Competitive Algorithms for Unbounded One-Way Trading

We study an online inventory trading problem where a user seeks to maximize the aggregate revenue of trading multiple inventories over a time horizon. The trading constraints and concave revenue functions are revealed sequentially in time, and the user needs to make irrevocable decisions. The problem has wide applications in various engineering domains. Existing works employ the primal-dual framework to design online algorithms with sub-optimal, albeit near-optimal, competitive ratios (CR). We exploit the problem structure to develop a new divide-and-conquer approach to solve the online multi-inventory problem by solving multiple calibrated single-inventory ones separately and combining their solutions. The approach achieves the optimal CR of łn θ + 1 if Nłeq łn θ + 1, where N is the number of inventories and θ represents the revenue function uncertainty; it attains a CR of 1/[1-e^-1/(łnθ+1) ] in [łn θ +1, łn θ +2) otherwise. The divide-and-conquer approach reveals novel structural insights for the problem, (partially) closes a gap in existing studies, and generalizes to broader settings. For example, it gives an algorithm with a CR within a constant factor to the lower bound for a generalized one-way trading problem with price elasticity with no previous results. When developing the above results, we also extend a recent CR-Pursuit algorithmic framework and introduce an online allocation problem with allowance augmentation, both of which can be of independent interest.

We study an online inventory trading problem where a user seeks to maximize the aggregate revenue of trading multiple inventories over a time horizon. The trading constraints and concave revenue functions are revealed sequentially in time, and the user needs to make irrevocable decisions. The problem has wide applications in various engineering domains. Existing works employ the primal-dual framework to design online algorithms with sub-optimal, albeit near-optimal, competitive ratios (CR). We exploit the problem structure to develop a new divide-and-conquer approach to solve the online multi-inventory problem by solving multiple calibrated single-inventory ones separately and combining their solutions. The approach achieves the optimal CR of $łn θ + 1$ if $Nłeq łn θ + 1$, where N is the number of inventories and θ represents the revenue function uncertainty; it attains a CR of $1/[1-e^-1/(łnθ+1) ] \in [łn θ +1, łn θ +2)$ otherwise. The divide-and-conquer approach reveals novel structural insights for the problem, (partially) closes a gap in existing studies, and generalizes to broader settings. For example, it gives an algorithm with a CR within a constant factor to the lower bound for a generalized one-way trading problem with price elasticity with no previous results. When developing the above results, we also extend a recent CR-Pursuit algorithmic framework and introduce an online allocation problem with allowance augmentation, both of which can be of independent interest.

Delayed Information and Action in On-Line Algorithms

This paper develops competitive bidding strategies for an online linear optimization problem with inventory management constraints in both cost minimization and profit maximization settings. In the minimization problem, a decision maker should satisfy its time-varying demand by either purchasing units of an asset from the market or producing them from a local inventory with limited capacity. In the maximization problem, a decision maker has a time-varying supply of an asset that may be sold to the market or stored in the inventory to be sold later. In both settings, the market price is unknown in each timeslot and the decision maker can submit a finite number of bids to buy/sell the asset. Once all bids have been submitted, the market price clears and the amount bought/sold is determined based on the clearing price and submitted bids. From this setup, the decision maker must minimize/maximize their cost/profit in the market, while also devising a bidding strategy in the face of an unknown clearing price. We propose DEMBID and SUPBID, two competitive bidding strategies for these online linear optimization problems with inventory management constraints for the minimization and maximization setting respectively. We then analyze the competitive ratios of the proposed algorithms and show that the performance of our algorithms approaches the best possible competitive ratio as the maximum number of bids increases. As a case study, we use energy data traces from Akamai data centers, renewable outputs from NREL, and energy prices from NYISO to show the effectiveness of our bidding strategies in the context of energy storage management for a large energy customer participating in a real-time electricity market.

