Ordinal classification tasks that require the allocation of limited resources are prevalent in various real-world scenarios. Examples include assessing disease severity in the context of medical resource allocation and categorizing the quality of machines as good, medium, or bad to schedule maintenance treatment within capacity constraints. We propose a comprehensive analytic framework for scenarios that, in addition to including ordinal classification problems, also have constraints on the number of classified samples of classes due to resource limitations. The framework uses a probability matrix generated by a trained ordinal classifier as the input for an optimization model with a minimum misclassification cost objective and resource allocation constraints. We illustrated the equivalence between the formulation of the resource allocation problem into samples and the transportation problem, enabling the utilization of established transportation heuristics for our solution. To demonstrate the effectiveness and applicability of the framework, we applied it with various ordinal machine-learning models to both tabular data and image datasets. The proposed framework performs significantly better than the alternative common approach of using non-ordinal classifiers, achieving an average cost reduction of 1% with ordinal decision tree-based models and 4.4% with ordinal neural networks. Our results show that the proposed framework can provide an effective limited-resource allocation for ordinal classification problems. Our code is available at https://github.com/liorRabkin/hybrid-cost-sensitive-ml-optimization.

We study truthful mechanisms for approximating the Maximin-Share (MMS) allocation of agents with additive valuations for indivisible goods. Algorithmically, constant factor approximations exist for the problem for any number of agents. When adding incentives to the mix, a jarring result by Amanatidis, Birmpas, Christodoulou, and Markakis [EC 2017] shows that the best possible approximation for two agents and $m$ items is $\lfloor \frac{m}{2} \rfloor$. We adopt a learning-augmented framework to investigate what is possible when some prediction on the input is given. For two agents, we give a truthful mechanism that takes agents' ordering over items as prediction. When the prediction is accurate, we give a $2$-approximation to the MMS (consistency), and when the prediction is off, we still get an $\lceil \frac{m}{2} \rceil$-approximation to the MMS (robustness). We further show that the mechanism's performance degrades gracefully in the number of ``mistakes" in the prediction; i.e., we interpolate (up to constant factors) between the two extremes: when there are no mistakes, and when there is a maximum number of mistakes. We also show an impossibility result on the obtainable consistency for mechanisms with finite robustness. For the general case of $n\ge 2$ agents, we give a 2-approximation mechanism for accurate predictions, with relaxed fallback guarantees. Finally, we give experimental results which illustrate when different components of our framework, made to insure consistency and robustness, come into play.

We explore various generalizations of the online matching problem in a bipartite graph G as the b-matching problem [8], the allocation problem [5], and the AdWords problem [13] in a beyond-worst-case setting. Specifically, we assume that G is a ( k, d )-bounded degree graph, introduced by Naor and Wajc [14]. Such graphs model natural properties on the degrees of advertisers and queries in the allocation and AdWords problems. While previous work only considers the scenario where k ≥ d , we consider the interesting intermediate regime of k ≤ d and prove a tight competitive ratio as a function of k, d (under the small-bid assumption) of τ ( k, d ) = 1 − (1 − k / d ) · (1 − 1 / d ) d − k for the b -matching and allocation problems. We exploit primal-dual schemes [6, 3] to design and analyze the corresponding tight upper and lower bounds. Finally, we show a separation between the allocation and AdWords problems. We demonstrate that τ ( k, d ) competitiveness is impossible for the AdWords problem even in ( k, d )-bounded degree graphs.

Online allocation is a broad class of problems where items arriving online have to be allocated to agents who have a fixed utility/cost for each assigned item so to maximize/minimize some objective. This framework captures a broad range of fundamental problems such as the Santa Claus problem (maximizing minimum utility), Nash welfare maximization (maximizing geometric mean of utilities), makespan minimization (minimizing maximum cost), minimization of $\ell_p$-norms, and so on. We focus on divisible items (i.e., fractional allocations) in this paper. Even for divisible items, these problems are characterized by strong super-constant lower bounds in the classical worst-case online model. In this paper, we study online allocations in the {\em learning-augmented} setting, i.e., where the algorithm has access to some additional (machine-learned) information about the problem instance. We introduce a {\em general} algorithmic framework for learning-augmented online allocation that produces nearly optimal solutions for this broad range of maximization and minimization objectives using only a single learned parameter for every agent. As corollaries of our general framework, we improve prior results of Lattanzi et al. (SODA 2020) and Li and Xian (ICML 2021) for learning-augmented makespan minimization, and obtain the first learning-augmented nearly-optimal algorithms for the other objectives such as Santa Claus, Nash welfare, $\ell_p$-minimization, etc. We also give tight bounds on the resilience of our algorithms to errors in the learned parameters, and study the learnability of these parameters.

