For two matroids ( M _1 ) and ( M _2 ) defined on the same ground set E, the online matroid intersection problem is to design an algorithm that constructs a large common independent set in an online fashion. The algorithm is presented with the ground set elements one-by-one in a uniformly random order. At each step, the algorithm must irrevocably decide whether to pick the element, while always maintaining a common independent set. While the natural greedy algorithm—pick an element whenever possible—is half competitive, nothing better was previously known; even for the special case of online bipartite matching in the edge arrival model. We present the first randomized online algorithm that has a ( 1 2 + ) competitive ratio in expectation, where ( >0 ) is a constant. The expectation is over the random order and the coin tosses of the algorithm. As a corollary, we also obtain the first linear time algorithm that beats half competitiveness for offline matroid intersection.
Goel and Mehta @cite_25 introduced the random vertex-arrival model. In this model, the adversary may choose the worst instance of a graph, but the online vertices arrive in a random order. The greedy algorithm is already @math -competitive for this problem, as the analysis reduces to @cite_7 . Later works @cite_26 @cite_28 showed that the ranking algorithm has a competitive ratio of at least @math , beating the bounds for adversarial vertex arrival model. There is still a gap between known upper and lower bounds, and closing this gap remains an open problem.
@cite_28 @cite_26 @cite_25 @cite_7
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In a seminal paper, Karp, Vazirani, and Vazirani show that a simple ranking algorithm achieves a competitive ratio of 1-1 e for the online bipartite matching problem in the standard adversarial model, where the ratio of 1-1 e is also shown to be optimal. Their result also implies that in the random arrivals model defined by Goel and Mehta, where the online nodes arrive in a random order, a simple greedy algorithm achieves a competitive ratio of 1-1 e. In this paper, we study the ranking algorithm in the random arrivals model, and show that it has a competitive ratio of at least 0.696, beating the 1-1 e ≈ 0.632 barrier in the adversarial model. Our result also extends to the i.i.d. distribution model of , removing the assumption that the distribution is known. Our analysis has two main steps. First, we exploit certain dominance and monotonicity properties of the ranking algorithm to derive a family of factor-revealing linear programs (LPs). In particular, by symmetry of the ranking algorithm in the random arrivals model, we have the monotonicity property on both sides of the bipartite graph, giving good "strength" to the LPs. Second, to obtain a good lower bound on the optimal values of all these LPs and hence on the competitive ratio of the algorithm, we introduce the technique of strongly factor-revealing LPs. In particular, we derive a family of modified LPs with similar strength such that the optimal value of any single one of these new LPs is a lower bound on the competitive ratio of the algorithm. This enables us to leverage the power of computer LP solvers to solve for large instances of the new LPs to establish bounds that would otherwise be difficult to attain by human analysis. We introduce a fully online model of maximum cardinality matching in which all vertices arrive online. On the arrival of a vertex, its incident edges to previously-arrived vertices are revealed. Each vertex has a deadline that is after all its neighbors’ arrivals. If a vertex remains unmatched until its deadline, the algorithm must then irrevocably either match it to an unmatched neighbor, or leave it unmatched. The model generalizes the existing one-sided online model and is motivated by applications including ride-sharing platforms, real-estate agency, etc. We show that the Ranking algorithm by (STOC 1990) is 0.5211-competitive in our fully online model for general graphs. Our analysis brings a novel charging mechanic into the randomized primal dual technique by (SODA 2013), allowing a vertex other than the two endpoints of a matched edge to share the gain. To our knowledge, this is the first analysis of Ranking that beats 0.5 on general graphs in an online matching problem, a first step towards solving the open problem by (STOC 1990) about the optimality of Ranking on general graphs. If the graph is bipartite, we show that the competitive ratio of Ranking is between 0.5541 and 0.5671. Finally, we prove that the fully online model is strictly harder than the previous model as no online algorithm can be 0.6317 In this paper we study the classic online matching problem, introduced in the seminal work of Karp, Vazirani and Vazirani (STOC 1990), in regular graphs. For such graphs, an optimal deterministic algorithm as well as efficient algorithms under stochastic input assumptions were known. In this work, we present a novel randomized algorithm with competitive ratio tending to one on this family of graphs, under adversarial arrival order. Our main contribution is a novel algorithm which achieves competitive ratio [EQUATION] in expectation on d-regular graphs. In contrast, we show that all previously-studied online algorithms have competitive ratio strictly bounded away from one. Moreover, we show the convergence rate of our algorithm's competitive ratio to one is nearly tight, as no algorithm achieves competitive ratio better