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.
On the other hand, for uniformly random edge arrival Konrad, Magniez, and Mathieu @cite_29 gave the first single pass algorithm that obtains a @math -competitive ratio for bipartite matching in the semi-streaming setting. Their algorithm crucially used the ability to revoke earlier decisions. One of the contributions in this paper is to show that a variant of the greedy algorithm, which appears simple in hindsight, achieves a competitive ratio better than half in the more restrictive online model.
@cite_29
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We present three semi-streaming algorithms for Maximum Bipartite Matching with one and two passes. Our one-pass semi-streaming algorithm is deterministic and returns a matching of size at least 1 2 + 0.005 times the optimal matching size in expectation, assuming that edges arrive one by one in (uniform) random order. Our first two-pass algorithm is randomized and returns a matching of size at least 1 2 + 0.019 times the optimal matching size in expectation (over its internal random coin flips) for any arrival order. These two algorithms apply the simple Greedy matching algorithm several times on carefully chosen subgraphs as a subroutine. Furthermore, we present a two-pass deterministic algorithm for any arrival order returning a matching of size at least 1 2 + 0.019 times the optimal matching size. This algorithm is built on ideas from the computation of semi-matchings. 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 study the online stochastic bipartite matching problem, in a form motivated by display ad allocation on the Internet. In the online, but adversarial case, the celebrated result of Karp, Vazirani and Vazirani gives an approximation ratio of @math , a very familiar bound that holds for many online problems; further, the bound is tight in this case. In the online, stochastic case when nodes are drawn repeatedly from a known distribution, the greedy algorithm matches this approximation ratio, but still, no algorithm is known that beats the @math bound.Our main result is a @math -approximation online algorithm for stochastic bipartite matching, breaking this @math barrier. Furthermore, we show that no online algorithm can produce a @math approximation for an arbitrarily small @math for this problem. Our algorithms are based on computing an optimal offline solution to the expected instance, and using this solution as a guideline in the process of online allocation. We employ a novel application of the idea of the power of two choices from load balancing: we compute two disjoint solutions to the expected instance, and use both of them in the online algorithm in a prescribed preference order. To identify these two disjoint solutions, we solve a max flow problem in a boosted flow graph, and then carefully decompose this maximum flow to two edge-disjoint (near-) matchings. In addition to guiding the online decision making, these two offline solutions are used to characterize an upper bound for the optimum in any scenario. This is done by identifying a cut whose value we can bound under the arrival distribution. At the end, we discuss extensions of our results to more general bipartite allocations that are important in a display ad application. We present the first poly-logarithmic competitive online algorithm for minimum metric bipartite matching. Via induction and a careful use of potential functions, we show that a simple randomized greedy algorithm is competitive on a hierarchically separated tree. Application of recent results on randomized embedding of metrics into trees yield the poly-logarithmic result for general metrics. 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 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 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. In this paper, we study the non-bipartite maximum matching problem in the semi-streaming model. The maximum matching problem in the semi-streaming model has received a significant amount of attention lately. While the problem has been somewhat well solved for bipartite graphs, the known algorithms for non-bipartite graphs use @math passes or @math time to compute a @math approximation. In this paper we provide the first FPTAS (polynomial in @math ) for the problem which is efficient in both the running time and the number of passes. We also show that we can estimate the size of the matching in @math passes using slightly superlinear space. To achieve both results, we use the structural properties of the matching polytope such as the laminarity of the tight sets and total dual integrality. The algorithms are iterative, and are based on the fractional packing and covering framework. However the formulations herein require exponentially many variables or constraints. We use laminarity, metric embeddings and graph sparsification to reduce the space required by the algorithms in between and across the iterations. This is the first use of these ideas in the semi-streaming model to solve a combinatorial optimization problem. This paper deals with the design of efficiently computable incentive-compatible mechanisms for combinatorial optimization problems with single-minded agents each possibly having multiple private parameters. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques. Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone fully polynomial time approximation scheme (FPTAS). This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multiunit auctions. The best previous result for such auctions was a 2-appro-xi-ma-tion. In addition, we present a monotone PTAS for the generalized assignment problem with any constant number of private parameters per agent. The most efficient way to solve packing integer programs (PIPs) is linear programming-based randomized rounding, which also is in general not monotone. We show that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like single-minded combinatorial auctions (CAs), unsplittable flow routing, and multicast routing. Our primal-dual approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle. We consider variants of the online stochastic bipartite matching problem motivated by Internet advertising display applications, as introduced in [Feldman J, Mehta A, Mirrokni VS, Muthukrishnan S (2009) Online stochastic matching: Beating 1 − 1 e. FOCS '09: Proc. 50th Annual IEEE Sympos. Foundations Comput. Sci. (IEEE, Washington, DC), 117–126]. In this setting, advertisers express specific interests into requests for impressions of different types. Advertisers are fixed and known in advance, whereas requests for impressions come online. The task is to assign each request to an interested advertiser (or to discard it) immediately upon its arrival. In the adversarial online model, the ranking algorithm of [Karp RM, Vazirani UV, Varirani VV (1990) An optimal algorithm for online bipartite matching. STOC '90: Proc. 22nd Annual ACM Sympos. Theory Comput. (ACM, New York), 352–358] provides a best possible randomized algorithm with competitive ratio 1 − 1 e ≈ 0.632. In the stochastic ... We prove the first non-trivial performance ratios strictly above 0.5 for weighted versions of the oblivious matching problem. Even for the unweighted version, since Aronson, Dyer, Frieze, and Suen first proved a non-trivial ratio above 0.5 in the mid-1990s, during the next twenty years several attempts have been made to improve this ratio, until Chan, Chen, Wu and Zhao successfully achieved a significant ratio of 0.523 very recently (SODA 2014). To the best of our knowledge, our work is the first in the literature that considers the node-weighted and edge-weighted versions of the problem in arbitrary graphs (as opposed to bipartite graphs). (1) For arbitrary node weights, we prove that a weighted version of the Ranking algorithm has ratio strictly above 0.5. We have discovered a new structural property of the ranking algorithm: if a node has two unmatched neighbors at the end of algorithm, then it will still be matched even when its rank is demoted to the bottom. This property allows us to form LP constraints for both the node-weighted and the unweighted oblivious matching problems. As a result, we prove that the ratio for the node-weighted case is at least 0.501512. Interestingly via the structural property, we can also improve slightly the ratio for the unweighted case to 0.526823 (from the previous best 0.523166 in SODA 2014). (2) For a bounded number of distinct edge weights, we show that ratio strictly above 0.5 can be achieved by partitioning edges carefully according to the weights, and running the (unweighted) Ranking algorithm on each part. Our analysis is based on a new primal-dual framework known as , in which dual feasibility is bypassed. Instead, only dual constraints corresponding to edges in an optimal matching are satisfied. Using this framework we also design and analyze an algorithm for the edge-weighted online bipartite matching problem with free disposal. We prove that for the case of bounded online degrees, the ratio is strictly above 0.5. Abstract We give an on-line deterministic algorithm for the weighted bipartite matching problem that achieves a competitive ratio of (2 n −1) in any metric space (where n is the number of vertices). This algorithm is optimal - there is no on-line deterministic algorithm that achieves a competitive ratio better than (2 n −1) in all metric spaces. We also study the stable marriage problem, where we are interested in the number of unstable pairs produced. We show that the simple “first come, first served” deterministic algorithm yields on the average O( n log n ) unstable pairs, but in the worst case no deterministic or randomized on-line algorithm can do better than ω( n 2 ) unstable pairs. This appears to be the first on-line problem for which provably one cannot do better with randomization; for most on-line problems studied in the past, randomization has helped in improving the performance. We study the online stochastic matching problem in a form motivated by Internet display advertisement. Recently, gave an algorithm that achieves 0.6702 competitive ratio, thus breaking through the 1-1 e barrier. One of the questions left open in their work is to obtain a better competitive ratio by generalizing their two suggested matchings (TSM) algorithm to d-suggested matchings (d-SM). We show that the best competitive ratio that can be obtained with the static analysis used in the d-SM algorithm is upper bounded by 0.76, even for the special case of d-regular graphs, thus suggesting that a dynamic analysis may be needed to improve the competitive ratio significantly. We make the first step in this direction by showing that the RANDOM algorithm, which assigns an impression to a randomly chosen eligible advertiser, achieves 1 - e-ddd d! = 1-O(1 √d) competitive ratio for d-regular graphs, which converges to 1 as d increases. On the hardness side, we improve the upper bound of 0.989 on the competitive ratio of any online algorithm obtained by to 1-1 (e + e2) ≅ 0.902. Finally, we show how to modify the TSM algorithm to obtain an improved 0.699 approximation for general bipartite graphs. We study the maximum weight matching problem in the semi-streaming model, and improve on the currently best one-pass algorithm due to Zelke [Proceedings of the 25th Annual Symposium on Theoretical Aspects of Computer Science, 2008, pp. 669–680] by devising a deterministic approach whose performance guarantee is 4.91+e. In addition, we study preemptive online algorithms, a class of algorithms related to one-pass semi-streaming algorithms, where we are allowed to maintain only a feasible matching in memory