We provide online algorithms for secretary matching in general weighted graphs, under the well-studied models of vertex and edge arrivals. In both models, edges are associated with arbitrary weights that are unknown from the outset, and are revealed online. Under vertex arrival, vertices arrive online in a uniformly random order; upon the arrival of a vertex $v$, the weights of edges from $v$ to all previously arriving vertices are revealed, and the algorithm decides which of these edges, if any, to include in the matching. Under edge arrival, edges arrive online in a uniformly random order; upon the arrival of an edge $e$, its weight is revealed, and the algorithm decides whether to include it in the matching or not. We provide a $5/12$-competitive algorithm for vertex arrival, and show it is tight. For edge arrival, we provide a $1/4$-competitive algorithm. Both results improve upon state of the art bounds for the corresponding settings. Interestingly, for vertex arrival, secretary matching in general graphs outperforms secretary matching in bipartite graphs with 1-sided arrival, where $1/e$ is the best possible guarantee.
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…
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…
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…
Most prior work on online matching problems has been with the flexibility of keeping some vertices unmatched. We study three related online matching problems with the constraint of matching every vertex, i.e., with no rejections. We adopt a model in which vertices arrive in uniformly random order and the non-negative edge-weights are arbitrary. For the capacitated online bipartite matching problem, in which the vertices of one side of the graph are offline and those of the other side arrive online, we give a 4.62-competitive algorithm when the capacity of each offline vertex is 2. For the onli…
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…
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…
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 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.
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…
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 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,…
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…
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…
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…
In combinatorial auctions, a designer must decide how to allocate a set of indivisible items amongst a set of bidders. Each bidder has a valuation function which gives the utility they obtain from any subset of the items. Our focus is specifically on welfare maximization, where the objective is to maximize the sum of valuations that the bidders place on the items that they were allocated (the valuation functions are assumed to be reported truthfully). We analyze an online problem in which the algorithm is not given the set of bidders in advance. Instead, the bidders are revealed sequentially i…
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…
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…
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…
We provide prophet inequality algorithms for online weighted matching in general (non-bipartite) graphs, under two well-studied arrival models, namely edge arrival and vertex arrival. The weight of each edge is drawn independently from an a-priori known probability distribution. Under edge arrival, the weight of each edge is revealed upon arrival, and the algorithm decides whether to include it in the matching or not. Under vertex arrival, the weights of all edges from the newly arriving vertex to all previously arrived vertices are revealed, and the algorithm decides which of these edges, if …
We consider the prophet inequality problem for (not necessarily bipartite) matching problems with independent edge values, under both edge arrivals and vertex arrivals. We show constant-factor prophet inequalities for the case where the online algorithm has only limited access to the value distributions through samples. First, we give a $16$-approximate prophet inequality for matching in general graphs under edge arrivals that uses only a single sample from each value distribution as prior information. Then, for bipartite matching and (one-sided) vertex arrivals, we show an improved bound of $…
We study a dynamic non-bipartite matching problem. There is a fixed set of agent types, and agents of a given type arrive and depart according to type-specific Poisson processes. Agent departures are not announced in advance. The value of a match is determined by the types of the matched agents. We present an online algorithm that is (1/8)-competitive with respect to the value of the optimal-in-hindsight policy, for arbitrary weighted graphs. Our algorithm treats agents heterogeneously, interpolating between immediate and delayed matching in order to thicken the market while still matching val…
We study the online stochastic matching problem. Consider a bipartite graph with offline vertices on one side, and with i.i.d.online vertices on the other side. The offline vertices and the distribution of online vertices are known to the algorithm beforehand. The realization of the online vertices, however, is revealed one at a time, upon which the algorithm immediately decides how to match it. For maximizing the cardinality of the matching, we give a $0.711$-competitive online algorithm, which improves the best previous ratio of $0.706$. When the offline vertices are weighted, we introduce a…
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 …
Design of {\it online algorithms} for assigning mobile users to basestations is considered with the objective of maximizing the sum-rate, when all users associated to any one basestation equally share each basestation's resources. Each user on its arrival reveals the rates it can obtain if connected to each of the basestations, and the problem is to assign each user to any one basestation irrevocably so that the sum-rate is maximized at the end of all user arrivals, without knowing the future user arrival or rate information or its statistics at each user arrival. Online algorithms with consta…
Online bipartite matching with edge arrivals remained a major open question for a long time until a recent negative result by [Gamlath et al. FOCS 2019], who showed that no online policy is better than the straightforward greedy algorithm, i.e., no online algorithm has a worst-case competitive ratio better than $0.5$. In this work, we consider the bipartite matching problem with edge arrivals in a natural stochastic framework, i.e., Bayesian setting where each edge of the graph is independently realized according to a known probability distribution. We focus on a natural class of prune & greed…
The online matching problem was introduced by Karp, Vazirani and Vazirani nearly three decades ago. In that seminal work, they studied this problem in bipartite graphs with vertices arriving only on one side, and presented optimal deterministic and randomized algorithms for this setting. In comparison, more general arrival models, such as edge arrivals and general vertex arrivals, have proven more challenging and positive results are known only for various relaxations of the problem. In particular, even the basic question of whether randomization allows one to beat the trivially-optimal determ…
An online truthful budgeted matching problem is considered for a bipartite graph, where the right vertices are available ahead of time, and individual left vertices arrive sequentially. On arrival of a left vertex, its edge utilities (or weights) to all the right vertices and a corresponding cost (or bid) are revealed. If a left vertex is matched to any of the right vertices, then it has to be paid at least as much as its cost. The problem is to match each left vertex instantaneously and irrevocably to any one of the right vertices, if at all, to find the maximum weight matching that is truthf…
We consider Online Minimum Bipartite Matching under the uniform metric. We show that Randomized Greedy achieves a competitive ratio equal to $(1+1/n) (H_{n+1}-1)$, which matches the lower bound. Comparing with the fact that RG achieves an optimal ratio of $Θ(\ln n)$ for the same problem but under the adversarial order, we find that the weaker arrival assumption of random order doesn't offer any extra algorithmic advantage for RG, or make the model strictly more tractable.
