Efficient Mechanism Design for Online Scheduling

We consider the following non-preemptive semi-online scheduling problem. Jobs with non-increasing sizes arrive one by one to be scheduled on two uniformly related machines, with the goal of minimizing the makespan. We analyze both the optimal overall competitive ratio, and the optimal competitive ratio as a function of the speed ratio (q ≥ 1) between the two machines. We show that the greedy algorithm LPT has optimal competitive ratio 1/4(1 + √17) ≈ 1.28 overall, but does not have optimal competitive ratio for every value of q. We determine the intervals of q where LPT is an algorithm of optimal competitive ratio, and design different algorithms of optimal competitive ratio for the intervals where it fails to be the best algorithm. As a result, we give a tight analysis of the competitive ratio for every speed ratio.KeywordsInput SequenceCompetitive RatioSpeed RatioIdentical MachineOnline ScheduleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We study competitive online algorithms for EV (electrical vehicle) charging under the scenario of an aggregator serving a large number of EVs together with its background load, using both its own renewable energy (for free) and the energy procured from the external grid. The goal of the aggregator is to minimize its peak procurement from the grid, subject to the constraint that each EV has to be fully charged before its deadline. Further, the aggregator can predict the future demand and the renewable energy supply with some levels of uncertainty. The key challenge here is how to develop a model that captures the prior knowledge from such prediction, and how to best utilize this prior knowledge to reduce the peak under future uncertainty. In this paper, we first propose a 2-level increasing precision model (2-IPM), to capture the system uncertainty. We develop a powerful computation approach that can compute the optimal competitive ratio under 2-IPM over any online algorithm, and also online algorithms that can achieve the optimal competitive ratio. A dilemma for online algorithm design is that an online algorithm with good competitive ratio may exhibit poor average-case performance. We then propose a new Algorithm-Robustification procedure that can convert an online algorithm with reasonable average-case performance to one with both the optimal competitive ratio and good average-case performance. The robustified version of a well-known heuristic algorithm, Receding Horizon Control (RHC), is found to demonstrate superior performance via trace-based simulations.

We consider the problem of on-line call admission and routing on trees and meshes. Previous work gave randomized on-line algorithms for these problems and proved that they have optimal (up to constant factors) competitive ratios. However, these algorithms can obtain very low profit with high probability. We investigate the question of devising for these problems on-line competitive algorithms that also guarantee a "good" solution with "good" probability. We give a new family of randomized algorithms with asymptotically optimal competitive ratios and "good" probability to get a profit close to the expectation. We complement these results by providing bounds on the probability of any optimally competitive randomized on-line algorithm for the problems we consider to get a profit close to the expectation. To the best of our knowledge, this is the first study of the relationship between the tail distribution and the competitive ratio of randomized on-line benefit algorithms.

Weighted group search on a line & implications to the priority evacuation problem

The principal problem in algorithmic mechanism design is in merging the incentive constraints imposed by selfish behavior with the algorithmic constraints imposed by computational intractability. This field is motivated by the observation that the preeminent approach for designing incentive compatible mechanisms, namely that of Vickrey, Clarke, and Groves; and the central approach for circumventing computational obstacles, that of approximation algorithms, are fundamentally incompatible: natural applications of the VCG approach to an approximation algorithm fails to yield an incentive compatible mechanism. We consider relaxing the desideratum of (ex post) incentive compatibility (IC) to Bayesian incentive compatibility (BIC), where truthtelling is a Bayes-Nash equilibrium (the standard notion of incentive compatibility in economics). For welfare maximization in single-parameter agent settings, we give a general black-box reduction that turns any approximation algorithm into a Bayesian incentive compatible mechanism with essentially the same approximation factor.

