Pranav Nuti
We study a variant of the secretary problem where candidates come from independent, not necessarily identical distributions known to us, and show that we can do at least as well as in the IID setting. This resolves a conjecture of Esfandiari et al.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We show that the matroid secretary problem is equivalent to correlated contention resolution in the online random-order model. Specifically, the matroid secretary conjecture is true if and only if every matroid admits an online random-order contention resolution scheme which, given an arbitrary (possibly correlated) prior distribution over subsets of the ground set, matches the balance ratio of the best offline scheme for that distribution up to a constant. We refer to such a scheme as universal. Our result indicates that the core challenge of the matroid secretary problem lies in resolving co…
The Knapsack Secretary Problem is Strictly Harder Than the Secretary Problem
研究背包秘书问题是否能达到经典秘书问题的最优1/e竞争比，通过构造困难实例族分析得出不存在(1/e - 0.0001)竞争比的算法，并给出一种将最佳已知竞争比从0.153提高到0.178的算法。
The most well-known conjecture in the context of matroid secretary problems claims the existence of a constant-factor approximation applicable to any matroid. Whereas this conjecture remains open, modified forms of it were shown to be true, when assuming that the assignment of weights to the secretaries is not adversarial but uniformly random (Soto [SODA 2011], Oveis Gharan and Vondrák [ESA 2011]). However, so far, there was no variant of the matroid secretary problem with adversarial weight assignment for which a constant-factor approximation was found. We address this point by presenting a 9…
We consider a generalization of the secretary problem where contracts are temporary, and for a fixed duration. This models online hiring of temporary employees, or online auctions for re-usable resources. The problem is related to the question of Finding a large independent set in a random unit interval graph.
Settling the Matroid Secretary Problem
本文提出一个序数、仅依赖比较与独立性预言机的 $e$-概率竞争算法，以期望多项式复杂度解决了拟阵秘书问题。
In this paper, we investigate two variants of the secretary problem. In these variants, we are presented with a sequence of numbers $X_i$ that come from distributions $\mathcal{D}_i$, and that arrive in either random or adversarial order. We do not know what the distributions are, but we have access to a single sample $Y_i$ from each distribution $\mathcal{D}_i$. After observing each number, we have to make an irrevocable decision about whether we would like to accept it or not with the goal of maximizing the probability of selecting the largest number. The random order version of this problem…
Some variations of the secretary problem
本文研究了秘书问题的两种变种，一种是允许面试者重复出现的变种，另一种是允许选择前三名面试者的变种，探讨了最优策略及其对选择性能的影响。
The secretary problem became one of the most prominent online selection problems due to its numerous applications in online mechanism design. The task is to select a maximum weight subset of elements subject to given constraints, where elements arrive one-by-one in random order, revealing a weight upon arrival. The decision whether to select an element has to be taken immediately after its arrival. The different applications that map to the secretary problem ask for different constraint families to be handled. The most prominent ones are matroid constraints, which both capture many relevant se…
We study the submodular secretary problem with a cardinality constraint. In this problem, $n$ candidates for secretaries appear sequentially in random order. At the arrival of each candidate, a decision maker must irrevocably decide whether to hire him. The decision maker aims to hire at most $k$ candidates that maximize a non-negative submodular set function. We propose an $(\mathrm{e} - 1)^2 / (\mathrm{e}^2 (1 + \mathrm{e}))$-competitive algorithm for this problem, which improves the best one known so far.