We study online secretary problems with returns in combinatorial packing domains with $n$ candidates that arrive sequentially over time in random order. The goal is to accept a feasible packing of candidates of maximum total value. In the first variant, each candidate arrives exactly twice. All $2n$ arrivals occur in random order. We propose a simple 0.5-competitive algorithm that can be combined with arbitrary approximation algorithms for the packing domain, even when the total value of candidates is a subadditive function. For bipartite matching, we obtain an algorithm with competitive ratio…
We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the…
The bipartite matching problem in the online and streaming settings has received a lot of attention recently. The classical vertex arrival setting, for which the celebrated Karp, Vazirani and Vazirani (KVV) algorithm achieves a $1-1/e$ approximation, is rather well understood: the $1-1/e$ approximation is optimal in both the online and semi-streaming setting, where the algorithm is constrained to use $n\cdot \log^{O(1)} n$ space. The more challenging the edge arrival model has seen significant progress recently in the online algorithms literature. For the strictly online model (no preemption) …
Online bipartite matching with one-sided arrival and its variants have been extensively studied since the seminal work of Karp, Vazirani, and Vazirani (STOC 1990). Motivated by real-life applications with dynamic market structures, e.g. ride-sharing, two generalizations of the classical one-sided arrival model are proposed to allow non-bipartite graphs and to allow all vertices to arrive online. Namely, online matching with general vertex arrival is introduced by Wang and Wong (ICALP 2015), and fully online matching is introduced by Huang et al. (JACM 2020). In this paper, we study the fractio…
Two related online problems: knapsack and truthful bipartite matching are considered. For these two problems, the common theme is how to `match' an arriving left vertex in an online fashion with any of the available right vertices, if at all, so as to maximize the sum of the value of the matched edges, subject to satisfying a sum-weight constraint on the matched left vertices. Assuming that the left vertices arrive in an uniformly random order (secretary model), two almost similar algorithms are proposed for the two problems, that are $2e$ competitive and $24$ competitive, respectively. The pr…
For online resource allocation problems, we propose a new demand arrival model where the sequence of arrivals contains both an adversarial component and a stochastic one. Our model requires no demand forecasting; however, due to the presence of the stochastic component, we can partially predict future demand as the sequence of arrivals unfolds. Under the proposed model, we study the problem of the online allocation of a single resource to two types of customers, and design online algorithms that outperform existing ones. Our algorithms are adjustable to the relative size of the stochastic comp…
Motivated by display advertising on the internet, the online stochastic matching problem is proposed by Feldman, Mehta, Mirrokni, and Muthukrishnan (FOCS 2009). Consider a stochastic bipartite graph with offline vertices on one side and with i.i.d. online vertices on the other side. The algorithm knows the offline vertices and the distribution of the online vertices in advance. Upon the arrival of each online vertex, its type is realized and the algorithm immediately and irrevocably decides how to match it. In the vertex-weighted version of the problem, each offline vertex is associated with a…
In this paper we present improved bounds for approximating maximum matchings in bipartite graphs in the streaming model. First, we consider the question of how well maximum matching can be approximated in a single pass over the input using $\tilde O(n)$ space, where $n$ is the number of vertices in the input graph. Two natural variants of this problem have been considered in the literature: (1) the edge arrival setting, where edges arrive in the stream and (2) the vertex arrival setting, where vertices on one side of the graph arrive in the stream together with all their incident edges. The la…
This work presents an optimally-competitive algorithm for the problem of maximum weighted online perfect bipartite matching with i.i.d. arrivals. In this problem, we are given a known set of workers, a distribution over job types, and non-negative utility weights for each pair of worker and job types. At each time step, a job is drawn i.i.d. from the distribution over job types. Upon arrival, the job must be irrevocably assigned to a worker and cannot be dropped. The goal is to maximize the expected sum of utilities after all jobs are assigned. We introduce DISPATCH, a 0.5-competitive, randomi…