at any point in time. We provide a lower bound of 4.967 on the competitive ratio of any such deterministic algorithm, and hence show that future improvements will have to store in memory a set of edges that is not necessarily a feasible matching. We conclude by presenting an empirical study, conducted in order to compare the practical performance of our approach to that of previously suggested algorithms. This paper deals with the design of efficiently computable incentive compatible, or truthful, mechanisms for combinatorial optimization problems with multi-parameter agents. We focus on approximation algorithms for NP-hard mechanism design problems. These algorithms need to satisfy certain monotonicity properties to ensure truthfulness. Since most of the known approximation techniques do not fulfill these properties, we study alternative techniques.Our first contribution is a quite general method to transform a pseudopolynomial algorithm into a monotone FPTAS. This can be applied to various problems like, e.g., knapsack, constrained shortest path, or job scheduling with deadlines. For example, the monotone FPTAS for the knapsack problem gives a very efficient, truthful mechanism for single-minded multi-unit auctions. The best previous result for such auctions was a 2-approximation. In addition, we present a monotone PTAS for the generalized assignment problem with any bounded number of parameters per agent.The most efficient way to solve packing integer programs (PIPs) is LP-based randomized rounding, which also is in general not monotone. We show that primal-dual greedy algorithms achieve almost the same approximation ratios for PIPs as randomized rounding. The advantage is that these algorithms are inherently monotone. This way, we can significantly improve the approximation ratios of truthful mechanisms for various fundamental mechanism design problems like single-minded combinatorial auctions (CAs), unsplittable flow routing and multicast routing. Our approximation algorithms can also be used for the winner determination in CAs with general bidders specifying their bids through an oracle. Motivated by the display ad allocation problem on the Internet, we study the online stochastic weighted matching problem. In this problem, given an edge-weighted bipartite graph, nodes of one side arrive online i.i.d. according to a known probability distribution. Recently, a sequence of results by Feldman et. al [14] and Manshadi et. al [20] result in a 0.702-approximation algorithm for the unweighted version of this problem, aka online stochastic matching, breaking the 1−1 e barrier. Those results, however, do no hold for the more general online stochastic weighted matching problem. Moreover, all of these results employ the idea of power of two choices. In this paper, we present the first approximation (0.667-competitive) algorithm for the online stochastic weighted matching problem beating the 1−1 e barrier. Moreover, we improve the approximation factor of the online stochastic matching by analyzing the more general framework of power of multiple choices. In particular, by computing a careful third pseudo-matching along with the two offline solutions, and using it in the online algorithm, we improve the approximation factor of the online stochastic matching for any bipartite graph to 0.7036. 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. We present the first deterministic 1+e approximation algorithm for finding a large matching in a bipartite graph in the semi-streaming model which requires only O((1 e)5) passes over the input stream. In this model, the input graph G=(V,E) is given as a stream of its edges in some arbitrary order, and storage of the algorithm is bounded by O(npolylog n) bits, where @math . The only previously known arbitrarily good approximation for general graphs is achieved by the randomized algorithm of McGregor (Proceedings of the International Workshop on Approximation Algorithms for Combinatorial Optimization Problems and Randomization and Computation, Berkeley, CA, USA, pp. 170–181, 2005), which uses Ω((1 e)1 e ) passes. We show that even for bipartite graphs, McGregor’s algorithm needs Ω(1 e) Ω(1 e) passes, thus it is necessarily exponential in the approximation parameter. The design as well as the analysis of our algorithm require the introduction of some new techniques. A novelty of our algorithm is a new deterministic assignment of matching edges to augmenting paths which is responsible for the complexity reduction, and gets rid of randomization. We repeatedly grow an initial matching using augmenting paths up to a length of 2k+1 for k=⌈2 e⌉. We terminate when the number of augmenting paths found in one iteration falls below a certain threshold also depending on k, that guarantees a 1+e approximation. The main challenge is to find those augmenting paths without requiring an excessive number of passes. In each iteration, using multiple passes, we grow a set of alternating paths in parallel, considering each edge as a possible extension as it comes along in the stream. Backtracking is used on paths that fail to grow any further. Crucial are the so-called position limits: when a matching edge is the ith matching edge in a path and it is then removed by backtracking, it will only be inserted into a path again at a position strictly lesser than i. This rule strikes a balance between terminating quickly on the one hand and giving the procedure enough freedom on the other hand. We present an approximation algorithm to find a weighted matching of a graph in the one-pass semi-streaming model. The semi-streaming model forbids random access to the input graph and restricts the memory to @math bits where n denotes the number of the vertices of the input graph. We obtain an approximation ratio of 5.58 while the previously best algorithm achieves a ratio of 5.82.