We consider the weighted completion time minimization problem for capacitated parallel machines, which is a fundamental problem in modern cloud computing environments. In our setting, the processed jobs may be of varying duration, require different resources, and be of unequal importance (weight). Each server (machine) can process multiple concurrent jobs up to its capacity. We study heuristic approaches with provable approximation guarantees and offer an algorithm that prioritizes the jobs with the smallest volume-by-weight ratio. We bound the algorithm's approximation ratio using a decreasing function of the ratio between the highest resource demand of any job and the server's capacity. Thereafter, we create a hybrid, constant approximation algorithm for two or more machines. We also develop a constant approximation algorithm for the case of a single machine. Via a numerical study and a mixed-integer linear program of the problem, we demonstrate the performance of the suggested algorithm with respect to the optimal solutions and alternative scheduling methods. We show that the suggested scheduling method can be applied to both offline and online problems that may arise in real-world settings. This research is the first, to the best of our knowledge, to propose a polynomial-time algorithm with a constant approximation ratio for minimizing the weighted sum of job completion times for capacitated parallel machines.

A contention resolution (CR) scheme is a basic algorithmic primitive, that deals with how to allocate items among a random set S of competing players, while maintaining various properties. We consider the most basic setting where a single item must be allocated to some player in S . Here, in a seminal work, Feige and Vondrak (2006) designed a fair CR scheme when the set S is chosen from a product distribution. We explore whether such fair schemes exist for arbitrary non-product distributions on sets of players S , and whether they admit simple algorithmic primitives. Moreover, can we go beyond fair allocation and design such schemes for all possible achievable allocations. We show that for any arbitrary distribution on sets of players S , and for any achievable allocation, there exist several natural CR schemes that can be succinctly described, are simple to implement and can be efficiently computed to any desired accuracy. We also characterize the space of achievable allocations for any distribution, give algorithms for computing an optimum fair allocation for arbitrary distributions, and describe other natural fair CR schemes for product distributions. These results are based on matrix scaling and various convex programming relaxations.

We consider the weighted completion time minimization problem for capacitated parallel machines, which is a fundamental problem in modern cloud computing environments. We study settings in which the processed jobs may have varying duration, resource requirements and importance (weight). Each server (machine) can process multiple concurrent jobs up to its capacity. Due to the problem’s 𝒩𝒫 -hardness, we study heuristic approaches with provable approximation guarantees. We first analyze an algorithm that prioritizes the jobs with the smallest volume-by-weight ratio. We bound its approximation ratio with a decreasing function of the ratio between the highest resource demand of any job to the server’s capacity. Then, we use the algorithm for scheduling jobs with resource demands equal to or smaller than 0.5 of the server’s capacity in conjunction with the classic weighted shortest processing time algorithm for jobs with resource demands higher than 0.5. We thus create a hybrid, constant approximation algorithm for two or more machines. We also develop a constant approximation algorithm for the case with a single machine. This research is the first, to the best of our knowledge, to propose a polynomial-time algorithm with a constant approximation ratio for minimizing the weighted sum of job completion times for capacitated parallel machines.

In recent years, the explosion of research on large-scale networks has been fueled to a large extent by the increasing availability of large, detailed network data sets. Specifically, exploration of social networks constitutes a growing field of research, as they generate a huge amount of data on a daily basis and are the main tool for networking, communications, and content sharing. Exploring these networks is resource-consuming (time, money, energy, etc.). Moreover, uncertainty is a crucial aspect of graph exploration since links costs are unknown in advance, e.g., creating a positive influence between two people in social networks. One approach to model this problem is the stochastic graph exploration problem [4], where, given a graph and a source vertex, rewards on vertices, and distributions for the costs of the edges. The goal is to probe a subset of the edges, so the total cost of the edges is at most some prespecified budget, and the sub-graph is connected, containing the source vertex, and maximizes the total reward of the spanned vertices. In this stochastic setting, an optimal probing strategy is likely to be adaptive, i.e., it may determine the next edge to probe based on the realized costs of the already probed edges. As computing such adaptive strategies is intractable [15], we focus on developing non-adaptive strategies, which fix a list of edges to probe in advance. A non-adaptive strategy would not be competitive versus the optimal adaptive one unless it uses a budget augmentation. The current results demand an augmentation factor, which depends logarithmically on the number of nodes. Such a factor is unrealistic in large-scale network scenarios. In this paper, we provide constant competitive non-adaptive strategies using only a constant budget augmentation for various scenarios.

