12,367 research outputs found

    On the discrepancy of random low degree set systems

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    Motivated by the celebrated Beck-Fiala conjecture, we consider the random setting where there are nn elements and mm sets and each element lies in tt randomly chosen sets. In this setting, Ezra and Lovett showed an O((tlogt)1/2)O((t \log t)^{1/2}) discrepancy bound in the regime when nmn \leq m and an O(1)O(1) bound when nmtn \gg m^t. In this paper, we give a tight O(t)O(\sqrt{t}) bound for the entire range of nn and mm, under a mild assumption that t=Ω(loglogm)2t = \Omega (\log \log m)^2. The result is based on two steps. First, applying the partial coloring method to the case when n=mlogO(1)mn = m \log^{O(1)} m and using the properties of the random set system we show that the overall discrepancy incurred is at most O(t)O(\sqrt{t}). Second, we reduce the general case to that of nmlogO(1)mn \leq m \log^{O(1)}m using LP duality and a careful counting argument

    k-Trails: Recognition, Complexity, and Approximations

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    The notion of degree-constrained spanning hierarchies, also called k-trails, was recently introduced in the context of network routing problems. They describe graphs that are homomorphic images of connected graphs of degree at most k. First results highlight several interesting advantages of k-trails compared to previous routing approaches. However, so far, only little is known regarding computational aspects of k-trails. In this work we aim to fill this gap by presenting how k-trails can be analyzed using techniques from algorithmic matroid theory. Exploiting this connection, we resolve several open questions about k-trails. In particular, we show that one can recognize efficiently whether a graph is a k-trail. Furthermore, we show that deciding whether a graph contains a k-trail is NP-complete; however, every graph that contains a k-trail is a (k+1)-trail. Moreover, further leveraging the connection to matroids, we consider the problem of finding a minimum weight k-trail contained in a graph G. We show that one can efficiently find a (2k-1)-trail contained in G whose weight is no more than the cheapest k-trail contained in G, even when allowing negative weights. The above results settle several open questions raised by Molnar, Newman, and Sebo

    Improved Approximation Algorithms for Stochastic Matching

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    In this paper we consider the Stochastic Matching problem, which is motivated by applications in kidney exchange and online dating. We are given an undirected graph in which every edge is assigned a probability of existence and a positive profit, and each node is assigned a positive integer called timeout. We know whether an edge exists or not only after probing it. On this random graph we are executing a process, which one-by-one probes the edges and gradually constructs a matching. The process is constrained in two ways: once an edge is taken it cannot be removed from the matching, and the timeout of node vv upper-bounds the number of edges incident to vv that can be probed. The goal is to maximize the expected profit of the constructed matching. For this problem Bansal et al. (Algorithmica 2012) provided a 33-approximation algorithm for bipartite graphs, and a 44-approximation for general graphs. In this work we improve the approximation factors to 2.8452.845 and 3.7093.709, respectively. We also consider an online version of the bipartite case, where one side of the partition arrives node by node, and each time a node bb arrives we have to decide which edges incident to bb we want to probe, and in which order. Here we present a 4.074.07-approximation, improving on the 7.927.92-approximation of Bansal et al. The main technical ingredient in our result is a novel way of probing edges according to a random but non-uniform permutation. Patching this method with an algorithm that works best for large probability edges (plus some additional ideas) leads to our improved approximation factors

    The Complexity of Scheduling for p-norms of Flow and Stretch

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    We consider computing optimal k-norm preemptive schedules of jobs that arrive over time. In particular, we show that computing the optimal k-norm of flow schedule, is strongly NP-hard for k in (0, 1) and integers k in (1, infinity). Further we show that computing the optimal k-norm of stretch schedule, is strongly NP-hard for k in (0, 1) and integers k in (1, infinity).Comment: Conference version accepted to IPCO 201

    Matroid Online Bipartite Matching and Vertex Cover

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    The Adwords and Online Bipartite Matching problems have enjoyed a renewed attention over the past decade due to their connection to Internet advertising. Our community has contributed, among other things, new models (notably stochastic) and extensions to the classical formulations to address the issues that arise from practical needs. In this paper, we propose a new generalization based on matroids and show that many of the previous results extend to this more general setting. Because of the rich structures and expressive power of matroids, our new setting is potentially of interest both in theory and in practice. In the classical version of the problem, the offline side of a bipartite graph is known initially while vertices from the online side arrive one at a time along with their incident edges. The objective is to maintain a decent approximate matching from which no edge can be removed. Our generalization, called Matroid Online Bipartite Matching, additionally requires that the set of matched offline vertices be independent in a given matroid. In particular, the case of partition matroids corresponds to the natural scenario where each advertiser manages multiple ads with a fixed total budget. Our algorithms attain the same performance as the classical version of the problems considered, which are often provably the best possible. We present 11/e1-1/e-competitive algorithms for Matroid Online Bipartite Matching under the small bid assumption, as well as a 11/e1-1/e-competitive algorithm for Matroid Online Bipartite Matching in the random arrival model. A key technical ingredient of our results is a carefully designed primal-dual waterfilling procedure that accommodates for matroid constraints. This is inspired by the extension of our recent charging scheme for Online Bipartite Vertex Cover.Comment: 19 pages, to appear in EC'1

