arXiv subjects

Sepehr Assadi

Publications and source records attributed to Sepehr Assadi.

81 records · Page 5Linked to original sources

arXiv2017

Combinatorial Auctions Do Need Modest Interaction

We study the necessity of interaction for obtaining efficient allocations in subadditive combinatorial auctions. This problem was originally introduced by Dobzinski, Nisan, and Oren (STOC'14) as the following simple market scenario: $m$ items are to be allocated among $n$ bidders in a distributed setting where bidders valuations are private and hence communication is needed to obtain an efficient allocation. The communication happens in rounds: in each round, each bidder, simultaneously with others, broadcasts a message to all parties involved and the central planner computes an allocation solely based on the communicated messages. Dobzinski et.al. showed that no non-interactive ($1$-round) protocol with polynomial communication (in the number of items and bidders) can achieve approximation ratio better than $Ω(m^{{1}/{4}})$, while for any $r \geq 1$, there exists $r$-round protocols that achieve $\widetilde{O}(r \cdot m^{{1}/{r+1}})$ approximation with polynomial communication; in particular, $O(\log{m})$ rounds of interaction suffice to obtain an (almost) efficient allocation. A natural question at this point is to identify the "right" level of interaction (i.e., number of rounds) necessary to obtain an efficient allocation. In this paper, we resolve this question by providing an almost tight round-approximation tradeoff for this problem: we show that for any $r \geq 1$, any $r$-round protocol that uses polynomial communication can only approximate the social welfare up to a factor of $Ω(\frac{1}{r} \cdot m^{{1}/{2r+1}})$. This in particular implies that $Ω(\frac{\log{m}}{\log\log{m}})$ rounds of interaction are necessary for obtaining any efficient allocation in these markets. Our work builds on the recent multi-party round-elimination technique of Alon, Nisan, Raz, and Weinstein (FOCS'15) and settles an open question posed by Dobzinski et.al. and Alon et. al.

cs.GT↗

arXiv2017

Tight Space-Approximation Tradeoff for the Multi-Pass Streaming Set Cover Problem

We study the classic set cover problem in the streaming model: the sets that comprise the instance are revealed one by one in a stream and the goal is to solve the problem by making one or few passes over the stream while maintaining a sublinear space $o(mn)$ in the input size; here $m$ denotes the number of the sets and $n$ is the universe size. Notice that in this model, we are mainly concerned with the space requirement of the algorithms and hence do not restrict their computation time. Our main result is a resolution of the space-approximation tradeoff for the streaming set cover problem: we show that any $α$-approximation algorithm for the set cover problem requires $\widetildeΩ(mn^{1/α})$ space, even if it is allowed polylog${(n)}$ passes over the stream, and even if the sets are arriving in a random order in the stream. This space-approximation tradeoff matches the best known bounds achieved by the recent algorithm of Har-Peled et.al. (PODS 2016) that requires only $O(α)$ passes over the stream in an adversarial order, hence settling the space complexity of approximating the set cover problem in data streams in a quite robust manner. Additionally, our approach yields tight lower bounds for the space complexity of $(1- ε)$-approximating the streaming maximum coverage problem studied in several recent works.

cs.DS↗

arXiv2017

On Estimating Maximum Matching Size in Graph Streams

We study the problem of estimating the maximum matching size in graphs whose edges are revealed in a streaming manner. We consider both insertion-only streams and dynamic streams and present new upper and lower bound results for both models. On the upper bound front, we show that an $α$-approximate estimate of the matching size can be computed in dynamic streams using $\widetilde{O}({n^2/α^4})$ space, and in insertion-only streams using $\widetilde{O}(n/α^2)$-space. On the lower bound front, we prove that any $α$-approximation algorithm for estimating matching size in dynamic graph streams requires $Ω(\sqrt{n}/α^{2.5})$ bits of space, even if the underlying graph is both sparse and has arboricity bounded by $O(α)$. We further improve our lower bound to $Ω(n/α^2)$ in the case of dense graphs. Furthermore, we prove that a $(1+ε)$-approximation to matching size in insertion-only streams requires RS$(n) \cdot n^{1-O(ε)}$ space; here, RS${n}$ denotes the maximum number of edge-disjoint induced matchings of size $Θ(n)$ in an $n$-vertex graph. It is a major open problem to determine the value of RS$(n)$, and current results leave open the possibility that RS$(n)$ may be as large as $n/\log n$. We also show how to avoid the dependency on the parameter RS$(n)$ in proving lower bound for dynamic streams and present a near-optimal lower bound of $n^{2-O(ε)}$ for $(1+ε)$-approximation in this model. Using a well-known connection between matching size and matrix rank, all our lower bounds also hold for the problem of estimating matrix rank. In particular our results imply a near-optimal $n^{2-O(ε)}$ bit lower bound for $(1+ε)$-approximation of matrix ranks for dense matrices in dynamic streams, answering an open question of Li and Woodruff (STOC 2016).

