We study truthful mechanisms for approximating the Maximin-Share (MMS) value 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 a 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, our mechanism gives a 2-approximation to the MMS (consistency), and when the prediction is off, our mechanism still obtains 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 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 ≥ 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 ensure consistency and robustness, come into play.
Ordinal classification tasks that require the allocation of limited resources are prevalent in various realworld 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/hybridcost-sensitive-ml-optimization.
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 ℓp-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 learning-augmented setting, i.e., where the algorithm has access to some additional (machine-learned) information about the problem instance. We introduce a 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, ℓp-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 explore various generalizations of the online matching problem in a bipartite graph G as the b-matching problem~, the allocation problem~, and the AdWords problem~ in a beyond-worst-case setting. Specifically, we assume that G is a (k, d)-bounded degree graph, introduced by Naor and Wajc . 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 $\tau(k,d) = 1 - (1-\nicefrac{k}{d})\cdot({1 - \nicefrac{1}{d}})^{d - k}$ for the b-matching and allocation problems. We exploit primal-dual schemes 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.
We study the following balls and bins stochastic process: There is a buffer with $B$ bins, and there is a stream of balls $X = \langle X_1, X_2, \ldots ,X_T \rangle$ such that $X_i$ is the number of balls that arrive before time $i$ but after time $i-1$. Once a ball arrives, it is stored in one of the unoccupied bins. If all the bins are occupied then the ball is thrown away. In each time step, we select a bin uniformly at random, clear it, and gain its content. Once the stream of balls ends, all the remaining balls in the buffer are cleared and added to our gain. We are interested in analyzing the expected gain of this randomized process with respect to that of an optimal gain-maximizing strategy, which gets the same online stream of balls, and clears a ball from a bin, if exists, at any step. We name this gain ratio the loss of serving in the dark. In this paper, we determine the exact loss of serving in the dark. We prove that the expected gain of the randomized process is worse by a factor of $\rho + \epsilon$ from that of the optimal gain-maximizing strategy where $\epsilon = O(\nicefrac{1}{B^{1/3}})$ and $\rho = \max_{\alpha > 1} \alpha e^\alpha/((\alpha-1)e^\alpha + e - 1) \approx 1.69996$. We also demonstrate that this bound is essentially tight as there are specific ball streams for which the above-mentioned gain ratio tends to $\rho$. Our stochastic process occurs naturally in packets scheduling and mechanisms design applications.
Classification tasks aiming to minimize misclassification costs that involve allocation of scarce resources are common in many real-world problems such as allocation of organ transplants to patients, budget allocations for direct advertising, and classification of machines that need maintenance when there is a maintenance capacity limit. We propose a comprehensive analytic framework for scenarios that, in addition to including multi-class classification problems with misclassification costs, also have constraints on the number of classified samples of classes due to resource limitations. To classify samples under the constraints, the framework uses a probability matrix generated by a trained cost-sensitive classifier as the input for an optimization model with a minimum cost objective and resource allocation constraints. To illustrate its effectiveness and applicability, the framework with a cost-sensitive neural network was applied in the context of a medical resources allocation case study. The proposed framework performs significantly better than the alternative common approach with a cost-insensitive classifier. Our results show that the proposed framework is capable of providing an effective limitedresource allocation for misclassification cost problems.
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 subgraph 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.
In this work, we consider online d-dimensional vector bin packing. It is known that no algorithm can have a competitive ratio of O(d/\log^2 d) in the absolute sense, although upper bounds for this problem have always been presented in the asymptotic sense. Since variants of bin packing are traditionally studied with respect to the asymptotic measure, and since the two measures are different, we focus on the asymptotic measure and prove new lower bounds of the asymptotic competitive ratio. The existing lower bounds prior to this work were known to be smaller than 3, even for very large d. Here, we significantly improved on the best known lower bounds of the asymptotic competitive ratio (and as a byproduct, on the absolute competitive ratio) for online vector packing of vectors with d \geq 3 dimensions, for every dimension d. To obtain these results, we use several different constructions, one of which is an adaptive construction with a lower bound of \Omega(\sqrt{d}). Our main result is that the lower bound of \Omega(d/\log^2 d) on the competitive ratio holds also in the asymptotic sense. This result holds also against randomized algorithms, and requires a careful adaptation of constructions for online coloring, rather than simple black-box reductions.
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 \mathcal{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.
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.
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((log d)/(log log d)) (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.
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(logn), and they conjectured the existence of online algorithms using Δ(1 + o(1)) colors for Δ = 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))Δ-edge-coloring algorithm for Δ = ω(logn) known a priori. Surprisingly, if Δ is not known ahead of time, we show that no $\big(\frac{e}{e-1} - \Omega(1) \big) \Delta$-edge-coloring algorithm exists. We then provide an optimal, $\big(\frac{e}{e-1}+o(1)\big)\Delta$-edge-coloring algorithm for unknown Δ = 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.
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(lognR) resource augmentation suffices to obtain O(1) approximation on trees and O(lognR) 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(lognR) 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 + ε budget, whose adaptivity gaps on trees and general graphs are 1 + ε and 8 + ε, respectively. Finally, we provide additional results on the structure and the complexity of nonadaptive and adaptive strategies.
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 , 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 α-competitive dynamic pricing scheme exists on the tree if and only if there exists α-competitive online algorithm on the tree that is lazy, local, and monotone. Given this characterization, the main technical difficulty is coming up with such an online algorithm.
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