge/hyperedge insertions and deletions) and distributed systems such as MapReduce. In the case of dynamic graph streams, we use this primitive to prove the following results: -- Matching: First, there exists an $\tilde{O}(k^2)$ space algorithm that returns an exact maximum matching on the assumption the cardinality is at most $k$. The best previous algorithm used $\tilde{O}(kn)$ space where $n$ is the number of vertices in the graph and we prove our result is optimal up to logarithmic factors. Read More

We study the problem of selling $n$ items to a single buyer with an additive valuation function. We consider the valuation of the items to be correlated, i.e. Read More

Cournot competition is a fundamental economic model that represents firms competing in a single market of a homogeneous good. Each firm tries to maximize its utility---a function of the production cost as well as market price of the product---by deciding on the amount of production. In today's dynamic and diverse economy, many firms often compete in more than one market simultaneously, i. Read More

As graphs continue to grow in size, we seek ways to effectively process such data at scale. The model of streaming graph processing, in which a compact summary is maintained as each edge insertion/deletion is observed, is an attractive one. However, few results are known for optimization problems over such dynamic graph streams. Read More

We study the power of fractional allocations of resources to maximize influence in a network. This work extends in a natural way the well-studied model by Kempe, Kleinberg, and Tardos (2003), where a designer selects a (small) seed set of nodes in a social network to influence directly, this influence cascades when other nodes reach certain thresholds of neighbor influence, and the goal is to maximize the final number of influenced nodes. Despite extensive study from both practical and theoretical viewpoints, this model limits the designer to a binary choice for each node, with no way to apply intermediate levels of influence. Read More

Adoption or rejection of ideas, products, and technologies in a society is often governed by simultaneous propagation of positive and negative influences. Consider a planner trying to introduce an idea in different parts of a society at different times. How should the planner design a schedule considering this fact that positive reaction to the idea in early areas has a positive impact on probability of success in later areas, whereas a flopped reaction has exactly the opposite impact? We generalize a well-known economic model which has been recently used by Chierichetti, Kleinberg, and Panconesi (ACM EC'12). Read More

A Fixed-Parameter Tractable (\FPT) $\rho$-approximation algorithm for a minimization (resp. maximization) parameterized problem $P$ is an FPT algorithm that, given an instance $(x, k)\in P$ computes a solution of cost at most $k \cdot \rho(k)$ (resp. $k/\rho(k)$) if a solution of cost at most (resp. Read More

In this paper, we consider the fault-tolerant $k$-median problem and give the \emph{first} constant factor approximation algorithm for it. In the fault-tolerant generalization of classical $k$-median problem, each client $j$ needs to be assigned to at least $r_j \ge 1$ distinct open facilities. The service cost of $j$ is the sum of its distances to the $r_j$ facilities, and the $k$-median constraint restricts the number of open facilities to at most $k$. Read More

In the {\em Movement Repairmen (MR)} problem we are given a metric space $(V, d)$ along with a set $R$ of $k$ repairmen $r_1, r_2, ... Read More

Moss and Rabani[12] study constrained node-weighted Steiner tree problems with two independent weight values associated with each node, namely, cost and prize (or penalty). They give an O(log n)-approximation algorithm for the prize-collecting node-weighted Steiner tree problem (PCST). They use the algorithm for PCST to obtain a bicriteria (2, O(log n))-approximation algorithm for the Budgeted node-weighted Steiner tree problem. Read More

In this paper we consider a generalization of the classical k-center problem with capacities. Our goal is to select k centers in a graph, and assign each node to a nearby center, so that we respect the capacity constraints on centers. The objective is to minimize the maximum distance a node has to travel to get to its assigned center. Read More

We introduce a new technique for designing fixed-parameter algorithms for cut problems, namely randomized contractions. We apply our framework to obtain the first FPT algorithm for the Unique Label Cover problem and new FPT algorithms with exponential speed up for the Steiner Cut and Node Multiway Cut-Uncut problems. More precisely, we show the following: - We prove that the parameterized version of the Unique Label Cover problem, which is the base of the Unique Games Conjecture, can be solved in 2^{O(k^2\log |\Sigma|)}n^4\log n deterministic time (even in the stronger, vertex-deletion variant) where k is the number of unsatisfied edges and |\Sigma| is the size of the alphabet. Read More

We study an extensive class of movement minimization problems which arise from many practical scenarios but so far have little theoretical study. In general, these problems involve planning the coordinated motion of a collection of agents (representing robots, people, map labels, network messages, etc.) to achieve a global property in the network while minimizing the maximum or average movement (expended energy). Read More