Whereas there are simple algorithms that are proven to be optimal for the Classical and the Multiple Choice Secretary Problem, the Matroid Secretary Problem is less thoroughly understood. This paper proposes the generalization of some simple algorithms from the Classical and Multiple Choice versions on the Matroid Secretary Problem. Out of two algorithms that make decisions based on samples, like the Dynkin's algorithm, one is proven to be an instance of Greedy Algorithm (Bahrani et al., 2022), while the other is not. A generalized version of the Virtual Algorithm (Babaioff et al., 2018) obtai…
Only recently progress has been made in obtaining $o(\log(\mathrm{rank}))$-competitive algorithms for the matroid secretary problem. More precisely, Chakraborty and Lachish (2012) presented a $O(\sqrt{\log(\mathrm{rank})})$-competitive procedure, and Lachish (2014) later presented a $O(\log\log(\mathrm{rank}))$-competitive algorithm. Both these algorithms and their analyses are very involved, which is also reflected in the extremely high constants in their competitive ratios. Using different tools, we present a considerably simpler $O(\log\log(\mathrm{rank}))$-competitive algorithm for the mat…
During the last decade, the matroid secretary problem (MSP) became one of the most prominent classes of online selection problems. Partially linked to its numerous applications in mechanism design, substantial interest arose also in the study of nonlinear versions of MSP, with a focus on the submodular matroid secretary problem (SMSP). So far, O(1)-competitive algorithms have been obtained for SMSP over some basic matroid classes. This created some hope that, analogously to the matroid secretary conjecture, one may even obtain O(1)-competitive algorithms for SMSP over any matroid. However, up …
In the online random-arrival model, an algorithm receives a sequence of n requests that arrive in a random order. The algorithm is expected to make an irrevocable decision with regard to each request based only on the observed history. We consider the following natural extension of this model: each request arrives k times, and the arrival order is a random permutation of the kn arrivals; the algorithm is expected to make a decision regarding each request only upon its last arrival. We focus primarily on the case when k=2, which can also be interpreted as each request arriving at, and departing…
In the secretary problem of Cayley (1875) and Moser (1956), $n$ non-negative, independent, random variables with common distribution are sequentially presented to a decision maker who decides when to stop and collect the most recent realization. The goal is to maximize the expected value of the collected element. In the $k$-choice variant, the decision maker is allowed to make $k \leq n$ selections to maximize the expected total value of the selected elements. Assuming that the values are drawn from a known distribution with finite support, we prove that the best regret---the expected gap betw…
The secretary problem is a classic model for online decision making. Recently, combinatorial extensions such as matroid or matching secretary problems have become an important tool to study algorithmic problems in dynamic markets. Here the decision maker must know the numerical value of each arriving element, which can be a demanding informational assumption. In this paper, we initiate the study of combinatorial secretary problems with ordinal information, in which the decision maker only needs to be aware of a preference order consistent with the values of arrived elements. The goal is to des…
Contention resolution schemes have proven to be a useful and unifying abstraction for a variety of constrained optimization problems, in both offline and online arrival models. Much of prior work restricts attention to product distributions for the input set of elements, and studies contention resolution for increasingly general packing constraints, both offline and online. In this paper, we instead focus on generalizing the input distribution, restricting attention to matroid constraints in both the offline and online random arrival models. In particular, we study contention resolution when t…
Babaioff et al. [BIK2007] introduced the matroid secretary problem in 2007, a natural extension of the classic single-choice secretary problem to matroids, and conjectured that a constant-competitive online algorithm exists. The conjecture still remains open despite substantial partial progress, including constant-competitive algorithms for numerous special cases of matroids, and an $O(\log \log \text{rank})$-competitive algorithm in the general case. Many of these algorithms follow principled frameworks. The limits of these frameworks are previously unstudied, and prior work establishes only …
