The Online Stochastic Generalized Assignment Problem

Alaei, Saeed; Hajiaghayi, MohammadTaghi; Liaghat, Vahid

doi:10.1007/978-3-642-40328-6_2

Cited by 38 publications

(36 citation statements)

References 17 publications

(39 reference statements)

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“…( [7,8]); otherwise, it is called adversarial stochastic input ( [7]). As for known adversarial distributions, in each round an item is sampled from a known distribution, which is allowed to change over time ( [2,3]). Another variant of this problem is when the edges have stochastic rewards.…”

Section: Related Workmentioning

confidence: 99%

“…2 Let f be an optimal fractional solution vector. Call DR[f, 2] to get an integral vector F. 3 Create the graph G F with F e copies of each edge e ∈ E and decompose it into two matchings. 4 Randomly permute the matchings to get a random ordered pair of matchings, say [M 1 , M 2 ].…”

Section: Analysis Of Algorithm Ewmentioning

confidence: 99%

“…Let f be the optimal solution vector. 2 Invoke DR[f, 3] to obtain the vector F. 3 Independently run EW 1 and EW 2 with probabilities q and 1 − q respectively on G F . The details of algorithm EW 1 and EW 2 and the proof of Theorem 2 are presented in the following sections.…”

Section: Algorithm 2: Ew[q]mentioning

confidence: 99%

“…2 When a vertex v comes for the first time, assign v to some u 1 with (u 1 , v) ∈ M 1 . 3 When v comes for the second time, assign v to some u 2 with (u 2 , v) ∈ M 2 . 4 When v comes for the third time, if e is either a large edge or a small edge of type Γ 1 then assign v to some u 3 with e = (u 3 , v) ∈ M 3 .…”

Section: Algorithm 3: Ew 1 [H]mentioning

confidence: 99%

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Online Stochastic Matching: New Algorithms and Bounds

et al. 2020

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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/e) 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 [12] to 0.705; and (b) the vertex-weighted case, where we improve the 0.7250 ratio of Jaillet and Lu [13] to 0.7299.We also consider an extension of stochastic rewards, a variant where each edge has an independent probability of being present. For the setting of stochastic rewards with non-integral arrival rates, we present a simple optimal non-adaptive algorithm with a ratio of 1 − 1/e. For the special case where each edge is unweighted and has a uniform constant probability of being present, we improve upon 1 − 1/e by proposing a strengthened LP benchmark.One of the key ingredients of our improvement is the following (offline) approach to bipartitematching 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 (Mathematics of Operations Research, 2013) and is a tractable structure for algorithm design and analysis. The appropriate random move preserves many of the removed constraints (approximately with high probability and exactly in expectation). This underlies some of our improvements and could be of independent interest. * A preliminary version of this appeared in the European Symposium on Algorithms (ESA), 2016 † Claim 4. For a large edge e, EW 1 [h] (3) with parameter h achieves a competitive ratio of R[EW 1 , 2/3] = 0.67529 + (1 − h) * 0.00446. Claim 5. For a small edge e of type Γ 1 , EW 1 [h] (3) achieves a competitive ratio of R[EW 1 , 1/3] = 0.751066, regardless of the value h. Claim 6. For a small edge e of type Γ 2 , EW 1 [h] (3) achieves a competitive ratio of R[EW 1 , 1/3] = 0.72933 + h * 0.040415. By setting h = 0.537815, the two types of small edges have the same ratio and we get that EW 1 [h] achieves (R[EW 1 , 2/3], R[EW 1 , 1/3]) = (0.679417, 0.751066). Thus, this proves Lemma 2. Proof of Claim 4Proof. Consider a large edge e = (u, v 1 ) in the graph G F . Let e ′ = (u, v 2 ) be the other small edge incident to u. Edges e and e ′ can appear in [M ...

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Section: Related Workmentioning

confidence: 99%

Section: Analysis Of Algorithm Ewmentioning

confidence: 99%

Section: Algorithm 2: Ew[q]mentioning

confidence: 99%

Section: Algorithm 3: Ew 1 [H]mentioning

confidence: 99%

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Online Stochastic Matching: New Algorithms and Bounds

et al. 2020

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“…In the worst case model, the celebrated result of Karp et al ([26], STOC'90) gives a (1 − 1/e)-competitive algorithm. Different variants of this problem have been extensively studied in the past decade, e.g., for the random arrival model see [16,25,32,36], for the full information model see [33,38], and for the prophetinequality model see [5,3,4]. We also refer the reader to the comprehensive survey by Mehta [35].…”

Section: Related Online Problemsmentioning

confidence: 99%

Online Degree-Bounded Steiner Network Design

Sina¹,

Ehsani²,

Hajiaghayi³

et al. 2015

Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms

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We initiate the study of degree-bounded network design problems in the online setting. The degree-bounded Steiner tree problem -which asks for a subgraph with minimum degree that connects a given set of vertices -is perhaps one of the most representative problems in this class. This paper deals with its well-studied generalization called the degree-bounded Steiner forest problem where the connectivity demands are represented by vertex pairs that need to be individually connected. In the classical online model, the input graph is given offline but the demand pairs arrive sequentially in online steps. The selected subgraph starts off as the empty subgraph, but has to be augmented to satisfy the new connectivity constraint in each online step. The goal is to be competitive against an adversary that knows the input in advance.We design a simple greedy-like algorithm that achieves a competitive ratio of O(log n) where n is the number of vertices. We show that no (randomized) algorithm can achieve a (multiplicative) competitive ratio o(log n); thus our result is asymptotically tight. We further show strong hardness results for the group Steiner tree and the edge-weighted variants of degree-bounded connectivity problems.Fürer and Raghavachari resolved the offline variant of degree-bounded Steiner forest in their paper in SODA'92. Since then, the natural family of degree-bounded network design problems has been extensively studied in the literature resulting in the development of many interesting tools and numerous papers on the topic. We hope that our approach in this paper, paves the way for solving the online variants of the classical problems in this family of network design problems.

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Prophet Inequality and Online Auctions

Hajiaghayi¹,

Liaghat²

2016

Encyclopedia of Algorithms

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The Online Stochastic Generalized Assignment Problem

Cited by 38 publications

References 17 publications

Online Stochastic Matching: New Algorithms and Bounds

Online Stochastic Matching: New Algorithms and Bounds

Online Degree-Bounded Steiner Network Design

Prophet Inequality and Online Auctions

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