than [EQUATION]. Finally, we show that our algorithm yields a similar competitive ratio with high probability, as well as guaranteeing each vertex a probability of being matched tending to one. Online matching has received significant attention over the last 15 years due to its close connection to Internet advertising. As the seminal work of Karp, Vazirani, and Vazirani has an optimal (1 - 1 epsilon) competitive ratio in the standard adversarial online model, much effort has gone into developing useful online models that incorporate some stochasticity in the arrival process. One such popular model is the "known I.I.D. model" where different customer-types arrive online from a known distribution. We develop algorithms with improved competitive ratios for some basic variants of this model with integral arrival rates, including: (a) the case of general weighted edges, where we improve the best-known ratio of 0.667 due to [Haeupler, Mirrokni and Zadimoghaddam WINE 2011] to 0.705; and (b) the vertex-weighted case, where we improve the 0.7250 ratio of [Jaillet and Lu Math. Oper. Res 2013] to 0.7299. We also consider two extensions, one is "known I.I.D." with non-integral arrival rate and stochastic rewards; the other is "known I.I.D." b-matching with non-integral arrival rate and stochastic rewards. We present a simple non-adaptive algorithm which works well simultaneously on the two extensions. One of the key ingredients of our improvement is the following (offline) approach to bipartite-matching polytopes with additional constraints. We first add several valid constraints in order to get a good fractional solution f; however, these give us less control over the structure of f. We next remove all these additional constraints and randomly move from f to a feasible point on the matching polytope with all coordinates being from the set 0, 1 k, 2 k,..., 1 for a chosen integer k. The structure of this solution is inspired by [Jaillet and Lu Math. Oper. Res 2013] and is a tractable structure for algorithm design and analysis. The appropriate random move preserves many of the removed constraints (approximately [exactly] with high probability [in expectation]). This underlies some of our improvements, and, we hope, could be of independent interest. We study the following vertex-weighted online bipartite matching problem: G(U, V, E) is a bipartite graph. The vertices in U have weights and are known ahead of time, while the vertices in V arrive online in an arbitrary order and have to be matched upon arrival. The goal is to maximize the sum of weights of the matched vertices in U. When all the weights are equal, this reduces to the classic online bipartite matching problem for which Karp, Vazirani and Vazirani gave an optimal (1−1 e)-competitive algorithm in their seminal work [10]. Our main result is an optimal (1−1 e)-competitive randomized algorithm for general vertex weights. We use random perturbations of weights by appropriately chosen multiplicative factors. Our solution constitutes the first known generalization of the algorithm in [10] in this model and provides new insights into the role of randomization in online allocation problems. It also effectively solves the problem of online budgeted allocations [14] in the case when an agent makes the same bid for any desired item, even if the bid is comparable to his budget - complementing the results of [14, 3] which apply when the bids are much smaller than the budgets. We study the problem of exploring an unknown undirected connected graph. Beginning in some start vertex, a searcher must visit each node of the graph by traversing edges. Upon visiting a vertex for the first time, the searcher learns all incident edges and their respective traversal costs. The goal is to find a tour of minimum total cost. Kalyanasundaram and Pruhs (Constructing competitive tours from local information, Theoretical Computer Science 130, pp. 125-138, 1994) proposed a sophisticated generalization of a Depth First Search that is 16-competitive on planar graphs. While the algorithm is feasible on arbitrary graphs, the question whether it has constant competitive ratio in general has remained open. Our main result is an involved lower bound construction that answers this question negatively. On the positive side, we prove that the algorithm has constant competitive ratio on any class of graphs with bounded genus. Furthermore, we provide a constant competitive algorithm for general graphs with a bounded number of distinct weights. In this paper, we study online algorithms when the input is not chosen adversarially, but consists of draws from some given probability distribution. While this model has been studied for online problems like paging and k-server, it is not known how to beat the Φ(log n) bound for online Steiner tree if at each time instant, the demand vertex is a uniformly random vertex from the graph. For the online Steiner tree problem, we show that if each demand vertex is an independent draw from some probability distribution π: V → [0, 1], a variant of the natural greedy algorithm achieves Eω[A(ω)] Eω[OPT (ω)] = O(1); moreover, this result can be extended to some other subadditive problems. Both assumptions that the input sequence consists of independent draws from π, and that π is known to the algorithm are both essential; we show (almost) logarithmic lower bounds if either assumption is violated. Moreover, we give preliminary results on extending the Steiner tree results above to