The J-choice K-best secretary problem, also known as the (J,K)-secretary problem, is a generalization of the classical secretary problem. An algorithm for the (J,K)-secretary problem is allowed to make J choices and the payoff to be maximized is the expected number of items chosen among the K best items. Previous works analyzed the case when the total number n of items is finite, and considered what happens when n grows. However, for general J and K, the optimal solution for finite n is difficult to analyze. Instead, we prove a formal connection between the finite model and the infinite 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 \in 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 sh…
Knapsack Secretary is not $1/e$-Competitive
研究背包秘书问题，通过构造困难实例证明不存在1/e竞争比算法，改进不可能性结果，竞争比至多为0.36437，还给出(1/5.10 - o(1))竞争比的简单算法，解决相关开放问题。
We revisit the knapsack-secretary problem (Babaioff et al.; APPROX 2007), a generalization of the classic secretary problem in which items have different sizes and multiple items may be selected if their total size does not exceed the capacity $B$ of a knapsack. Previous works show competitive ratios of $1/(10e)$ (Babaioff et al.), $1/8.06$ (Kesselheim et al.; STOC 2014), and $1/6.65$ (Albers, Khan, and Ladewig; APPROX 2019) for the general problem but no definitive answers for the achievable competitive ratio; the best known impossibility remains $1/e$ as inherited from the classic secretary …
The classical analysis of online algorithms, due to its worst-case nature, can be quite pessimistic when the input instance at hand is far from worst-case. Often this is not an issue with machine learning approaches, which shine in exploiting patterns in past inputs in order to predict the future. However, such predictions, although usually accurate, can be arbitrarily poor. Inspired by a recent line of work, we augment three well-known online settings with machine learned predictions about the future, and develop algorithms that take them into account. In particular, we study the following on…
A matroid $\mathcal{M}$ on a set $E$ of elements has the $α$-partition property, for some $α>0$, if it is possible to (randomly) construct a partition matroid $\mathcal{P}$ on (a subset of) elements of $\mathcal{M}$ such that every independent set of $\mathcal{P}$ is independent in $\mathcal{M}$ and for any weight function $w:E\to\mathbb{R}_{\geq 0}$, the expected value of the optimum of the matroid secretary problem on $\mathcal{P}$ is at least an $α$-fraction of the optimum on $\mathcal{M}$. We show that the complete binary matroid, ${\cal B}_d$ on $\mathbb{F}_2^d$ does not satisfy the $α$-p…
Prophet Secretary and Matching: the Significance of the Largest Item
本研究针对prophet secretary问题及其匹配变体，利用最大项的特性设计算法，分别打破了盲策略和1-1/e的竞争比壁垒，同时提升了加权随机匹配查询-承诺模型的最优比率。
Prophet and Secretary at the Same Time
针对在线算法对分布错误设定的鲁棒性问题，在最优停止场景中探讨停止规则同时适配先知不等式（已知分布）与秘书问题（未知分布）的可能性，提出具非平凡联合保证的最优算法并确定不可达逼近比，结果适用于固定及泊松到达情况。
The classical secretary problem investigates the question of how to hire the best secretary from $n$ candidates who come in a uniformly random order. In this work we investigate a parallel generalizations of this problem introduced by Feldman and Tennenholtz [14]. We call it shared $Q$-queue $J$-choice $K$-best secretary problem. In this problem, $n$ candidates are evenly distributed into $Q$ queues, and instead of hiring the best one, the employer wants to hire $J$ candidates among the best $K$ persons. The $J$ quotas are shared by all queues. This problem is a generalized version of $J$-choi…
The Secretary Problem with Predictions
本文研究带价值预测的秘书问题，提出在预测准确时近乎最优、失准时仍保持常数竞争比的算法，并将保证扩展到多选情形。
In classical secretary problems, a sequence of $n$ elements arrive in a uniformly random order, and we want to choose a single item, or a set of size $K$. The random order model allows us to escape from the strong lower bounds for the adversarial order setting, and excellent algorithms are known in this setting. However, one worrying aspect of these results is that the algorithms overfit to the model: they are not very robust. Indeed, if a few "outlier" arrivals are adversarially placed in the arrival sequence, the algorithms perform poorly. E.g., Dynkin's popular $1/e$-secretary algorithm fai…
A short note about the learning-augmented secretary problem