The "green" tariff laws in effect in Ukraine were adopted without the necessary technical and economic justification. These laws provide RES owners a wide range of preferences, which in total lead to hypertrophied losses of the Ukrainian energy market. In the process of the state developing a set of measures to radically improve the existing situation, it is necessary to have a wide range of methods of system analysis, including the determination of price indices on the energy market. In the paper a new method and a corresponding mathematical model for determining price indices on energy markets are proposed and tested on real data in conditions of solar and wind power plants functioning in the integrated energy system structure and when "green" tariff laws actions. The model determines the size of the electricity production sector, where competitive relations are maintained. Then the total costs of the energy market are calculated for the summary production of electricity in the power system at both market and fixed prices. The new price is defined as the ratio of total costs to the capacity of electricity production in the sector with competitive relations. The price index is calculated as the ratio of the new price to the market price recorded on the energy market in the base year. Calculations based on real data of the Ukraine’s IES for 2021 showed that if there were no RES in the energy system structure, the market price for electricity would be 2.17 times lower than the actual price. Research of the price index on the energy market of Ukraine at the development of its IES in accordance with the approved government documents showed that at the level of 2030 the average price for electricity will increase by 3.74 times. The country's economy will not be able to withstand such an electricity prices increasing. To solve this problem of national importance, a fundamentally new approach must be implemented. One of the effective solutions is the organization of RES functioning based on the principle of self-sufficiency, when RES ensures frequency and power stability with its own energy. Keywords: market price, price index, energy system, energy market, preference, payback.

The peak-demand charge motivates large-load customers to flatten their demand curves, while their self-owned renewable generations aggravate demand fluctuations. Thus, it is attractive to utilize energy storage for shaping real-time loads and reducing electricity bills. In this paper, we propose the first peak-aware competitive online algorithm for leveraging stored energy (e.g., in fuel cells) to minimize peak-demand charges. Our algorithm decides the discharging quantity slot by slot to maintain the optimal worst-case performance guarantee (namely, competitive ratio) among all deterministic online algorithms. Interestingly, we show that the best competitive ratio can be computed by solving a linear number of linear-fractional problems. We can also extend our competitive algorithm and analysis to improve the average-case performance and consider short-term prediction.

We develop a new approach for online network design and obtain improved competitive ratios for several problems. Our approach gives natural deterministic algorithms and simple analyses. At the heart of our work is a novel application of embeddings into hierarchically well-separated trees (HSTs) to the analysis of online network design algorithms --- we charge the cost of the algorithm to the cost of the optimal solution on any HST embedding of the terminals. This analysis technique is widely applicable to many problems and gives a unified framework for online network design. In a sense, our work brings together two of the main approaches to online network design. The first uses greedy-like algorithms and analyzes them using dual-fitting. The second uses tree embeddings and results in randomized $O(\log n)$-competitive algorithms, where $n$ is the total number of vertices in the graph. Our approach uses deterministic greedy-like algorithms but analyzes them via HST embeddings of the terminals. Our proofs are simpler as we do not need to carefully construct dual solutions and we get $O(\log k)$ competitive ratios, where $k$ is the number of terminals. In this paper, we apply our approach to obtain deterministic $O(\log k)$-competitive online algorithms for the following problems. - Steiner network with edge duplication. Previously, only a randomized $O(\log n)$-competitive algorithm was known. - Rent-or-buy. Previously, only deterministic $O(\log^2 k)$-competitive and randomized $O(\log k)$-competitive algorithms by Awerbuch, Azar and Bartal (2004) were known. - Connected facility location. Previously, only a randomized $O(\log^2 k)$-competitive algorithm by San Felice, Williamson and Lee (2014) was known. - Prize-collecting Steiner forest. We match the competitive ratio first achieved by Qian and Williamson (2011) and give a simpler analysis.