the related "expected ratio" measure Eω[ω(ω) OPT (ω)]. Finally, we use these ideas to give an average-case analysis of the Universal TSP problem. We study an online assignment problem, motivated by Adwords Allocation, in which queries are to be assigned to bidders with budget constraints. We analyze the performance of the Greedy algorithm (which assigns each query to the highest bidder) in a randomized input model with queries arriving in a random permutation. Our main result is a tight analysis of Greedy in this model showing that it has a competitive ratio of 1 - 1 e for maximizing the value of the assignment. We also consider the more standard i.i.d. model of input, and show that our analysis holds there as well. This is to be contrasted with the worst case analysis of [MSVV05] which shows that Greedy has a ratio of 1 2, and that the optimal algorithm presented there has a ratio of 1 - 1 e. The analysis of Greedy is important in the Adwords setting because it is the natural allocation algorithm for an auction-style process. From a theoretical perspective, our result simplifies and generalizes the classic algorithm of Karp, Vazirani and Vazirani for online bipartite matching. Our results include a new proof to show that the Ranking alforithm of [KVV90] has a ratio of 1 - 1 e in the worst case. It has been recently discovered [KV07] (independent of our results) that one of the crucial lemmas in [KVV90], related to a certain reduction, is incorrect. Our proof is direct, in that it does not go via such a reduction, which also enables us to generalize the analysis to our online assignment problem. The greedy sequential algorithm for maximal independent set (MIS) loops over the vertices in an arbitrary order adding a vertex to the resulting set if and only if no previous neighboring vertex has been added. In this loop, as in many sequential loops, each iterate will only depend on a subset of the previous iterates (i.e. knowing that any one of a vertex's previous neighbors is in the MIS, or knowing that it has no previous neighbors, is sufficient to decide its fate one way or the other). This leads to a dependence structure among the iterates. If this structure is shallow then running the iterates in parallel while respecting the dependencies can lead to an efficient parallel implementation mimicking the sequential algorithm. In this paper, we show that for any graph, and for a random ordering of the vertices, the dependence length of the sequential greedy MIS algorithm is polylogarithmic (O(log^2 n) with high probability). Our results extend previous results that show polylogarithmic bounds only for random graphs. We show similar results for greedy maximal matching (MM). For both problems we describe simple linear-work parallel algorithms based on the approach. The algorithms allow for a smooth tradeoff between more parallelism and reduced work, but always return the same result as the sequential greedy algorithms. We present experimental results that demonstrate efficiency and the tradeoff between work and parallelism. We consider the optimal online packet scheduling problem in a single-user energy harvesting wireless communication system, where energy is harvested from natural renewable sources, making future energy arrivals instants and amounts random in nature. The most general case of arbitrary energy arrivals is considered where neither the future energy arrival instants or amount, nor their distribution is known. The problem considered is to adaptively change the transmission rate according to the causal energy arrival information, such that the time by which all packets are delivered is minimized. We assume that all bits have arrived and are ready at the source before the transmission begins. For a minimization problem, the utility of an online algorithm is tested by finding its competitive ratio or competitiveness that is defined to be the maximum of the ratio of the gain of the online algorithm with the optimal offline algorithm over all input sequences. We derive a lower and upper bound on the competitive ratio of any online algorithm to minimize the total transmission time in an energy harvesting system. The upper bound is obtained using a lazy' transmission policy that chooses its transmission power to minimize the transmission time assuming that no further energy arrivals are going to occur in future. The lazy transmission policy is shown to be strictly two-competitive. We also derive an adversarial lower bound that shows that competitive ratio of any online algorithm is at least 1.325. Copyright © 2017, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. Motivated by the fact that in several cases a matching in a graph is stable if and only if it is produced by a greedy algorithm, we study the problem of computing a maximum weight greedy matching on weighted graphs, termed GREEDY-MATCHING. In wide contrast to the maximum weight matching problem, for which many efficient algorithms are known, we prove that GREEDYMATCHING is strongly NP-hard and APX-complete, and thus it does not admit a PTAS unless P=NP, even on graphs with maximum degree at most 3 and with at most three different integer edge weights. Furthermore we prove that GREEDYMATCHING is strongly NP-hard if the input graph is in addition bipartite. Moreover we consider three natural parameters of the problem, for which we establish a sharp threshold behavior between NP-hardness and computational tractability. On the positive side, we present a