We prove that no online algorithm (even randomized, against an oblivious adversary) is better than 1/2-competitive for welfare maximization with coverage valuations, unless $NP = RP$. Since the Greedy algorithm is known to be 1/2-competitive for monotone submodular valuations, of which coverage is a special case, this proves that Greedy provides the optimal competitive ratio. On the other hand, we prove that Greedy in a stochastic setting with i.i.d.items and valuations satisfying diminishing returns is $(1-1/e)$-competitive, which is optimal even for coverage valuations, unless $NP=RP$. For online budget-additive allocation, we prove that no algorithm can be 0.612-competitive with respect to a natural LP which has been used previously for this problem.

We study the admission control problem in general networks. Communication requests arrive over time, and the online algorithm accepts or rejects each request while maintaining the capacity limitations of the network. The admission control problem has been usually analyzed as a benefit problem, where the goal is to devise an online algorithm that accepts the maximum number of requests possible. The problem with this objective function is that even algorithms with optimal competitive ratios may reject almost all of the requests, when it would have been possible to reject only a few. This could be inappropriate for settings in which rejections are intended to be rare events.In this paper, we consider preemptive online algorithms whose goal is to minimize the number of rejected requests. Each request arrives together with the path it should be routed on. We show an O(log2(mc))-competitive randomized algorithm for the weighted case, where m is the number of edges in the graph and c is the maximum edge capacity. For the unweighted case, we give an O(log m log c)-competitive randomized algorithm. This settles an open question of Blum, Kalai and Kleinberg raised in [10]. We note that allowing preemption and handling requests with given paths are essential for avoiding trivial lower bounds.The admission control problem is a generalization of the online set cover with repetitions problem, whose input is a family of m subsets of a ground set of n elements. Elements of the ground set are given to the online algorithm one by one, possibly requesting each element a multiple number of times. (If each element arrives at most once, this corresponds to the online set cover problem.) The algorithm must cover each element by different subsets, according to the number of times it has been requested.We give an O(log m log n)-competitive randomized algorithm for the online set cover with repetitions problem. This matches a recent lower bound of Ω(log m log n) given by Feige and Korman for the competitive ratio of any randomized polynomial time algorithm, under the BPP ≠ NP assumption. Given any constant e > 0, an O(log m logn)-competitive deterministic bicriteria algorithm is shown that covers each element by at least (1-e)k sets, where k is the number of times the element is covered by the optimal solution.

Opaque sales are a selling mechanism in which a seller offers multiple products but the buyer does not know which specific product they will receive until after the purchase. This mechanism is widely used in the tourism industry, such as staycation packages with undisclosed hotel options or tours to different destinations, and in e-commerce through mystery boxes. In many such settings, buyers operate under a unit demand constraint, meaning they derive benefit from only one item despite multiple options being available. In this paper, we formulate and solve a multi-item mechanism design problem for opaque sales. Our goal is to identify a distribution-free mechanism that maximizes the competitive ratio, defined as the ratio between the revenue generated by the mechanism and the maximum revenue attainable with full knowledge of the buyer's valuations. Despite the problem's infinite dimensionality, we show that the optimal mechanism admits a semi-analytical form, characterized by the solution of an auxiliary convex optimization problem whose size scales linearly with the number of items. Furthermore, we demonstrate that this mechanism can be equivalently implemented through a menu of infinite level-access lotteries, where the buyer pays an amount to access a randomly assigned subset of items, with higher payments granting access to a larger selection. We then analyze the impact of menu size limitations on the competitive ratio and conclude with a prototypical example illustrating our approach.

We study online combinatorial auctions with production costs proposed by Blum et al. using the online primal dual framework. In this model, buyers arrive online, and the seller can produce multiple copies of each item subject to a non-decreasing marginal cost per copy. The goal is to allocate items to maximize social welfare less total production cost. For arbitrary (strictly convex and differentiable) production cost functions, we characterize the optimal competitive ratio achievable by online mechanisms/algorithms. We show that online posted pricing mechanisms, which are incentive compatible, can achieve competitive ratios arbitrarily close to the optimal, and construct lower bound instances on which no online algorithms, not necessarily incentive compatible, can do better. Our positive results improve or match the results in several previous work, e.g., Bartal et al., Blum et al., and Buchbinder and Gonen. Our lower bounds apply to randomized algorithms and resolve an open problem by Buchbinder and Gonen.