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arXiv2016

Tight Bounds for Single-Pass Streaming Complexity of the Set Cover Problem

We resolve the space complexity of single-pass streaming algorithms for approximating the classic set cover problem. For finding an $α$-approximate set cover (for any $α= o(\sqrt{n})$) using a single-pass streaming algorithm, we show that $Θ(mn/α)$ space is both sufficient and necessary (up to an $O(\log{n})$ factor); here $m$ denotes number of the sets and $n$ denotes size of the universe. This provides a strong negative answer to the open question posed by Indyk et al. (2015) regarding the possibility of having a single-pass algorithm with a small approximation factor that uses sub-linear space. We further study the problem of estimating the size of a minimum set cover (as opposed to finding the actual sets), and establish that an additional factor of $α$ saving in the space is achievable in this case and that this is the best possible. In other words, we show that $Θ(mn/α^2)$ space is both sufficient and necessary (up to logarithmic factors) for estimating the size of a minimum set cover to within a factor of $α$. Our algorithm in fact works for the more general problem of estimating the optimal value of a covering integer program. On the other hand, our lower bound holds even for set cover instances where the sets are presented in a random order.

cs.DS↗

arXiv2015

Algorithms for Provisioning Queries and Analytics

Provisioning is a technique for avoiding repeated expensive computations in what-if analysis. Given a query, an analyst formulates $k$ hypotheticals, each retaining some of the tuples of a database instance, possibly overlapping, and she wishes to answer the query under scenarios, where a scenario is defined by a subset of the hypotheticals that are "turned on". We say that a query admits compact provisioning if given any database instance and any $k$ hypotheticals, one can create a poly-size (in $k$) sketch that can then be used to answer the query under any of the $2^{k}$ possible scenarios without accessing the original instance. In this paper, we focus on provisioning complex queries that combine relational algebra (the logical component), grouping, and statistics/analytics (the numerical component). We first show that queries that compute quantiles or linear regression (as well as simpler queries that compute count and sum/average of positive values) can be compactly provisioned to provide (multiplicative) approximate answers to an arbitrary precision. In contrast, exact provisioning for each of these statistics requires the sketch size to be exponential in $k$. We then establish that for any complex query whose logical component is a positive relational algebra query, as long as the numerical component can be compactly provisioned, the complex query itself can be compactly provisioned. On the other hand, introducing negation or recursion in the logical component again requires the sketch size to be exponential in $k$. While our positive results use algorithms that do not access the original instance after a scenario is known, we prove our lower bounds even for the case when, knowing the scenario, limited access to the instance is allowed.

cs.DB↗

arXiv2015

Dynamic Sketching for Graph Optimization Problems with Applications to Cut-Preserving Sketches