该研究针对学习增强秘书问题，提出新的难度构造，证明即使候选者真实值为不随n缩放的常数，兼具1-一致性和1/e-鲁棒性的算法仍无法实现，补充了已有相关结论。
In the Prophet Secretary problem, samples from a known set of probability distributions arrive one by one in a uniformly random order, and an algorithm must irrevocably pick one of the samples as soon as it arrives. The goal is to maximize the expected value of the sample picked relative to the expected maximum of the distributions. This is one of the most simple and fundamental problems in online decision making that models the process selling one item to a sequence of costumers. For a closely related problem called the Prophet Inequality where the order of the random variables is adversarial…
We consider online resource allocation problems where given a set of requests our goal is to select a subset that maximizes a value minus cost type of objective function. Requests are presented online in random order, and each request possesses an adversarial value and an adversarial size. The online algorithm must make an irrevocable accept/reject decision as soon as it sees each request. The "profit" of a set of accepted requests is its total value minus a convex cost function of its total size. This problem falls within the framework of secretary problems. Unlike previous work in that area,…
The Secretary Problem with Predictions and a Chosen Order
本文提出了一种结合预测与到达顺序控制的随机算法，改进了随机顺序和选定顺序秘书问题的竞争比。
We study the matroid secretary problems with submodular valuation functions. In these problems, the elements arrive in random order. When one element arrives, we have to make an immediate and irrevocable decision on whether to accept it or not. The set of accepted elements must form an {\em independent set} in a predefined matroid. Our objective is to maximize the value of the accepted elements. In this paper, we focus on the case that the valuation function is a non-negative and monotonically non-decreasing submodular function. We introduce a general algorithm for such {\em submodular matroid…
Suppose that $n$ items arrive online in random order and the goal is to select $k$ of them such that the expected sum of the selected items is maximized. The decision for any item is irrevocable and must be made on arrival without knowing future items. This problem is known as the $k$-secretary problem, which includes the classical secretary problem with the special case $k=1$. It is well-known that the latter problem can be solved by a simple algorithm of competitive ratio $1/e$ which is optimal for $n \to \infty$. Existing algorithms beating the threshold of $1/e$ either rely on involved sel…
First, for the for the submodular $k$-secretary problem with shortlists [1], we provide a near optimal $1-1/e-ε$ approximation using shortlist of size $O(k poly(1/ε))$. In particular, we improve the size of shortlist used in \cite{us} from $O(k 2^{poly(1/ε)})$ to $O(k poly(1/ε))$. As a result, we provide a near optimal approximation algorithm for random-order streaming of monotone submodular functions under cardinality constraints, using memory $O(k poly(1/ε))$. It exponentially improves the running time and memory of \cite{us} in terms of $1/ε$. Next we generalize the problem to matroid const…
We study various generalizations of the secretary problem with submodular objective functions. Generally, a set of requests is revealed step-by-step to an algorithm in random order. For each request, one option has to be selected so as to maximize a monotone submodular function while ensuring feasibility. For our results, we assume that we are given an offline algorithm computing an $α$-approximation for the respective problem. This way, we separate computational limitations from the ones due to the online nature. When only focusing on the online aspect, we can assume $α= 1$. In the submodular…
The secretary problem is probably the purest model of decision making under uncertainty. In this paper we ask which advice can we give the algorithm to improve its success probability? We propose a general model that unifies a broad range of problems: from the classic secretary problem with no advice, to the variant where the quality of a secretary is drawn from a known distribution and the algorithm learns each candidate's quality on arrival, to more modern versions of advice in the form of samples, to an ML-inspired model where a classifier gives us noisy signal about whether or not the curr…
Multilevel Dynamic Thinning for Matroid Secretary
本文提出多级动态稀疏化方法，将拟阵秘书问题的单一参考集扩展为嵌套层级，实现任意拟阵上的 $(e+\eps)$-概率竞争算法，最坏查询复杂度 $O_\eps(n^2)$。
