scholarly journals Efficient Algorithms for Learning Revenue-Maximizing Two-Part Tariffs

2020 ◽  
Author(s):  
Maria-Florina Balcan ◽  
Siddharth Prasad ◽  
Tuomas Sandholm
Keyword(s):  
Cell Phone ◽  
High Probability ◽  
Optimal Solution ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Problem Instance ◽  
Lump Sum ◽  
Multiple Markets ◽  
Pricing Scheme ◽  
Generalization Theory

A two-part tariff is a pricing scheme that consists of an up-front lump sum fee and a per unit fee. Various products in the real world are sold via a menu, or list, of two-part tariffs---for example gym memberships, cell phone data plans, etc. We study learning high-revenue menus of two-part tariffs from buyer valuation data, in the setting where the mechanism designer has access to samples from the distribution over buyers' values rather than an explicit description thereof. Our algorithms have clear direct uses, and provide the missing piece for the recent generalization theory of two-part tariffs. We present a polynomial time algorithm for optimizing one two-part tariff. We also present an algorithm for optimizing a length-L menu of two-part tariffs with run time exponential in L but polynomial in all other problem parameters. We then generalize the problem to multiple markets. We prove how many samples suffice to guarantee that a two-part tariff scheme that is feasible on the samples is also feasible on a new problem instance with high probability. We then show that computing revenue-maximizing feasible prices is hard even for buyers with additive valuations. Then, for buyers with identical valuation distributions, we present a condition that is sufficient for the two-part tariff scheme from the unsegmented setting to be optimal for the market-segmented setting. Finally, we prove a generalization result that states how many samples suffice so that we can compute the unsegmented solution on the samples and still be guaranteed that we get a near-optimal solution for the market-segmented setting with high probability.

scholarly journals Scheduling Jobs and a Variable Maintenance on a Single Machine with Common Due-Date Assignment

2014 ◽  
Vol 2014 ◽  
pp. 1-5 ◽  
Author(s):  
Long Wan
Keyword(s):  
Single Machine ◽  
Optimal Solution ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Due Date ◽  
Due Date Assignment ◽  
Common Due Date ◽  
Optimal Polynomial ◽  
Special Case ◽  
Identical Jobs

We investigate a common due-date assignment scheduling problem with a variable maintenance on a single machine. The goal is to minimize the total earliness, tardiness, and due-date cost. We derive some properties on an optimal solution for our problem. For a special case with identical jobs we propose an optimal polynomial time algorithm followed by a numerical example.

scholarly journals Communication-Based Decomposition Mechanisms for Decentralized MDPs

2008 ◽  
Vol 32 ◽  
pp. 169-202 ◽  
Author(s):  
C. V. Goldman ◽  
S. Zilberstein
Keyword(s):  
Search Algorithm ◽  
Single Agent ◽  
Real Life ◽  
Optimal Solution ◽  
Approximate Solutions ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Markov Decision Problem ◽  
Heuristic Search Algorithm ◽  
Markov Decision

Multi-agent planning in stochastic environments can be framed formally as a decentralized Markov decision problem. Many real-life distributed problems that arise in manufacturing, multi-robot coordination and information gathering scenarios can be formalized using this framework. However, finding the optimal solution in the general case is hard, limiting the applicability of recently developed algorithms. This paper provides a practical approach for solving decentralized control problems when communication among the decision makers is possible, but costly. We develop the notion of communication-based mechanism that allows us to decompose a decentralized MDP into multiple single-agent problems. In this framework, referred to as decentralized semi-Markov decision process with direct communication (Dec-SMDP-Com), agents operate separately between communications. We show that finding an optimal mechanism is equivalent to solving optimally a Dec-SMDP-Com. We also provide a heuristic search algorithm that converges on the optimal decomposition. Restricting the decomposition to some specific types of local behaviors reduces significantly the complexity of planning. In particular, we present a polynomial-time algorithm for the case in which individual agents perform goal-oriented behaviors between communications. The paper concludes with an additional tractable algorithm that enables the introduction of human knowledge, thereby reducing the overall problem to finding the best time to communicate. Empirical results show that these approaches provide good approximate solutions.

scholarly journals A Maximal Tractable Class of Soft Constraints

2004 ◽  
Vol 22 ◽  
pp. 1-22 ◽  
Author(s):  
D. Cohen ◽  
M. Cooper ◽  
P. Jeavons ◽  
A. Krokhin
Keyword(s):  
Artificial Intelligence ◽  
Computational Complexity ◽  
Polynomial Time ◽  
Optimal Solution ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Soft Constraints ◽  
Soft Constraint ◽  
Binary Constraints ◽  
Binary Constraint

