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125
First Steps in Tropical Geometry
 CONTEMPORARY MATHEMATICS
"... Tropical algebraic geometry is the geometry of the tropical semiring (R, min, +). Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete descr ..."
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Cited by 72 (10 self)
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Tropical algebraic geometry is the geometry of the tropical semiring (R, min, +). Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus’ Theorem.
Barrier coverage with wireless sensors
 In ACM MobiCom
, 2005
"... When a sensor network is deployed to detect objects penetrating a protected region, it is not necessary to have every point in the deployment region covered by a sensor. It is enough if the penetrating objects are detected at some point in their trajectory. If a sensor network guarantees that every ..."
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Cited by 67 (8 self)
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When a sensor network is deployed to detect objects penetrating a protected region, it is not necessary to have every point in the deployment region covered by a sensor. It is enough if the penetrating objects are detected at some point in their trajectory. If a sensor network guarantees that every penetrating object will be detected by at least £ distinct sensors before it crosses the barrier of wireless sensors, we say the network provides £barrier coverage. In this paper, we develop theoretical foundations for £barrier coverage. We propose efficient algorithms using which one can quickly determine, after deploying the sensors, whether the deployment region is £barrier covered. Next, we establish the optimal deployment pattern to achieve £barrier coverage when deploying sensors deterministically. Finally, we consider barrier coverage with high probability when sensors are deployed randomly. The major challenge, when dealing with probabilistic barrier coverage, is to derive critical conditions using which one can compute the minimum number of sensors needed to ensure barrier coverage with high probability. Deriving critical conditions for £barrier coverage is, however, still an open problem. We derive critical conditions for a weaker notion of barrier coverage, called weak £barrier coverage.
On the Capacity of Information Networks
"... An outer bound on the rate region of noisefree information networks is given. This outer bound combines properties of entropy with a strong information inequality derived from the structure of the network. This blend of information theoretic and graph theoretic arguments generates many interestin ..."
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Cited by 58 (7 self)
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An outer bound on the rate region of noisefree information networks is given. This outer bound combines properties of entropy with a strong information inequality derived from the structure of the network. This blend of information theoretic and graph theoretic arguments generates many interesting results. For example, the capacity of directed cycles is characterized. Also, a gap between the sparsity of an undirected graph and its capacity is shown. Extending this result, it is shown that multicommodity flow solutions achieve the capacity in an infinite class of undirected graphs, thereby making progress on a conjecture of Li and Li. This result is in sharp contrast to the situation with directed graphs, where a family of graphs are presented in which the gap between the capacity and the rate achievable using multicommodity flows is linear in the size of the graph.
Knapsack Auctions
 Proceedings of the Seventeenth Annual ACMSIAM Symposium on Discrete Algorithms (SODA
, 2006
"... We consider a game theoretic knapsack problem that has application to auctions for selling advertisements on Internet search engines. Consider n agents each wishing to place an object in the knapsack. Each agent has a private valuation for having their object in the knapsack and each object has a pu ..."
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Cited by 56 (9 self)
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We consider a game theoretic knapsack problem that has application to auctions for selling advertisements on Internet search engines. Consider n agents each wishing to place an object in the knapsack. Each agent has a private valuation for having their object in the knapsack and each object has a publicly known size. For this setting, we consider the design of auctions in which agents have an incentive to truthfully reveal their private valuations. Following the framework of Goldberg et al. [10], we look to design an auction that obtains a constant fraction of the profit obtainable by a natural optimal pricing algorithm that knows the agents ’ valuations and object sizes. We give an auction that obtains a constant factor approximation in the nontrivial special case where the knapsack has unlimited capacity. We then reduce the limited capacity version of the problem to the unlimited capacity version via an approximately efficient auction (i.e., one that maximizes the social welfare). This reduction follows from generalizable principles. 1
Designing localized algorithms for barrier coverage
 Proc. ACM MobiCom
, 2007
"... Global barrier coverage that requires much fewer sensors than full coverage, is known to be an appropriate model of coverage for movement detection applications such as intrusion detection. However, it has been proved that given a sensor deployment, sensors can not locally determine whether the depl ..."
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Cited by 35 (3 self)
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Global barrier coverage that requires much fewer sensors than full coverage, is known to be an appropriate model of coverage for movement detection applications such as intrusion detection. However, it has been proved that given a sensor deployment, sensors can not locally determine whether the deployment provides global barrier coverage, making it impossible to develop localized algorithms, thus limiting its use in practice. In this paper, we introduce the concept of local barrier coverage to address this limitation. Motivated by the observation that movements are likely to follow a shorter path in crossing a belt region, local barrier coverage guarantees the detection of all movements whose trajectory is confined to a slice of the belt region of deployment. We prove that it is possible for individual sensors to locally determine the existence of local barrier coverage, even when the region of deployment is arbitrarily curved. Although local barrier coverage does not always guarantee global barrier coverage, we show that for thin belt regions, local barrier coverage almost always provides global barrier coverage. To demonstrate that local barrier coverage can be used to design localized algorithms, we develop a novel sleepwakeup algorithm for maximizing the network lifetime, called Localized Barrier Coverage Protocol (LBCP). We show that LBCP provides close to optimal enhancement in network lifetime, while providing global barrier coverage most of the time. It outperforms an existing algorithm called Randomized Independent Sleeping (RIS) by up to 6 times.