We consider the online vector bin packing problem where n items specified by d-dimensional vectors must be packed in the fewest number of identical d-dimensional bins. Azar et al. (STOC'13) showed that for any online algorithm A, there exist instances I, such that A(I), the number of bins used by A to pack I, is Ω(d/log^2 d) times OPT(I), the minimal number of bins to pack I. However in those instances, OPT(I) was only O(log d), which left open the possibility of improved algorithms with better asymptotic competitive ratio when OPT(I) ≫ d. We rule this out by showing that for any arbitrary function q(·) and any randomized online algorithm A, there exist instances I such that E[A(I)] ≥ c· d/log^3d · OPT(I) + q(d), for some universal constant c.

In this paper, we consider the problem of assigning 2-dimensional vector jobs to identical machines online so to minimize the maximum load on any dimension of any machine. For arbitrary number of dimensions d , this problem is known as vector scheduling, and recent research has established the optimal competitive ratio as O (cid:0) log d loglog d (cid:1) (Im et al. FOCS 2015, Azar et al. SODA 2018). But, these results do not shed light on the situation for small number of dimensions, particularly for d = 2 which is of practical interest. In this case, a trivial analysis shows that the classic list scheduling greedy algorithm has a competitive ratio of 3. We show the following improvements over this baseline in this paper: We give an improved, and tight, analysis of the list scheduling algorithm establishing a competitive ratio of 8 / 3 for two dimensions. If the value of opt is known, we improve the competitive ratio to 9 / 4 using a variant of the classic best fit algorithm for two dimensions. For any fixed number of dimensions, we design an algorithm that is provably the best possible against a fractional optimum solution. This algorithm provides a proof of concept that we can simulate the optimal algorithm online up to the integrality gap of the natural LP relaxation of the problem.

Exploring large-scale networks is a time consuming and expensive task which is usually operated in a complex and uncertain environment. A crucial aspect of network exploration is the development of suitable strategies that decide which nodes and edges to probe at each stage of the process. To model this process, we introduce the stochastic graph exploration problem. The input is an undirected graph G = (V, E) with a source vertex s, stochastic edge costs drawn from a distribution πe, e ∈ E, and rewards on vertices of maximum value R. The goal is to find a set F of edges of total cost at most B such that the subgraph of G induced by F is connected, contains s, and maximizes the total reward. This problem generalizes the stochastic knapsack problem and other stochastic probing problems recently studied. Our focus is on the development of efficient nonadaptive strategies that are competitive against the optimal adaptive strategy. A major challenge is the fact that the problem has an Ω(n) adaptivity gap even on a tree of n vertices. This is in sharp contrast with O(1) adaptivity gap of the stochastic knapsack problem, which is a special case of our problem. We circumvent this negative result by showing that O(log nR) resource augmentation suffices to obtain O(1) approximation on trees and O(log nR) approximation on general graphs. To achieve this result, we reduce stochastic graph exploration to a memoryless process - the minesweeper problem - which assigns to every edge a probability that the process terminates when the edge is probed. For this problem, interesting in its own, we present an optimal polynomial time algorithm on trees and an O(log nR) approximation for general graphs. We study also the problem in which the maximum cost of an edge is a logarithmic fraction of the budget. We show that under this condition, there exist polynomial-time oblivious strategies that use 1 + budget, whose adaptivity gaps on trees and general graphs are 1 + and 8 + , respectively. Finally, we provide additional results on the structure and the complexity of nonadaptive and adaptive strategies.