On-line scheduling of parallel jobs with runtime restrictions

We show that efficient anonymous incentive compatible (dominant strategy) mechanisms for public goods eliminate externalities, i.e., each individual is unable to change the welfare of anyone else. The characterization is used to derive existence and non-existence results for models with a finite number of individuals and to explain existence results in the continuum. A similar characterization and conclusions are demonstrated for private goods in (J Econ Theory 85:169–225, 1999). However, unlike private goods, elimination of externalities with public goods implies that individuals cannot change the outcome. Hence, such mechanisms provide only weak incentives for truth-telling.

On the Coexistence Of Stability and Incentive Compatibility in Fractional Matchings

We study several variants of a fixed length ski rental problem and related scheduling problems with rejection. A ski season consists of m days, and an equipment of cost 1 is to be used during these days. The equipment can be bought on any day, in which case it can be used without any additional cost starting that day and until the vacation ends. On each day, the algorithm is informed with the current non-negative cost of renting the equipment. As long as the algorithm did not buy the equipment, it must rent it every day of the vacation, paying the rental cost of each day of rental. We consider the case of arbitrary, non-increasing, and non-decreasing rental costs. We consider the case where the season cannot end before the mth day, and the case that it can end without prior notice. We propose optimal online algorithms for all values of m for all variants. The optimal competitive ratios are either defined by solutions of equations (closed formulas or finite recurrences) or sets of mathematical programs, and tend to 2 as m grows.

We introduce the online stochastic Convex Programming (CP) problem, a very general version of stochastic online problems which allows arbitrary concave objectives and convex feasibility constraints. Many well-studied problems like online stochastic packing and covering, online stochastic matching with concave returns, etc. form a special case of online stochastic CP. We present fast algorithms for these problems, which achieve near-optimal regret guarantees for both the i.i.d. and the random permutation models of stochastic inputs. When applied to the special case online packing, our ideas yield a simpler and faster primal-dual algorithm for this well studied problem, which achieves the optimal competitive ratio. Our techniques make explicit the connection of primal-dual paradigm and online learning to online stochastic CP.

The problem of online matching with stochastic rewards is a generalization of the online bipartite matching problem where each edge has a probability of success. When a match is made it succeeds with the probability of the corresponding edge. Introducing this model, Mehta and Panigrahi (FOCS 2012) focused on the special case of identical edge probabilities. Comparing against a deterministic offline LP, they showed that the Ranking algorithm of Karp et al. (STOC 1990) is 0.534 competitive and proposed a new online algorithm with an improved guarantee of 0.567 for vanishingly small probabilities. For the case of vanishingly small but heterogeneous probabilities Mehta et al. (SODA 2015), gave a 0.534 competitive algorithm against the same LP benchmark. For the more general vertex-weighted version of the problem, to the best of our knowledge, no results being 1/2 were previously known even for identical probabilities. We focus on the vertex-weighted version and give two improvements. First, we show that a natural generalization of the Perturbed-Greedy algorithm of Aggarwal et al. (SODA 2011), is (1-1/e) competitive when probabilities decompose as a product of two factors, one corresponding to each vertex of the edge. This is the best achievable guarantee as it includes the case of identical probabilities and in particular, the classical online bipartite matching problem. Second, we give a deterministic 0.596 competitive algorithm for the previously well studied case of case of fully heterogeneous but vanishingly small edge probabilities. A key contribution of our approach is the use of novel path-based formulations and a generalization of the primal-dual scheme of Devanur et al. (SODA 2013). These allow us to compare against the natural benchmark of clairvoyant (offline) algorithms that know the sequence of arrivals and the edge probabilities in advance, but not the outcomes of potential matches. These ideas may be of independent interest in other online settings with post-allocation stochasticity.