One-way trading is a fundamental problem in the online algorithms. A seller has some product to be sold to a sequence of buyers \(\{u_1, u_2, \dots \}\) in an online fashion and each buyer \(u_i\) is associated with his accepted unit price \(p_i\), which is known to the seller on the arrival of \(u_i\). The seller needs to decide the amount of products to be sold to \(u_i\) at the then-prevailing price \(p_i\). The objective is to maximize the total revenue of the seller. In this paper, we study the unbounded one-way trading, i.e., the highest unit price among all buyers is unbounded. We also assume that the highest prices of buyers follow some distribution with monotone hazard rate, which is well-adopted in Economics. We investigate two variants, (1) the distribution is on the highest price among all buyers, and (2) a general variant that the prices of buyers is independent and identically distributed. To measure the performance of the algorithms, the expected competitive ratios, \(\mathrm {E}[OPT]/\mathrm {E}[ALG]\) and \(\mathrm {E}[OPT/ALG]\), are considered and constant-competitive algorithms are given if the distributions satisfy the monotone hazard rate.

This paper aims to evaluate and improve the usefulness of publicly available electricity market prices for real-time optimal dispatching (RTOD) of a privately owned energy storage system (ESS) in a competitive electricity market. The RTOD algorithm seeks to maximize the revenue by exploiting arbitrage opportunities available due to the inter-temporal variation of electricity prices in the day-ahead market. The pre-dispatch prices, issued by the Ontario independent electricity system operator, and the corresponding ex-post hourly Ontario energy prices are employed as the forecast and the actual prices. A compressed-air ESS is sized and employed for evaluations due to its lower capital expenditure and its ability to be positively influenced by the availability of waste heat. First, the conventional RTOD algorithm is developed by formulating a mixed integer linear programming problem. It is demonstrated that the forecast inaccuracy of publicly available market prices significantly reduces the ESS revenue. Then, a new adaptive algorithm is proposed and evaluated which adapts the objective function of the optimization problem online based on historical market prices available before real-time. The outcomes reveal that the proposed adaptive RTOD can significantly increase the ESS revenue compared to the conventional algorithm as well as the back-casting method proposed in prior studies.

We study an online task assignment problem for multi-robot systems where robots can do multiple tasks during their mission and the tasks arrive dynamically in groups. Each robot can do at most one task from a group and the total number of tasks a robot can do is bounded by its limited battery life. There is a payoff for assigning each robot to a task and the objective is to maximize the total payoff. A special case, where each group has one task and each robot can do one task is the online maximum weighted bipartite matching problem (MWBMP). For online MWBMP, it is known that, under some assumptions on the payoffs, a greedy algorithm has a competitive ratio of 1 over 3. Our key result is to prove that for the general problem, under the same assumptions on the payoff as in MWBMP and an assumption on the number of tasks arising in each group, a repeated auction algorithm, where each group of tasks is (near) optimally allocated to the available group of robots has a guaranteed competitive ratio. We also prove that (a) without the assumptions on the payoffs, it is impossible to design an algorithm with any performance guarantee and (b) without the assumption on the task profile, the algorithms that can guarantee a feasible allocation (if one exists) have arbitrarily bad performance in the worst case. Additionally, we present simulation results depicting the average case performance of the repeated greedy auction algorithm.

We study the k -server problem in the resource augmentation setting, i.e., when the performance of the online algorithm with k servers is compared to the offline optimal solution with h ≤ k servers. The problem is very poorly understood beyond uniform metrics. For this special case, the classic k -server algorithms are roughly (1+1/ϵ)-competitive when k =(1+ϵ) h , for any ϵ > 0. Surprisingly, however, no o ( h )-competitive algorithm is known even for HSTs of depth 2 and even when k / h is arbitrarily large. We obtain several new results for the problem. First, we show that the known k -server algorithms do not work even on very simple metrics. In particular, the Double Coverage algorithm has competitive ratio Ω ( h ) irrespective of the value of k , even for depth-2 HSTs. Similarly, the Work Function Algorithm, which is believed to be optimal for all metric spaces when k = h , has competitive ratio Ω ( h ) on depth-3 HSTs even if k =2 h . Our main result is a new algorithm that is O (1)-competitive for constant depth trees, whenever k =(1+ϵ) h for any ϵ > 0. Finally, we give a general lower bound that any deterministic online algorithm has competitive ratio at least 2.4 even for depth-2 HSTs and when k / h is arbitrarily large. This gives a surprising qualitative separation between uniform metrics and depth-2 HSTs for the ( h , k )-server problem.