Vizing's celebrated theorem asserts that any graph of maximum degree Δ admits an edge coloring using at most Δ+1 colors. In contrast, Bar-Noy, Motwani and Naor showed over a quarter century ago that the trivial greedy algorithm, which uses 2Δ-1 colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to low-degree graphs, with Δ=O(log n), and they conjectured the existence of online algorithms using Δ(1+o(1)) colors for Δ=ω(log n). Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS'03 and Bahmani et al., SODA'10). We resolve the above conjecture for adversarial vertex arrivals in bipartite graphs, for which we present a (1+o(1))Δ-edge-coloring algorithm for Δ=ω(log n) known a priori. Surprisingly, if Δ is not known ahead of time, we show that no (e/(e-1) - Ω(1)) Δ-edge-coloring algorithm exists. We then provide an optimal, (e/(e-1)+o(1)) Δ-edge-coloring algorithm for unknown Δ=ω(log n). To obtain our results, we study a nonstandard fractional relaxation for edge coloring, for which we present optimal fractional online algorithms and a near-lossless online rounding scheme, yielding our optimal randomized algorithms.

Vizing’s celebrated theorem asserts that any graph of maximum degree ∆ admits an edge coloring using at most ∆ + 1 colors. In contrast, Bar-Noy, Naor and Motwani showed over a quarter century that the trivial greedy algorithm, which uses 2∆ − 1 colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to low-degree graphs, with ∆ = O(log n), and they conjectured the existence of online algorithms using ∆(1 + o(1)) colors for ∆ = ω(logn). Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS’03 and Bahmani et al., SODA’10). We resolve the above conjecture for adversarial vertex arrivals in bipartite graphs, for which we present a (1+ o(1))∆-edge-coloring algorithm for ∆ = ω(logn) known a priori. Surprisingly, if ∆ is not known ahead of time, we show that no ( e e−1 −Ω(1) ) ∆-edge-coloring algorithm exists. We then provide an optimal, ( e e−1 + o(1) ) ∆-edge-coloring algorithm for unknown ∆ = ω(logn). Key to our results, and of possible independent interest, is a novel fractional relaxation for edge coloring, for which we present optimal fractional online algorithms and a near-lossless online rounding scheme, yielding our optimal randomized algorithms. Email address: ilanrcohen@gmail.com. Email address: pbh15@mails.tsinghua.edu.cn. Work done in part while the author was visiting Carnegie Mellon University. Email address: dwajc@cs.cmu.edu. Work done in part while the author was visiting EPFL. Supported in part by NSF grants CCF-1618280, CCF-1814603, CCF-1527110, NSF CAREER award CCF1750808 and Sloan Research Fellowship.

In this paper we consider the k-server problem where events are generated by selfish agents, known as the selfish k-server problem. In this setting, there is a set of k servers located in some metric space. Selfish agents arrive in an online fashion, each has a request located on some point in the metric space, and seeks to serve his request with the server of minimum distance to the request. If agents choose to serve their request with the nearest server, this mimics the greedy algorithm which has an unbounded competitive ratio. We propose an algorithm that associates a surcharge with each server independently of the agent to arrive (and therefore, yields a truthful online mechanism). An agent chooses to serve his request with the server that minimizes the distance to the request plus the associated surcharge to the server. This paper extends [Ilan Reuven Cohen et al., 2015], which gave an optimal k-competitive dynamic pricing scheme for the selfish k-server problem on the line. We give a k-competitive dynamic pricing algorithm for the selfish k-server problem on tree metric spaces, which matches the optimal online (non truthful) algorithm. We show that an alpha-competitive dynamic pricing scheme exists on the tree if and only if there exists alpha-competitive online algorithm on the tree that is lazy and monotone. Given this characterization, the main technical difficulty is coming up with such an online algorithm.

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.

We study randomized algorithms for the online vector bin packing and vector scheduling problems. For vector bin packing, we achieve a competitive ratio of (O) over tilde (d(1/B)), where d is the number of dimensions and B the size of a bin. This improves the previous bound of (O) over tilde (d(1/(B) - (1))) by a polynomial factor, and is tight up to logarithmic factors. For vector scheduling, we show a lower bound of Omega(log d/log log d) on the competitive ratio of randomized algorithms, which is the first result for randomized algorithms and is asymptotically tight. Finally, we analyze the widely used "power of two choices' algorithm for vector scheduling, and show that its competitive ratio is O(log log n + log d/log log d), which is optimal up to the additive O (log log n) term that also appears in the scalar version of this algorithm.