In this paper, we introduce a new model for sublinear algorithms called \emph{dynamic sketching}. In this model, the underlying data is partitioned into a large \emph{static} part and a small \emph{dynamic} part and the goal is to compute a summary of the static part (i.e, a \emph{sketch}) such that given any \emph{update} for the dynamic part, one can combine it with the sketch to compute a given function. We say that a sketch is \emph{compact} if its size is bounded by a polynomial function of the length of the dynamic data, (essentially) independent of the size of the static part. A graph optimization problem $P$ in this model is defined as follows. The input is a graph $G(V,E)$ and a set $T \subseteq V$ of $k$ terminals; the edges between the terminals are the dynamic part and the other edges in $G$ are the static part. The goal is to summarize the graph $G$ into a compact sketch (of size poly$(k)$) such that given any set $Q$ of edges between the terminals, one can answer the problem $P$ for the graph obtained by inserting all edges in $Q$ to $G$, using only the sketch. We study the fundamental problem of computing a maximum matching and prove tight bounds on the sketch size. In particular, we show that there exists a (compact) dynamic sketch of size $O(k^2)$ for the matching problem and any such sketch has to be of size $Ω(k^2)$. Our sketch for matchings can be further used to derive compact dynamic sketches for other fundamental graph problems involving cuts and connectivities. Interestingly, our sketch for matchings can also be used to give an elementary construction of a \emph{cut-preserving vertex sparsifier} with space $O(kC^2)$ for $k$-terminal graphs; here $C$ is the total capacity of the edges incident on the terminals. Additionally, we give an improved lower bound (in terms of $C$) of $Ω(C/\log{C})$ on size of cut-preserving vertex sparsifiers.

cs.DS↗

arXiv2015

Fast Convergence in the Double Oral Auction

A classical trading experiment consists of a set of unit demand buyers and unit supply sellers with identical items. Each agent's value or opportunity cost for the item is their private information and preferences are quasi-linear. Trade between agents employs a double oral auction (DOA) in which both buyers and sellers call out bids or offers which an auctioneer recognizes. Transactions resulting from accepted bids and offers are recorded. This continues until there are no more acceptable bids or offers. Remarkably, the experiment consistently terminates in a Walrasian price. The main result of this paper is a mechanism in the spirit of the DOA that converges to a Walrasian equilibrium in a polynomial number of steps, thus providing a theoretical basis for the above-described empirical phenomenon. It is well-known that computation of a Walrasian equilibrium for this market corresponds to solving a maximum weight bipartite matching problem. The uncoordinated but rational responses of agents thus solve in a distributed fashion a maximum weight bipartite matching problem that is encoded by their private valuations. We show, furthermore, that every Walrasian equilibrium is reachable by some sequence of responses. This is in contrast to the well known auction algorithms for this problem which only allow one side to make offers and thus essentially choose an equilibrium that maximizes the surplus for the side making offers. Our results extend to the setting where not every agent pair is allowed to trade with each other.

cs.GT↗

arXiv2015

Online Assignment of Heterogeneous Tasks in Crowdsourcing Markets

We investigate the problem of heterogeneous task assignment in crowdsourcing markets from the point of view of the requester, who has a collection of tasks. Workers arrive online one by one, and each declare a set of feasible tasks they can solve, and desired payment for each feasible task. The requester must decide on the fly which task (if any) to assign to the worker, while assigning workers only to feasible tasks. The goal is to maximize the number of assigned tasks with a fixed overall budget. We provide an online algorithm for this problem and prove an upper bound on the competitive ratio of this algorithm against an arbitrary (possibly worst-case) sequence of workers who want small payments relative to the requester's total budget. We further show an almost matching lower bound on the competitive ratio of any algorithm in this setting. Finally, we propose a different algorithm that achieves an improved competitive ratio in the random permutation model, where the order of arrival of the workers is chosen uniformly at random. Apart from these strong theoretical guarantees, we carry out experiments on simulated data which demonstrates the practical applicability of our algorithms.

cs.DS↗

arXiv2015

Tight Bounds for Linear Sketches of Approximate Matchings

We resolve the space complexity of linear sketches for approximating the maximum matching problem in dynamic graph streams where the stream may include both edge insertion and deletion. Specifically, we show that for any $ε> 0$, there exists a one-pass streaming algorithm, which only maintains a linear sketch of size $\tilde{O}(n^{2-3ε})$ bits and recovers an $n^ε$-approximate maximum matching in dynamic graph streams, where $n$ is the number of vertices in the graph. In contrast to the extensively studied insertion-only model, to the best of our knowledge, no non-trivial single-pass streaming algorithms were previously known for approximating the maximum matching problem on general dynamic graph streams. Furthermore, we show that our upper bound is essentially tight. Namely, any linear sketch for approximating the maximum matching to within a factor of $O(n^ε)$ has to be of size $n^{2-3ε-o(1)}$ bits. We establish this lower bound by analyzing the corresponding simultaneous number-in-hand communication model, with a combinatorial construction based on Ruzsa-Szemerédi graphs.

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