    New developments in iterated rounding

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    Iterated rounding is a relatively recent technique in algorithm design, that despite its simplicity has led to several remarkable new results and also simpler proofs of many previous results. We will briefly survey some applications of the method, including some recent developments and giving a high level overview of the ideas

    Local Guarantees in Graph Cuts and Clustering

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    Correlation Clustering is an elegant model that captures fundamental graph cut problems such as Min sts-t Cut, Multiway Cut, and Multicut, extensively studied in combinatorial optimization. Here, we are given a graph with edges labeled ++ or - and the goal is to produce a clustering that agrees with the labels as much as possible: ++ edges within clusters and - edges across clusters. The classical approach towards Correlation Clustering (and other graph cut problems) is to optimize a global objective. We depart from this and study local objectives: minimizing the maximum number of disagreements for edges incident on a single node, and the analogous max min agreements objective. This naturally gives rise to a family of basic min-max graph cut problems. A prototypical representative is Min Max sts-t Cut: find an sts-t cut minimizing the largest number of cut edges incident on any node. We present the following results: (1)(1) an O(n)O(\sqrt{n})-approximation for the problem of minimizing the maximum total weight of disagreement edges incident on any node (thus providing the first known approximation for the above family of min-max graph cut problems), (2)(2) a remarkably simple 77-approximation for minimizing local disagreements in complete graphs (improving upon the previous best known approximation of 4848), and (3)(3) a 1/(2+ε)1/(2+\varepsilon)-approximation for maximizing the minimum total weight of agreement edges incident on any node, hence improving upon the 1/(4+ε)1/(4+\varepsilon)-approximation that follows from the study of approximate pure Nash equilibria in cut and party affiliation games

    On the number of matroids

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    We consider the problem of determining mnm_n, the number of matroids on nn elements. The best known lower bound on mnm_n is due to Knuth (1974) who showed that loglogmn\log \log m_n is at least n3/2logn1n-3/2\log n-1. On the other hand, Piff (1973) showed that loglogmnnlogn+loglogn+O(1)\log\log m_n\leq n-\log n+\log\log n +O(1), and it has been conjectured since that the right answer is perhaps closer to Knuth's bound. We show that this is indeed the case, and prove an upper bound on loglogmn\log\log m_n that is within an additive 1+o(1)1+o(1) term of Knuth's lower bound. Our proof is based on using some structural properties of non-bases in a matroid together with some properties of independent sets in the Johnson graph to give a compressed representation of matroids.Comment: Final version, 17 page

    An entropy argument for counting matroids

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    We show how a direct application of Shearers' Lemma gives an almost optimum bound on the number of matroids on nn elements.Comment: Short note, 4 page

    Sequential item pricing for unlimited supply

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    We investigate the extent to which price updates can increase the revenue of a seller with little prior information on demand. We study prior-free revenue maximization for a seller with unlimited supply of n item types facing m myopic buyers present for k < log n days. For the static (k = 1) case, Balcan et al. [2] show that one random item price (the same on each item) yields revenue within a \Theta(log m + log n) factor of optimum and this factor is tight. We define the hereditary maximizers property of buyer valuations (satisfied by any multi-unit or gross substitutes valuation) that is sufficient for a significant improvement of the approximation factor in the dynamic (k > 1) setting. Our main result is a non-increasing, randomized, schedule of k equal item prices with expected revenue within a O((log m + log n) / k) factor of optimum for private valuations with hereditary maximizers. This factor is almost tight: we show that any pricing scheme over k days has a revenue approximation factor of at least (log m + log n) / (3k). We obtain analogous matching lower and upper bounds of \Theta((log n) / k) if all valuations have the same maximum. We expect our upper bound technique to be of broader interest; for example, it can significantly improve the result of Akhlaghpour et al. [1]. We also initiate the study of revenue maximization given allocative externalities (i.e. influences) between buyers with combinatorial valuations. We provide a rather general model of positive influence of others' ownership of items on a buyer's valuation. For affine, submodular externalities and valuations with hereditary maximizers we present an influence-and-exploit (Hartline et al. [13]) marketing strategy based on our algorithm for private valuations. This strategy preserves our approximation factor, despite an affine increase (due to externalities) in the optimum revenue.Comment: 18 pages, 1 figur
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