Online algorithms are a hallmark of worst case optimization under uncertainty. On the other hand, in practice, the input is often far from worst case, and has some predictable characteristics. A recent line of work has shown how to use machine learned predictions to circumvent strong lower bounds on competitive ratios in classic online problems such as ski rental and caching. We study how predictive techniques can be used to break through worst case barriers in online scheduling. The makespan minimization problem with restricted assignments is a classic problem in online scheduling theory. Worst case analysis of this problem gives Ω(log m) lower bounds on the competitive ratio in the online setting. We identify a robust quantity that can be predicted and then used to guide online algorithms to achieve better performance. Our predictions are compact in size, having dimension linear in the number of machines, and can be learned using standard off the shelf methods. The performance guarantees of our algorithms depend on the accuracy of the predictions, given predictions with error η, we show how to construct O(log η) competitive fractional assignments. We then give an online algorithm that rounds any fractional assignment into an integral schedule. Our algorithm is O((log log m)3)-competitive and we give a nearly matching Ω(log log m) lower bound for online rounding algorithms.1 Altogether, we give algorithms that, equipped with predictions with error η, achieve O(log η (log log m)3) competitive ratios, breaking the Ω(log m) lower bound even for moderately accurate predictions.

We study the advice complexity and the random bit complexity of the online knapsack problem: Given a knapsack of unit capacity, and n items that arrive in successive time steps, an online algorithm has to decide for every item whether it gets packed into the knapsack or not. The goal is to maximize the value of the items in the knapsack without exceeding its capacity. In the model of advice complexity of online problems, one asks how many bits of advice about the unknown parts of the input are both necessary and sufficient to achieve a specific competitive ratio. It is well-known that even the unweighted online knapsack problem does not admit any competitive deterministic online algorithm. We show that a single bit of advice helps a deterministic algorithm to become 2-competitive, but that Ω(log n) advice bits are necessary to further improve the deterministic competitive ratio. This is the first time that such a phase transition for the number of advice bits has been observed for any problem. We also show that, surprisingly, instead of an advice bit, a single random bit allows for a competitive ratio of 2, and any further amount of randomness does not improve this. Moreover, we prove that, in a resource augmentation model, i.e., when allowing a little overpacking of the knapsack, a constant number of advice bits suffices to achieve a near-optimal competitive ratio. We also study the weighted version of the problem proving that, with O(log n) bits of advice, we can get arbitrarily close to an optimal solution and, using asymptotically fewer bits, we are not competitive.

On-Line Load Balancing of Temporary Tasks

Earlier publications indicate that the price of young wines cannot be estimated accurately through weather conditions. Considering that a majority of fine wine is sold in the form of wine futures, even before the wine is bottled, determining realistic prices for these wine futures accurately is one of the most critical decisions. Surveys of leading wine merchants and industry experts exhibit significant departures from the realized market prices. We develop a pricing model for fine wines using weather, market, and expert reviews. The financial exchange for fine wines called Liv‐ex has already adopted our estimated prices and tagged them as “realistic prices.” Our approach combines temperature, rainfall, market fluctuations, and tasting expert scores and leads to accurate estimations that the wine industry has not seen before. Our study shows that higher temperatures, lower levels of precipitation, appreciation in the Liv‐ex 100 index as a market indicator, and higher barrel scores increase market prices. We conduct a comprehensive set of robustness checks and show that the mean absolute deviation of actual market prices from our estimated prices is substantially smaller than any academic benchmark. Our realistic prices help create transparency in this highly opaque market. When compared with the realized prices, our realistic prices guide buyers (e.g., distributors, restaurateurs, merchants) in their purchase decisions as they can determine whether a wine is underpriced or overpriced.