Given a graph $G$ and an integer $k$, the Feedback Vertex Set (FVS) problem asks if there is a vertex set $T$ of size at most $k$ that hits all cycles in the graph. The fixed-parameter tractability status of FVS in directed graphs was a long-standing open problem until Chen et al. (STOC '08) showed that it is FPT by giving a $4^{k}k!n^{O(1)}$ time algorithm. Read More

In this paper we propose a game-theoretic model to analyze events similar to the 2009 \emph{DARPA Network Challenge}, which was organized by the Defense Advanced Research Projects Agency (DARPA) for exploring the roles that the Internet and social networks play in incentivizing wide-area collaborations. The challenge was to form a group that would be the first to find the locations of ten moored weather balloons across the United States. We consider a model in which $N$ people (who can form groups) are located in some topology with a fixed coverage volume around each person's geographical location. Read More

Given a directed graph $G$, a set of $k$ terminals and an integer $p$, the \textsc{Directed Vertex Multiway Cut} problem asks if there is a set $S$ of at most $p$ (nonterminal) vertices whose removal disconnects each terminal from all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous problem where $S$ is a set of at most $p$ edges. These two problems indeed are known to be equivalent. Read More

We focus on designing combinatorial algorithms for the Capacitated Network Design problem (Cap-SNDP). The Cap-SNDP is the problem of satisfying connectivity requirements when edges have costs and hard capacities. We begin by showing that the Group Steiner tree problem (GST) is a special case of Cap-SNDP even when there is connectivity requirement between only one source-sink pair. Read More

Coalition formation is a key topic in multi-agent systems. Coalitions enable agents to achieve goals that they may not have been able to achieve on their own. Previous work has shown problems in coalitional games to be computationally hard. Read More

In this paper, we reduce Prize-Collecting Steiner TSP (PCTSP), Prize-Collecting Stroll (PCS), Prize-Collecting Steiner Tree (PCST), Prize-Collecting Steiner Forest (PCSF) and more generally Submodular Prize-Collecting Steiner Forest (SPCSF) on planar graphs (and more generally bounded-genus graphs) to the same problems on graphs of bounded treewidth. More precisely, we show any $\alpha$-approximation algorithm for these problems on graphs of bounded treewidth gives an $(\alpha + \epsilon)$-approximation algorithm for these problems on planar graphs (and more generally bounded-genus graphs), for any constant $\epsilon > 0$. Since PCS, PCTSP, and PCST can be solved exactly on graphs of bounded treewidth using dynamic programming, we obtain PTASs for these problems on planar graphs and bounded-genus graphs. Read More

We study bargaining games between suppliers and manufacturers in a network context. Agents wish to enter into contracts in order to generate surplus which then must be divided among the participants. Potential contracts and their surplus are represented by weighted edges in our bipartite network. Read More

In this paper, we consider Steiner forest and its generalizations, prize-collecting Steiner forest and k-Steiner forest, when the vertices of the input graph are points in the Euclidean plane and the lengths are Euclidean distances. First, we present a simpler analysis of the polynomial-time approximation scheme (PTAS) of Borradaile et al. [12] for the Euclidean Steiner forest problem. Read More

We give the first polynomial-time approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus. As a first step, we show how to build a Steiner forest spanner for such graphs. The crux of the process is a clustering procedure called prize-collecting clustering that breaks down the input instance into separate subinstances which are easier to handle; moreover, the terminals in different subinstances are far from each other. Read More

In general, the games are played on a host graph, where each node is a selfish independent agent (player) and each edge has a fixed link creation cost \alpha. Together the agents create a network (a subgraph of the host graph) while selfishly minimizing the link creation costs plus the sum of the distances to all other players (usage cost). In this paper, we pursue two important facets of the network creation game. Read More

The k-forest problem is a common generalization of both the k-MST and the dense-$k$-subgraph problems. Formally, given a metric space on $n$ vertices $V$, with $m$ demand pairs $\subseteq V \times V$ and a ``target'' $k\le m$, the goal is to find a minimum cost subgraph that connects at least $k$ demand pairs. In this paper, we give an $O(\min\{\sqrt{n},\sqrt{k}\})$-approximation algorithm for $k$-forest, improving on the previous best ratio of $O(n^{2/3}\log n)$ by Segev & Segev. Read More

In this paper we extend the theory of bidimensionality to two families of graphs that do not exclude fixed minors: map graphs and power graphs. In both cases we prove a polynomial relation between the treewidth of a graph in the family and the size of the largest grid minor. These bounds improve the running times of a broad class of fixed-parameter algorithms. Read More

We prove that a random 3-SAT instance with clause-to-variable density less than 3.52 is satisfiable with high probability. The proof comes through an algorithm which selects (and sets) a variable depending on its degree and that of its complement. Read More