In this paper, we exploit linear programming duality in the online setting, where input arrives on the fly, from the unique perspective of designing lower bounds (i.e., hardness results) on the competitive ratio. In particular, we provide a systematic method (as opposed to ad hoc case analysis that is typically done) for obtaining online deterministic and randomized lower bounds on the competitive ratio for a wide variety of problems. We show the usefulness of our approach by providing new, tight hardness results for three diverse online problems: the Vector Bin Packing problem, Ad-auctions (and various online matching problems), and the Capital Investment problem. Our methods are sufficiently general that they can also be used to reconstruct existing lower bounds. Our approach is in stark contrast to previous works, which exploit linear programming duality to obtain positive results, often via the useful primal-dual scheme. We design a general recipe with the opposite aim of obtaining negative results via duality. The general idea behind our approach is to construct a parameterized family of primal linear programs based on a candidate collection of input sequences for proving the lower bound, where the objective function corresponds to optimizing the competitive ratio. Solving the parameterized family of primal linear programs optimally would yield a valid lower bound, but is a challenging task and limits the tools that can be applied, since analysis must be done precisely and exact optimality needs to be proved. To this end, we consider the corresponding parameterized family of dual linear programs and provide feasible solutions, where the objective function yields a lower bound on the competitive ratio. This opens up additional doors for analysis, including some of the techniques we employ (e.g., continuous analysis, differential equations, etc.), as we need not be so careful about exact optimality. We are confident that our methods can be successfully applied to produce many more lower bounds for a wide array of online problems.

Background: To test whether poor quality day-3 embryos can undergo successful blastulation and implantation.Methods: A prospective cohort study was conducted. Whether or not a good quality embryo was transferred on day-3, poor quality (rejected) embryos were further cultured and followed. The clinical outcome of each embryo was assessed.Results: A total of 694 rejected embryos (from 205 patients) were included, with a blastulation rate of 21.2% (147 embryos) compared to 64.2% general blastulation rate reported by our laboratory (P < 0.01). In a multivariate logistic regression model, only their grade on day-3 significantly affected blastulation (P = 0.01). A total of 97 embryos attained eligibility for fresh transfer or cryopreservation, only 6 of which resulted from a day-3 embryo scored < 2. Of these, 52 were transferred, resulting in 21 pregnancies (16 clinical and 5 chemical). In summary, 694 cultured embryos yielded 16 clinical pregnancies; a 2.3% clinical pregnancy rate.Conclusions: Low score day-3 embryos can result in successful blastulation and clinical pregnancies. However, the normal blastulation rate is poor.

We present online algorithms for covering and packing problems with (non-linear) convex objectives. The convex covering problem is defined as: min xϵ R + n f(x) s.t. Ax ≥ 1, where f:R + n → R + is a monotone convex function, and A is an m×n matrix with non-negative entries. In the online version, a new row of the constraint matrix, representing a new covering constraint, is revealed in each step and the algorithm is required to maintain a feasible and monotonically non-decreasing assignment x over time. We also consider a convex packing problem defined as: max yϵR+ m Σ j=1 m yj - g(A T y), where g:R + n →R + is a monotone convex function. In the online version, each variable yj arrives online and the algorithm must decide the value of yj on its arrival. This represents the Fenchel dual of the convex covering program, when g is the convex conjugate of f. We use a primal-dual approach to give online algorithms for these generic problems, and use them to simplify, unify, and improve upon previous results for several applications.

Online d-dimensional vector packing models many settings such as minimizing resources in data centers where jobs have multiple resource requirements (CPU, Memory, etc.). However, no online d-dimensional vector packing algorithm can achieve a competitive ratio better than d. Fortunately, in many natural applications, vectors are relatively small, and thus the lower bound does not hold. For sufficiently small vectors, an O(log d)-competitive algorithm was known. We improve this to a constant competitive ratio, arbitrarily close to e a 2.718, given that vectors are sufficiently small.We give improved results for the two dimensional case. For arbitrarily small vectors, the First Fit algorithm for two dimensional vector packing is no better than 2-competitive. We present a natural family of First Fit variants, and for optimized parameters get a competitive ratio a 1.48 for sufficiently small vectors.We improve upon the 1.48 competitive ratio -- not via a First Fit variant -- and give a competitive ratio arbitrarily close to 4/3 for packing small, two dimensional vectors. We show that no algorithm can achieve better than a 4/3 competitive ratio for two dimensional vectors, even if one allows the algorithm to split vectors among arbitrarily many bins.