Prophet Inequalities vs. Approximating Optimum Online

Niazadeh, Rad; Saberi, Amin; Shameli, Ali

doi:10.1007/978-3-030-04612-5_24

Cited by 7 publications

(9 citation statements)

References 28 publications

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“…We next turn to the approximability of the optimal online policy, where we might hope to achieve higher approximation guarantees. For the classic prophet inequality problem, Niazadeh et al [25] show that pricing-based policies yield no better approximation of the optimal online policy than they do of the optimal offline policy. For the stationary prophet inequality problem, the same is not true; while our inapproximability result of Theorem 1.1 implies that no competitive ratio beyond 1 /2 is possible, an algorithm of [3] yields a 1− 1 /e ≈ 0.632 approximation of the optimal online policy.…”

Section: Our Contributionsmentioning

confidence: 99%

“…The classic single-item prophet inequality problem has been generalized to selling more complicated combinatorical structures, including, e.g., multiple items [1,7,18], knapsacks [12,16], matroids and their intersections [4,8,12,16,20], matchings [12,14,15,16,17], and arbitrary downward-closed families [27]. Most of this work has focused on approximating the offline optimum algorithm, with recent work also studying the (in)approximability of the optimal online algorithm by poly-time algorithms [2,26] and particularly by posted-price policies [25]. Much of the interest in prophet inequality problems, and specifically pricing-based policies, has been fueled by their implication of truthful mechanisms which approximately maximize social welfare and revenue, first observed in [8] (see the surveys [10,19,23], and [11] for the "opposite" direction).…”

Section: Introductionmentioning

confidence: 99%

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The Stationary Prophet Inequality Problem

Kristen¹,

Saberi²,

Shameli³

et al. 2021

Preprint

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We study a continuous and infinite time horizon counterpart to the classic prophet inequality, which we term the stationary prophet inequality problem. Here, copies of a good arrive and perish according to Poisson point processes. Buyers arrive similarly and make take-it-or-leave-it offers for unsold items. The objective is to maximize the (infinite) time average revenue of the seller.Our main results are pricing-based policies which (i) achieve a 1/2-approximation of the optimal offline policy, which is best possible, and (ii) achieve a better than (1−1/e)-approximation of the optimal online policy. Result (i) improves upon bounds implied by recent work of Collina et al. (WINE'20), and is the first optimal prophet inequality for a stationary problem. Result (ii) improves upon a 1 − 1/e bound implied by recent work of Aouad and Saritaç (EC'20), and shows that this prevalent bound in online algorithms is not optimal for this problem.

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Section: Our Contributionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Stationary Prophet Inequality Problem

Kristen¹,

Saberi²,

Shameli³

et al. 2021

Preprint

Self Cite

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“…That is, rather than using the standard prophet benchmark, we use the arguably more realistic benchmark of the best order-aware online algorithm. This is part of a growing interest in alternative benchmarks to the prophet benchmark Kessel et al [2021], Niazadeh et al [2018], Papadimitriou et al [2021]. For example, Niazadeh et al [2018] quantify the loss due to single-threshold algorithms by the worst-case ratio between the best single-threshold algorithm and the best general online algorithm (single-threshold or not), both under a known order, and show that this 1/2 ratio is tight.…”

Section: Introductionmentioning

confidence: 99%

“…This is part of a growing interest in alternative benchmarks to the prophet benchmark Kessel et al [2021], Niazadeh et al [2018], Papadimitriou et al [2021]. For example, Niazadeh et al [2018] quantify the loss due to single-threshold algorithms by the worst-case ratio between the best single-threshold algorithm and the best general online algorithm (single-threshold or not), both under a known order, and show that this 1/2 ratio is tight. As another example, Papadimitriou et al [2021] consider the problem of online matching in bipartite graphs.…”

Section: Introductionmentioning

confidence: 99%

“…The order competitive ratio is defined as the worst-case ratio between the performance of the best order-unaware algorithm and the best order-aware algorithm. Note that this new measure uses the same benchmark as the one in Papadimitriou et al [2021], Niazadeh et al [2018], but unlike Papadimitriou et al [2021], our algorithms are order-unaware, thus enjoy all the advantages and robustness of order-unawareness. To get a sense of the problem, let us go back to the classic prophet inequality result.…”

Section: Introductionmentioning

confidence: 99%

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On the Significance of Knowing the Arrival Order in Prophet Inequality

Ezra¹,

Feldman²,

Gravin³

et al. 2022

Preprint

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In a prophet inequality problem, n boxes arrive online, each containing some value that is drawn independently from a known distribution. Upon the arrival of a box, its value is realized, and an online algorithm decides, immediately and irrevocably, whether to accept it or proceed to the next box. Clearly, an online algorithm that knows the arrival order may be more powerful than an online algorithm that is unaware of the order. Despite the growing interest in the role of the arrival order on the performance of online algorithms, the effect of knowledge of the order has been overlooked thus far.Our goal in this paper is to quantify the loss due to unknown order. We define the order competitive ratio as the worst-case ratio between the performance of the best order-unaware and the best order-aware algorithms. We study the order competitive ratio for two objective functions, namely (i) max-expectation: maximizing the expected accepted value, and (ii) max-probability: maximizing the probability of accepting the box with the largest value. For the max-expectation objective, we're golden: we give a deterministic order-unaware algorithm that achieves an order competitive ratio of the inverse of the golden ratio (i.e., 1/φ ≈ 0.618). For the max-probability objective, we give a deterministic order-unaware algorithm that achieves an order competitive ratio of ln 1 λ ≈ 0.806 (where λ is the unique solution to x 1−x = ln 1 x ). Both results are tight. Our algorithms are inevitably adaptive and go beyond single-threshold algorithms.

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The Importance of Knowing the Arrival Order in Combinatorial Bayesian Settings

Ezra,

Garbuz

2023

Lecture Notes in Computer Science

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Prophet Inequalities vs. Approximating Optimum Online

Cited by 7 publications

References 28 publications

The Stationary Prophet Inequality Problem

The Stationary Prophet Inequality Problem

On the Significance of Knowing the Arrival Order in Prophet Inequality

The Importance of Knowing the Arrival Order in Combinatorial Bayesian Settings

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