Efficient computation of the Weighted Clustering Coefficient

The clustering coefficient of an unweighted network has been extensively used to quantify how tightly connected is the neighbor around a node and it has been widely adopted for assessing the quality of nodes in a social network. The computation of the clustering coefficient is challenging since it requires to count the number of triangles in the graph. Several recent works proposed efficient sampling, streaming and MapReduce algorithms that allow to overcome this computational bottleneck. As a matter of fact, the intensity of the interaction between nodes, that is usually represented with weights on the edges of the graph, is also an important measure of the statistical cohesiveness of a network. Recently various notions of weighted clustering coefficient have been proposed but all those techniques are hard to implement on large-scale graphs. In this work we show how standard sampling techniques can be used to obtain efficient estimators for the most commonly used measures of weighted clustering coefficient. Furthermore we also propose a novel graph-theoretic notion of clustering coefficient in weighted networks. © 2016, Copyright © Taylor & Francis Group, LL

Approximately Efficient Double Auctions with Strong Budget Balance

Mechanism design for one-sided markets is an area of extensive research in economics and, since more than a decade, in computer science as well. Two-sided markets, on the other hand, have not received the same attention despite the numerous applications to web advertisement, stock exchange, and frequency spectrum allocation. This work studies double auctions, in which unit-demand buyers and unit-supply sellers act strategically. An ideal goal in double auction design is to maximize the social welfare of buyers and sellers with individually rational (IR), incentive compatible (IC) and strongly budget-balanced (SBB) mechanisms. The first two properties are standard. SBB requires that the payments charged to the buyers are entirely handed to the sellers. This property is crucial in all the contexts that do not allow the auctioneer retaining a share of buyers' payments or subsidizing the market. Unfortunately, this goal is known to be unachievable even for the special case of bilateral trade, where there is only one buyer and one seller. Therefore, in subsequent papers, meaningful trade-offs between these requirements have been investigated. Our main contribution is the first IR, IC and SBB mechanism that provides an O(1)-approximation to the optimal social welfare. This result holds for any number of buyers and sellers with arbitrary, independent distributions. Moreover, our result continues to hold when there is an additional matroid constraint on the sets of buyers who may get allocated an item. To prove our main result, we devise an extension of sequential posted price mechanisms to two-sided markets. In addition to this, we improve the best-known approximation bounds for the bilateral trade problem

Modeling and Evaluation of Multisource Streaming Strategies in P2P VoD Systems

In recent years, multimedia content distribution has largely been moved to the Internet, inducing broadcasters, operators and service providers to upgrade with large expenses their infrastructures. In this context, streaming solutions that rely on user devices such as set-top boxes (STBs) to offload dedicated streaming servers are particularly appropriate. In these systems, contents are usually replicated and scattered over the network established by STBs placed at users' home, and the video-on-demand (VoD) service is provisioned through streaming sessions established among neighboring STBs following a Peer-to-Peer fashion. Up to now the majority of research works have focused on the design and optimization of content replicas mechanisms to minimize server costs. The optimization of replicas mechanisms has been typically performed either considering very crude system performance indicators or analyzing asymptotic behavior. In this work, instead, we propose an analytical model that complements previous works providing fairly accurate predictions of system performance (i.e., blocking probability). Our model turns out to be a highly scalable, flexible, and extensible tool that may be helpful both for designers and developers to efficiently predict the effect of system design choices in large scale STB-VoD system

Zeros of Unilateral Quaternionic Polynomials

The purpose of this paper is to show how the problem of finding the zeros of unilateral n-order quaternionic polynomials can be solved by determining the eigen-vectors of the corresponding companion matrix. This approach, probably superfluous in the case of quadratic equations for which a closed formula can be given, becomes truly useful for (unilateral) n-order polynomials. To understand the strehgth of this method, we compare it with the Niven algorithm and show where this (full) matrix approach improves previous methods based on the use of the Niven algorithm. For the convenience of the readers, we explicitly solve some examples of second and third order unilateral quaternionic polynomials. The leading idea of the practical solution method proposed in this work can be summarized in following three steps: translating the quaternionic polynomial in the eigenvalue problem for its companion matrix, finding its eigenvectors, and, finally, giving the quaternionic solution of the unilateral polynomial in terms of the components of such eigenvectors. A brief discussion on bilateral quaternionic quadratic equations is also presented.Comment: 14 page

Pandora's Box Problem with Order Constraints

The Pandora's Box Problem, originally formalized by Weitzman in 1979, models selection from set of random, alternative options, when evaluation is costly. This includes, for example, the problem of hiring a skilled worker, where only one hire can be made, but the evaluation of each candidate is an expensive procedure. Weitzman showed that the Pandora's Box Problem admits an elegant, simple solution, where the options are considered in decreasing order of reservation value,i.e., the value that reduces to zero the expected marginal gain for opening the box. We study for the first time this problem when order - or precedence - constraints are imposed between the boxes. We show that, despite the difficulty of defining reservation values for the boxes which take into account both in-depth and in-breath exploration of the various options, greedy optimal strategies exist and can be efficiently computed for tree-like order constraints. We also prove that finding approximately optimal adaptive search strategies is NP-hard when certain matroid constraints are used to further restrict the set of boxes which may be opened, or when the order constraints are given as reachability constraints on a DAG. We complement the above result by giving approximate adaptive search strategies based on a connection between optimal adaptive strategies and non-adaptive strategies with bounded adaptivity gap for a carefully relaxed version of the problem