For many online problems, it is known that the uniform arrival order enables the design of algorithms with much better performance guarantees than under worst-case. The quintessential example is the secretary problem. If the sequence of elements is presented in uniformly random order there is an algorithm that picks the maximum value with probability 1/e, whereas no non-trivial performance guarantee is possible if the elements arrive in worst-case order. This work initiates an investigation into relaxations of the random-ordering hypothesis in online algorithms, by focusing on the secretary pr…
We study a twist on the classic secretary problem, which we term the secretary ranking problem: elements from an ordered set arrive in random order and instead of picking the maximum element, the algorithm is asked to assign a rank, or position, to each of the elements. The rank assigned is irrevocable and is given knowing only the pairwise comparisons with elements previously arrived. The goal is to minimize the distance of the rank produced to the true rank of the elements measured by the Kendall-Tau distance, which corresponds to the number of pairs that are inverted with respect to the tru…
Optimal Algorithms for Free Order Multiple-Choice Secretary
针对自由顺序多选秘书问题，提出熵为 $\Theta(\log\log n)$ 的顺序分布与确定性多阈值算法，同时达到近优竞争比和最优随机性，并证明匹配下界。
We present a number of positive and negative results for variants of the matroid secretary problem. Most notably, we design a constant-factor competitive algorithm for the "random assignment" model where the weights are assigned randomly to the elements of a matroid, and then the elements arrive on-line in an adversarial order (extending a result of Soto \cite{Soto11}). This is under the assumption that the matroid is known in advance. If the matroid is unknown in advance, we present an $O(\log r \log n)$-approximation, and prove that a better than $O(\log n / \log \log n)$ approximation is im…
The \emph{Temp Secretary Problem} was recently introduced by Fiat et al. It is a generalization of the Secretary Problem, in which commitments are temporary for a fixed duration. We present a simple online algorithm with improved performance guarantees for cases already considered by Fiat et al.\ and give competitive ratios for new generalizations of the problem. In the classical setting, where candidates have identical contract durations $γ\ll 1$ and we are allowed to hire up to $B$ candidates simultaneously, our algorithm is $(\frac{1}{2} - O(\sqrtγ))$-competitive. For large $B$, the bound i…
In the secretary problem we are faced with an online sequence of elements with values. Upon seeing an element we have to make an irrevocable take-it-or-leave-it decision. The goal is to maximize the probability of picking the element of maximum value. The most classic version of the problem is that in which the elements arrive in random order and their values are arbitrary. However, by varying the available information, new interesting problems arise. Also the case in which the arrival order is adversarial instead of random leads to interesting variants that have been considered in the literat…
In submodular $k$-secretary problem, the goal is to select $k$ items in a randomly ordered input so as to maximize the expected value of a given monotone submodular function on the set of selected items. In this paper, we introduce a relaxation of this problem, which we refer to as submodular $k$-secretary problem with shortlists. In the proposed problem setting, the algorithm is allowed to choose more than $k$ items as part of a shortlist. Then, after seeing the entire input, the algorithm can choose a subset of size $k$ from the bigger set of items in the shortlist. We are interested in unde…
The $k$-Fold Matroid Secretary Problem
本文提出了一种在 $k$-fold matroid 联合环境下，具有 $(1-O(\sqrt{\log(n)/k}))$ 竞争性的算法，扩展了 Kleinberg'05 的结果。
We extend the standard online worst-case model to accommodate past experience which is available to the online player in many practical scenarios. We do this by revealing a random sample of the adversarial input to the online player ahead of time. The online player competes with the expected optimal value on the part of the input that arrives online. Our model bridges between existing online stochastic models (e.g., items are drawn i.i.d. from a distribution) and the online worst-case model. We also extend in a similar manner (by revealing a sample) the online random-order model. We study the …
Beyond matroids: Secretary Problem and Prophet Inequality with general constraints
本文在任意向下封闭集合系统上推广先知不等式和秘书问题，给出{0,1}值下O(log n)竞争比算法及一般值下O(log n log r)竞争比算法，解决了Kleinberg提出的开放问题。