randomized approximation algorithm (RGMA) for GREEDYMATCHING on a special class of weighted graphs, called bush graphs. We highlight an unexpected connection between RGMA and the approximation of maximum cardinality matching in unweighted graphs via randomized greedy algorithms. We show that, if the approximation ratio of RGMA is θ, then for every ϵ > 0 the randomized MRG algorithm of ( 1995) gives a (ρ - ϵ)-approximation for the maximum cardinality matching. We conjecture that a tight bound for ρ is 2 3; we prove our conjecture true for four subclasses of bush graphs. Proving a tight bound for the approximation ratio of MRG on unweighted graphs (and thus also proving a tight value for ρ) is a long-standing open problem (Poloczek and Szegedy 2012). This unexpected relation of our RGMA algorithm with the MRG algorithm may provide new insights for solving this problem. We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit ‘single-letter’ characterization of the per-vertex mutual information between the vertex labels and the graph, when the graph average degree diverges. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and -in particular- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent of the vertex labels. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph @math from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit single-letter' characterization of the per-vertex mutual information between the vertex labels and the graph. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and --in particular-- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing. We consider the online stochastic matching problem proposed by [Feldman J, Mehta A, Mirrokni VS, Muthukrishnan S (2009) Online stochastic matching: Beating 1-1 e. Annual IEEE Sympos. Foundations Comput. Sci. 117--126] as a model of display ad allocation. We are given a bipartite graph; one side of the graph corresponds to a fixed set of bins, and the other side represents the set of possible ball types. At each time step, a ball is sampled independently from the given distribution and it needs to be matched upon its arrival to an empty bin. The goal is to maximize the number of allocations. We present an online algorithm for this problem with a competitive ratio of 0.702. Before our result, algorithms with a competitive ratio better than 1-1 e were known under the assumption that the expected number of arriving balls of each type is integral. A key idea of the algorithm is to collect statistics about the decisions of the optimum offline solution using Monte Carlo sampling and use those statistics to guide the decisions of the online algorithm. We also show that our algorithm achieves a competitive ratio of 0.705 when the rates are integral. On the hardness side, we prove that no online algorithm can have a competitive ratio better than 0.823 under the known distribution model (and henceforth under the permutation model). This improves upon the 5 6 hardness result proved by Goel and Mehta [Goel G, Mehta A (2008) Online budgeted matching in random input models with applications to adwords. ACM-SIAM Symposium Discrete Algorithms 982--991] for the permutation model. We examine several online matching problems, with applications to Internet advertising reservation systems. Consider an edge-weighted bipartite graph G, with partite sets L, R. We develop an 8-competitive algorithm for the following secretary problem: Initially given R, and the size of L, the algorithm receives the vertices of L sequentially, in a random order. When a vertex l L is seen, all edges incident to l are revealed, together with their weights. The algorithm must immediately either match l to an available vertex of R, or decide that l will remain unmatched. Dimitrov and Plaxton show a 16-competitive algorithm for the transversal matroid secretary problem, which is the special case with weights on vertices, not edges. (Equivalently, one may assume that for each l L, the weights on all edges incident to l are identical.) We use a similar algorithm, but simplify and improve the analysis to obtain a better competitive ratio for the more general problem. Perhaps of more interest is the fact that our analysis is easily extended to obtain competitive algorithms for similar problems, such as to find disjoint sets of edges in hypergraphs where edges arrive online. We also introduce secretary problems with adversarially chosen groups. Finally, we give a 2e-competitive algorithm for the secretary problem on graphic matroids, where, with edges appearing online, the goal is to find a maximum-weight acyclic subgraph of a given graph. We exhibit a poly(log k)-competitive randomized algorithm for the k-server problem on any metric space. The best previous result independent of the geometry of the underlying metric space is the 2k–1 competitive ratio established for the deterministic work function algorithm by Koutsoupias and Papadimitriou (1995). Even for the special case when the underlying metric space is the real line, the best known competitive ratio was k. Since deterministic algorithms can do no better than k on any metric space with at least k+1 points, this establishes that for every metric space on which the problem is non-trivial, randomized algorithms give an exponential improvement over deterministic algorithms. Our algorithm maintains an approximation of the underlying metric space by a distribution over HSTs. The granularity and accuracy of the approximation is adjusted dynamically