Many researchers in artificial intelligence are beginning to explore the use of soft constraints to express a set of (possibly conflicting) problem requirements. A soft constraint is a function defined on a collection of variables which associates some measure of desirability with each possible combination of values for those variables. However, the crucial question of the computational complexity of finding the optimal solution to a collection of soft constraints has so far received very little attention. In this paper we identify a class of soft binary constraints for which the problem of finding the optimal solution is tractable. In other words, we show that for any given set of such constraints, there exists a polynomial time algorithm to determine the assignment having the best overall combined measure of desirability. This tractable class includes many commonly-occurring soft constraints, such as 'as near as possible' or 'as soon as possible after', as well as crisp constraints such as 'greater than'. Finally, we show that this tractable class is maximal, in the sense that adding any other form of soft binary constraint which is not in the class gives rise to a class of problems which is NP-hard.

scholarly journals An Efficient Polynomial Time Algorithm for a Class of Generalized Linear Multiplicative Programs with Positive Exponents

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Bo Zhang ◽  
YueLin Gao ◽  
Xia Liu ◽  
XiaoLi Huang
Keyword(s):  
Polynomial Time ◽  
Outer Space ◽  
Optimal Solution ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Acceleration Technique ◽  
Region Division ◽  
Positive Exponent ◽  
Linear Programming Problems ◽  
Multiplicative Programs

This paper explains a region-division-linearization algorithm for solving a class of generalized linear multiplicative programs (GLMPs) with positive exponent. In this algorithm, the original nonconvex problem GLMP is transformed into a series of linear programming problems by dividing the outer space of the problem GLMP into finite polynomial rectangles. A new two-stage acceleration technique is put in place to improve the computational efficiency of the algorithm, which removes part of the region of the optimal solution without problems GLMP in outer space. In addition, the global convergence of the algorithm is discussed, and the computational complexity of the algorithm is investigated. It demonstrates that the algorithm is a complete polynomial time approximation scheme. Finally, the numerical results show that the algorithm is effective and feasible.

scholarly journals The Minimum Spectral Radius of an Edge-Removed Network: A Hypercube Perspective

2017 ◽  
Vol 2017 ◽  
pp. 1-8
Author(s):  
Yingbo Wu ◽  
Tianrui Zhang ◽  
Shan Chen ◽  
Tianhui Wang
Keyword(s):  
Spectral Radius ◽  
Polynomial Time ◽  
Minimization Problem ◽  
Heuristic Algorithms ◽  
Optimal Solution ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Regular Networks ◽  
Insight Into ◽  
Symmetric Networks

The spectral radius minimization problem (SRMP), which aims to minimize the spectral radius of a network by deleting a given number of edges, turns out to be crucial to containing the prevalence of an undesirable object on the network. As the SRMP is NP-hard, it is very unlikely that there is a polynomial-time algorithm for it. As a result, it is proper to focus on the development of effective and efficient heuristic algorithms for the SRMP. For that purpose, it is appropriate to gain insight into the pattern of an optimal solution to the SRMP by means of checking some regular networks. Hypercubes are a celebrated class of regular networks. This paper empirically studies the SRMP for hypercubes with two/three/four missing edges. First, for each of the three subproblems of the SRMP, a candidate for the optimal solution is presented. Second, it is shown that the candidate is optimal for small-sized hypercubes, and it is shown that the proposed candidate is likely to be optimal for medium-sized hypercubes. The edges in each candidate are evenly distributed over the network, which may be a common feature of all symmetric networks and hence is instructive in designing effective heuristic algorithms for the SRMP.

scholarly journals Reallocating Multiple Facilities on the Line

2019 ◽  
Author(s):  
Dimitris Fotakis ◽  
Loukas Kavouras ◽  
Panagiotis Kostopanagiotis ◽  
Philip Lazos ◽  
Stratis Skoulakis ◽  
...  
Keyword(s):  
Real Line ◽  
Optimal Solution ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Facility Locations ◽  
Interesting Connection ◽  
Server Problem ◽  
Special Case ◽  
Single Facility ◽  
Connection Cost

We study the multistage K-facility reallocation problem on the real line, where we maintain K facility locations over T stages, based on the stage-dependent locations of n agents. Each agent is connected to the nearest facility at each stage, and the facilities may move from one stage to another, to accommodate different agent locations. The objective is to minimize the connection cost of the agents plus the total moving cost of the facilities, over all stages. K-facility reallocation problem was introduced by (B.D. Kaijzer and D. Wojtczak, IJCAI 2018), where they mostly focused on the special case of a single facility. Using an LP-based approach, we present a polynomial time algorithm that computes the optimal solution for any number of facilities. We also consider online K-facility reallocation, where the algorithm becomes aware of agent locations in a stage-by stage fashion. By exploiting an interesting connection to the classical K-server problem, we present a constant-competitive algorithm for K = 2 facilities.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

scholarly journals Metric Dimension Parameterized By Treewidth

Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit
Keyword(s):  
Computable Function ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Size ◽  
Metric Dimension ◽  
Resolving Set ◽  
Input Graph ◽  
Outerplanar Graphs ◽  
Bounded Treewidth ◽  
Fpt Algorithm

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.

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