The Secretary Problem with Predicted Additive Gap
本文研究带预测加性差距的秘书问题，证明仅凭最高权重与第k高权重之差这一弱信息即可将竞争比提升至0.4，突破经典1/e障碍，并给出鲁棒性与一致性保证。
Tight Lower Bounds for the Multi-Secretary Problem via Bellman Certificates
针对多秘书问题中带间隙支撑的有界密度分布，证明最优遗憾至少为(\log T)^2，与现有上界匹配，并引入贝尔曼证书框架用于下界构造。
The secretary and the prophet inequality problems are central to the field of Stopping Theory. Recently, there has been a lot of work in generalizing these models to multiple items because of their applications in mechanism design. The most important of these generalizations are to matroids and to combinatorial auctions (extends bipartite matching). Kleinberg-Weinberg \cite{KW-STOC12} and Feldman et al. \cite{feldman2015combinatorial} show that for adversarial arrival order of random variables the optimal prophet inequalities give a $1/2$-approximation. For many settings, however, it's conceiv…
In the Secretary Problem, one has to hire the best among n candidates. The candidates are interviewed, one at a time, at a random order, and one has to decide on the spot, whether to hire a candidate or continue interviewing. It is well known that the best candidate can be hired with a probability of 1/e (Dynkin, 1963). Recent works extend this problem to settings in which multiple candidates can be hired, subject to some constraint. Here, one wishes to hire a set of candidates maximizing a given set function. Almost all extensions considered in the literature assume the objective set function…
The Secretary Problem with a Stochastic Precursor
本文研究了带随机前导的秘书问题，展示了预测仅因其到达时间而有价值。在随机顺序模型中，单个均匀时间的前导可使成功概率达到至少1/2，优于经典1/e的基准。在对抗性顺序模型中，足够集中的前导可恢复常数成功保证。
In the prophet secretary problem, $n$ values are drawn independently from known distributions, and presented in a uniformly random order. A decision-maker must accept or reject each value when it is presented, and may accept at most $k$ values in total. The objective is to maximize the expected sum of accepted values. We analyze the performance of static threshold policies, which accept the first $k$ values exceeding a fixed threshold (or all such values, if fewer than $k$ exist). We show that an appropriate threshold guarantees $γ_k = 1 - e^{-k}k^k/k!$ times the value of the offline optimal s…
Beating Competitive Ratio 4 for Graphic Matroid Secretary
本研究突破图拟阵秘书问题的4竞争比壁垒，提出竞争比3.95的新算法（简单图下达3.77），并证明常围长下可实现任意接近e的竞争比且该界是紧的，为强拟阵秘书猜想提供支撑。
Secretary Problems with Interactive Ordinal Queries
本文研究秘书问题中利用有限次序数查询改进决策，涵盖单选择与拟阵模型，给出各场景紧界，并证明查询可显著提升成功概率至最优。
In the Matroid Secretary Problem, introduced by Babaioff et al. [SODA 2007], the elements of a given matroid are presented to an online algorithm in random order. When an element is revealed, the algorithm learns its weight and decides whether or not to select it under the restriction that the selected elements form an independent set in the matroid. The objective is to maximize the total weight of the chosen elements. In the most studied version of this problem, the algorithm has no information about the weights beforehand. We refer to this as the zero information model. In this paper we stud…
In the \textit{Matroid Secretary Problem} (MSP), the elements of the ground set of a Matroid are revealed on-line one by one, each together with its value. An algorithm for the MSP is \textit{Matroid-Unknown} if, at every stage of its execution: (i) it only knows the elements that have been revealed so far and their values, and (ii) it has access to an oracle for testing whether or not any subset of the elements that have been revealed so far is an independent set. An algorithm is \textit{Known-Cardinality} if, in addition to (i) and (ii), it also initially knows the cardinality of the ground …
Secretary, Prophet, and Stochastic Probing via Big-Decisions-First
本文研究了在不确定性下三个基本问题：推荐问题、先知不等式和随机探测，针对一般向下闭合约束，提出大决策优先原则，解决二次差距问题，给出两种问题的$\tildeΩ(\log^2 n)$难性和第三种问题的$O(\log n)$近似算法。
In the matroid secretary problem, the elements of a matroid $\mathcal{M}$ arrive in random order. Once we observe an item we need to irrevocably decide whether or not to accept it. The set of selected elements should form an independent set of the matroid. The goal is to maximize the total sum of the values assigned to these elements. We introduce a version of this problem motivated by the shortlist model in [Agrawal et al.]. In this setting, the algorithm is allowed to choose a subset of items as part of a shortlist, possibly more than $k=rk(\mathcal{M})$ items. Then, after seeing the entire …
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。