according to the aggregate behavior of the HST algorithms. In short: We try to obtain more accurate approximations at the locations and scales where the gactionh is happening. Thus a crucial component of our approach is the O((log k)^2)-competitive randomized algorithm for HSTs obtained in our previous work with Bubeck, Cohen, Lee, and Ma.dry, and its "multiscale information theory" perspective. Nowadays, in the world of limited attention, the techniques that maximize the spread of social influence are more than welcomed. Companies try to maximize their profits on sales by providing customers with free samples believing in the power of word-of-mouth marketing, governments and non-governmental organizations often want to introduce positive changes in the society by appropriately selecting individuals or election candidates want to spend least budget yet still win the election. In this work we propose the use of evolutionary algorithm as a mean for selecting seeds in social networks. By framing the problem as genetic algorithm challenge we show that it is possible to outperform well-known greedy algorithm in the problem of influence maximization for the linear threshold model in both: quality (up to 16 better) and efficiency (up to 35 times faster). We implemented these two algorithms by using GPGPU approach showing that also the evolutionary algorithm can benefit from GPU acceleration making it efficient and scaling better than the greedy algorithm. As the experiments conducted by using three real world datasets reveal, the evolutionary approach proposed in this paper outperforms the greedy algorithm in terms of the outcome and it also scales much better than the greedy algorithm when the network size is increasing. The only drawback in the GPGPU approach so far is the maximum size of the network that can be processed - it is limited by the memory of the GPU card. We believe that by showing the superiority of the evolutionary approach over the greedy algorithm, we will motivate the scientific community to look for an idea to overcome this limitation of the GPU approach - we also suggest one of the possible paths to explore. Since the proposed approach is based only on topological features of the network, not on the attributes of nodes, the applications of it are broader than the ones that are dataset-specific. We give a simple proof that the ranking algorithm of Karp, Vazirani and Vazirani [KVV90] is 1-1 e competitive for the online bipartite matching problem. The proof is via a randomized primal-dual argument. Primal-dual algorithms have been successfully used for many online algorithm problems, but the dual constraints are always satisfied deterministically. This is the first instance of a non-trivial randomized primal-dual algorithm in which the dual constraints only hold in expectation. The approach also generalizes easily to the vertex-weighted version considered by [AGKM11]. Further we show that the proof is very similar to the deterministic primal-dual argument for the online budgeted allocation problem with small bids (also called the AdWords problem) of [MSVV05]. In this work, we study the notion of competing campaigns in a social network and address the problem of influence limitation where a "bad" campaign starts propagating from a certain node in the network and use the notion of limiting campaigns to counteract the effect of misinformation. The problem can be summarized as identifying a subset of individuals that need to be convinced to adopt the competing (or "good") campaign so as to minimize the number of people that adopt the "bad" campaign at the end of both propagation processes. We show that this optimization problem is NP-hard and provide approximation guarantees for a greedy solution for various definitions of this problem by proving that they are submodular. We experimentally compare the performance of the greedy method to various heuristics. The experiments reveal that in most cases inexpensive heuristics such as degree centrality compare well with the greedy approach. We also study the influence limitation problem in the presence of missing data where the current states of nodes in the network are only known with a certain probability and show that prediction in this setting is a supermodular problem. We propose a prediction algorithm that is based on generating random spanning trees and evaluate the performance of this approach. The experiments reveal that using the prediction algorithm, we are able to tolerate about 90 missing data before the performance of the algorithm starts degrading and even with large amounts of missing data the performance degrades only to 75 of the performance that would be achieved with complete data. We consider the online bipartite matching problem in the unknown distribution input model. We show that the Ranking algorithm of [KVV90] achieves a competitive ratio of at least 0.653. This is the first analysis to show an algorithm which breaks the natural 1 - 1 e -barrier' in the unknown distribution model (our analysis in fact works in the stricter, random order model) and answers an open question in [GM08]. We also describe a family of graphs on which Ranking does no better than 0.727 in the random order model. Finally, we show that for graphs which have k > 1 disjoint perfect matchings, Ranking achieves a competitive ratio of at least 1 - √(1 k - 1 k2 + 1 n) -- in particular Ranking achieves a factor of 1 - o(1) for graphs with ω(1) disjoint perfect matchings.