Computing common intervals of K permutations, with applications to modular decomposition of graphs
, 2008
"... We introduce a new approach to compute the common intervals of K permutations based on a very simple and general notion of generators of common intervals. This formalism leads to simple and efficient algorithms to compute the set of all common intervals of K permutations, that can contain a quadrat ..."
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Cited by 33 (13 self)
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We introduce a new approach to compute the common intervals of K permutations based on a very simple and general notion of generators of common intervals. This formalism leads to simple and efficient algorithms to compute the set of all common intervals of K permutations, that can contain a quadratic number of intervals, as well as a linear space basis of this set of common intervals. Finally, we show how our results on permutations can be used for computing the modular decomposition of graphs.
EdgeCut Bounds On Network Coding Rates
 Journal of Network and Systems Management
, 2006
"... Abstract — Two bounds on network coding rates are reviewed that generalize edgecut bounds on routing rates. The simpler bound is a bidirected cutset bound which generalizes and improves upon a flow cutset bound that is standard in networking. It follows that routing is rateoptimal if routing ach ..."
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Cited by 23 (2 self)
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Abstract — Two bounds on network coding rates are reviewed that generalize edgecut bounds on routing rates. The simpler bound is a bidirected cutset bound which generalizes and improves upon a flow cutset bound that is standard in networking. It follows that routing is rateoptimal if routing achieves the standard flow cutset bound. The second bound improves on the cutset bound, and it involves progressively removing edges from a network graph and checking whether certain strengthened dseparation conditions are satisfied. I.
Strong barrier coverage of wireless sensor networks
 in Proc. of The ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc
, 2008
"... Constructing sensor barriers to detect intruders crossing a randomlydeployed sensor network is an important problem. Early results have shown how to construct sensor barriers to detect intruders moving along restricted crossing paths in rectangular areas. We present a complete solution to this prob ..."
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Cited by 18 (7 self)
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Constructing sensor barriers to detect intruders crossing a randomlydeployed sensor network is an important problem. Early results have shown how to construct sensor barriers to detect intruders moving along restricted crossing paths in rectangular areas. We present a complete solution to this problem for sensors that are distributed according to a Poisson point process. In particular, we present an efficient distributed algorithm to construct sensor barriers on long strip areas of irregular shape without any constraint on crossing paths. Our approach is as follows: We first show that in a rectangular area of width w and length ℓ with w = Ω(log ℓ), if the sensor density reaches a certain value, then there exist, with high probability, multiple disjoint sensor barriers across the entire length of the area such that intruders cannot cross the area undetected. On the other hand, if w = o(log ℓ), then with high probability there is a crossing path not covered by any sensor regardless of the sensor density. We then devise, based on this result, an efficient distributed algorithm to construct multiple disjoint barriers in a large sensor network to cover a long boundary area of an irregular shape. Our algorithm approximates the area by dividing it into horizontal rectangular segments interleaved by vertical thin strips. Each segment and vertical strip independently computes the barriers in its own area. Constructing “horizontal ” barriers in each segment connected by“vertical ” barriers in neighboring vertical strips, we achieve continuous barrier coverage for the whole region. Our approach significantly reduces delay, communication overhead, and computation costs compared to centralized approaches. Finally, we implement our algorithm and carry out a number of experiments to demonstrate the effectiveness of constructing barrier coverage.
From convex optimization to randomized mechanisms: Toward optimal combinatorial auctions
 In Proceedings of the 43rd annual ACM Symposium on Theory of Computing (STOC
, 2011
"... We design an expected polynomialtime, truthfulinexpectation, (1 − 1/e)approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are matroid rank sums (MRS), which encompass mostconcreteexamplesofsubmodular ..."
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Cited by 17 (4 self)
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We design an expected polynomialtime, truthfulinexpectation, (1 − 1/e)approximation mechanism for welfare maximization in a fundamental class of combinatorial auctions. Our results apply to bidders with valuations that are matroid rank sums (MRS), which encompass mostconcreteexamplesofsubmodularfunctionsstudiedinthiscontext,includingcoveragefunctions, matroid weightedrank functions, and convex combinations thereof. Our approximation factor is the best possible, even for known and explicitly given coverage valuations, assuming P ̸ = NP. Ours is the first truthfulinexpectation and polynomialtime mechanism to achieve a constantfactor approximation for an NPhard welfare maximization problem in combinatorial auctions with heterogeneous goods and restricted valuations. Our mechanism is an instantiation of a new framework for designing approximation mechanisms based on randomized rounding algorithms. A typical such algorithm first optimizes over a fractional relaxation of the original problem, and then randomly rounds the fractional solution to an integral one. With rare exceptions, such algorithms cannot be converted into truthful mechanisms. The highlevel idea of our mechanism design framework is to optimize directly