Robust hierarchical k-center clustering

One of the most popular and widely used methods for data clustering is hierarchical clustering. This clustering technique has proved useful to reveal interesting structure in the data in several applications ranging from computational biology to computer vision. Robustness is an important feature of a clustering technique if we require the clustering to be stable against small perturbations in the input data. In most applications, getting a clustering output that is robust against adversarial outliers or stochastic noise is a necessary condition for the applicability and effectiveness of the clustering technique. This is even more critical in hierarchical clustering where a small change at the bottom of the hierarchy may propagate all the way through to the top. Despite all the previous work [2, 3, 6, 8], our theoretical understanding of robust hierarchical clustering is still limited and several hierarchical clustering algorithms are not known to satisfy such robustness properties. In this paper, we study the limits of robust hierarchical k-center clustering by introducing the concept of universal hierarchical clustering and provide (almost) tight lower and upper bounds for the robust hierarchical k-center clustering problem with outliers and variants of the stochastic clustering problem. Most importantly we present a constant-factor approximation for optimal hierarchical k-center with at most z outliers using a universal set of at most O(z2) set of outliers and show that this result is tight. Moreover we show the necessity of using a universal set of outliers in order to compute an approximately optimal hierarchical k-center with a diffierent set of outliers for each k

Fully decentralized computation of aggregates over data streams

In several emerging applications, data is collected in massive streams at several distributed points of observation. A basic and challenging task is to allow every node to monitor a neighbourhood of interest by issuing continuous aggregate queries on the streams observed in its vicinity. This class of algorithms is fully decentralized and diffusive in nature: collecting all data at few central nodes of the network is unfeasible in networks of low capability devices or in the presence of massive data sets. The main difficulty in designing diffusive algorithms is to cope with duplicate detections. These arise both from the observation of the same event at several nodes of the network and/or receipt of the same aggregated information along multiple paths of diffusion. In this paper, we consider fully decentralized algorithms that answer locally continuous aggregate queries on the number of distinct events, total number of events and the second frequency moment in the scenario outlined above. The proposed algorithms use in the worst case or on realistic distributions sublinear space at every node. We also propose strategies that minimize the communication needed to update the aggregates when new events are observed. We experimentally evaluate for the efficiency and accuracy of our algorithms on realistic simulated scenarios

A mazing 2+ε approximation for unsplittable flow on a path

We study the problem of unsplittable flow on a path (UFP), which arises naturally in many applications such as bandwidth allocation, job scheduling, and caching. Here we are given a path with nonnegative edge capacities and a set of tasks, which are characterized by a subpath, a demand, and a profit. The goal is to find the most profitable subset of tasks whose total demand does not violate the edge capacities. Not surprisingly, this problem has received a lot of attention in the research community. If the demand of each task is at most a small-enough fraction δ of the capacity along its subpath (δ-small tasks), then it has been known for a long time [Chekuri et al., ICALP 2003] how to compute a solution of value arbitrarily close to the optimum via LP rounding. However, much remains unknown for the complementary case, that is, when the demand of each task is at least some fraction δ > 0 of the smallest capacity of its subpath (δ-large tasks). For this setting, a constant factor approximation is known, improving on an earlier logarithmic approximation [Bonsma et al., FOCS 2011]. In this article, we present a polynomial-time approximation scheme (PTAS) for δ-large tasks, for any constant δ > 0. Key to this result is a complex geometrically inspired dynamic program. Each task is represented as a segment underneath the capacity curve, and we identify a proper maze-like structure so that each corridor of the maze is crossed by only O(1) tasks in the optimal solution. The maze has a tree topology, which guides our dynamic program. Our result implies a 2 + ε approximation for UFP, for any constant ε > 0, improving on the previously best 7 + ε approximation by Bonsma et al. We remark that our improved approximation algorithm matches the best known approximation ratio for the considerably easier special case of uniform edge capacities

Revenue maximizing envy-free fixed-price auctions with budgets

Traditional incentive-compatible auctions [6,16] for selling multiple goods to unconstrained and budgeted bidders can discriminate between bidders by selling identical goods at different prices. For this reason, Feldman et al. [7] dropped incentive compatibility and turned the attention to revenue maximizing envy-free item-pricing allocations for budgeted bidders. Envy-free allocations were suggested by classical papers [9,15]. The key property of such allocations is that no one envies the allocation and the price charged to anyone else. In this paper we consider this classical notion of envy-freeness and study fixed-price mechanisms which use nondiscriminatory uniform prices for all goods. Feldman et al. [7] gave an item-pricing mechanism that obtains 1/2 of the revenue obtained from any envy-free fixed-price mechanism for identical goods. We improve over this result by presenting an FPTAS for the problem that returns an (1 − ε)-approximation of the revenue obtained by any envy-free fixed-price mechanism for any ε > 0 and runs in polynomial time in the number of bidders n and 1/ ε even for exponential supply of goods m. Next, we consider the case of budgeted bidders with matching-type preferences on the set of goods, i.e., the valuation of each bidder for each item is either v i or 0. In this more general case, we prove that it is impossible to approximate the optimum revenue within O( min (n,m)1/2 − ε ) for any ε > 0 unless P = NP. On the positive side, we are able to extend the FPTAS for identical goods to budgeted bidders in the case of constant number of different types of goods. Our FPTAS gives also a constant approximation with respect to the general envy-free auction