arXiv subjects

Hossein Esfandiari

Publications and source records attributed to Hossein Esfandiari.

41 records · Page 3Linked to original sources

arXiv2015

Prophet Secretary

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 0.5. On the other hand, for the basic secretary problem, the optimal strategy achieves the tight competitive ratio of 1/e. In this paper, we introduce Prophet Secretary, a natural combination of the prophet inequality and the secretary problems. An example motivation for our problem is as follows. Consider a seller that has an item to sell on the market to a set of arriving customers. The seller knows the types of customers that may be interested in the item and he has a price distribution for each type: the price offered by a customer of a type is anticipated to be drawn from the corresponding distribution. However, the customers arrive in a random order. Upon the arrival of a customer, the seller makes an irrevocable decision whether to sell the item at the offered price. We address the question of finding a strategy for selling the item at a high price. We show that by using a uniform threshold one cannot break the 0.5 barrier. However, we show that i) using n distinct non-adaptive thresholds one can obtain a competitive ratio that goes to (1-1/e) as n grows; and ii) no online algorithm can achieve a competitive ratio better than 0.75. Our results improve the (asymptotic) approximation guarantee of single-item sequential posted pricing mechanisms from 0.5 to (1-1/e) when the order of agents (customers) is chosen randomly.

cs.GT↗

arXiv2015

Applications of Uniform Sampling: Densest Subgraph and Beyond

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., for dynamic graph streams. They present a $(0.5-ε)$-approximation algorithm using $\tilde{O}(n)$ space, where factors of $ε$ and $\log(n)$ are suppressed in the $\tilde{O}$ notation. However, the update time of this algorithm is large. To remedy this, they also provide a $(0.25-ε)$-approximation algorithm using $\tilde{O}(n)$ space with update time $\tilde{O}(1)$. In this paper we improve the algorithms by Bhattacharya et al. by providing a $(1-ε)$-approximation algorithm using $\tilde{O}(n)$ space. Our algorithm is conceptually simple - it samples $\tilde{O}(n)$ edges uniformly at random, and finds the densest subgraph on the sampled graph. We also show how to perform this sampling with update time $\tilde{O}(1)$. In addition to this, we show that given oracle access to the edge set, we can implement our algorithm in time $\tilde{O}(n)$ on a graph in the standard RAM model. To the best of our knowledge this is the fastest $(0.5-ε)$-approximation algorithm for the densest subgraph problem in the RAM model given such oracle access. Further, we extend our results to a general class of graph optimization problems that we call heavy subgraph problems. This class contains many interesting problems such as densest subgraph, directed densest subgraph, densest bipartite subgraph, $d$-cut and $d$-heavy connected component. Our result, by characterizing heavy subgraph problems, partially addresses open problem 13 at the IITK Workshop on Algorithms for Data Streams in 2006 regarding the effects of subsampling in this context.

cs.DS↗

arXiv2015

Low-Risk Mechanisms for the Kidney Exchange Game

In this paper we consider the pairwise kidney exchange game. This game naturally appears in situations that some service providers benefit from pairwise allocations on a network, such as the kidney exchanges between hospitals. Ashlagi et al. present a $2$-approximation randomized truthful mechanism for this problem. This is the best known result in this setting with multiple players. However, we note that the variance of the utility of an agent in this mechanism may be as large as $Ω(n^2)$, which is not desirable in a real application. In this paper we resolve this issue by providing a $2$-approximation randomized truthful mechanism in which the variance of the utility of each agent is at most $2+ε$. Interestingly, we could apply our technique to design a deterministic mechanism such that, if an agent deviates from the mechanism, she does not gain more than $2\lceil \log_2 m\rceil$. We call such a mechanism an almost truthful mechanism. Indeed, in a practical scenario, an almost truthful mechanism is likely to imply a truthful mechanism. We believe that our approach can be used to design low risk or almost truthful mechanisms for other problems.

cs.GT↗

arXiv2015

Kernelization via Sampling with Applications to Dynamic Graph Streams

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. Our algorithm has $\tilde{O}(1)$ update time. Second, there exists an $\tilde{O}(n^2/α^3)$ space algorithm that returns an $α$-approximation for matchings of arbitrary size. (Assadi et al. (2015) showed that this was optimal and independently and concurrently established the same upper bound.) We generalize both results for weighted matching. Third, there exists an $\tilde{O}(n^{4/5})$ space algorithm that returns a constant approximation in graphs with bounded arboricity. -- Vertex Cover and Hitting Set: There exists an $\tilde{O}(k^d)$ space algorithm that solves the minimum hitting set problem where $d$ is the cardinality of the input sets and $k$ is an upper bound on the size of the minimum hitting set. We prove this is optimal up to logarithmic factors. Our algorithm has $\tilde{O}(1)$ update time. The case $d=2$ corresponds to minimum vertex cover. Finally, we consider a larger family of parameterized problems (including $b$-matching, disjoint paths, vertex coloring among others) for which our subgraph sampling primitive yields fast, small-space dynamic graph stream algorithms. We then show lower bounds for natural problems outside this family.

cs.DS↗

arXiv2013

On the Erdős-Gyárfás conjecture in claw-free graphs

The Erdős-Gyárfás conjecture states that every graph with minimum degree at least three has a cycle whose length is a power of 2. Since this conjecture has proven to be far from reach, Hobbs asked if the Erdős-Gyárfás conjecture holds in claw-free graphs. In this paper, we obtain some results on this question, in particular for cubic claw-free graphs.

math.CO↗