Approximation algorithms for the MAXSPACE advertisement problem

In the MAXSPACE problem, given a set of ads A, one wants to schedule a subset A' of A into K slots B_1, ..., B_K of size L. Each ad A_i in A has a size s_i and a frequency w_i. A schedule is feasible if the total size of ads in any slot is at most L, and each ad A_i in A' appears in exactly w_i slots. The goal is to find a feasible schedule that maximizes the sum of the space occupied by all slots. We introduce a generalization called MAXSPACE-R in which each ad A_i also has a release date r_i >= 1, and may only appear in a slot B_j with j >= r_i. We also introduce a generalization of MAXSPACE-R called MAXSPACE-RD in which each ad A_i also has a deadline d_i <= K, and may only appear in a slot B_j with r_i <= j <= d_i. These parameters model situations where a subset of ads corresponds to a commercial campaign with an announcement date that may expire after some defined period. We present a 1/9-approximation algorithm for MAXSPACE-R and a polynomial-time approximation scheme for MAXSPACE-RD when K is bounded by a constant. This is the best factor one can expect, since MAXSPACE is NP-hard, even if K = 2

Local-Search Based Heuristics for Advertisement Scheduling

In the MAXSPACE problem, given a set of ads A, one wants to place a subset A' of A into K slots B_1, ..., B_K of size L. Each ad A_i in A has size s_i and frequency w_i. A schedule is feasible if the total size of ads in any slot is at most L, and each ad A_i in A' appears in exactly w_i slots. The goal is to find a feasible schedule that maximizes the space occupied in all slots. We introduce MAXSPACE-RDWV, a MAXSPACE generalization with release dates, deadlines, variable frequency, and generalized profit. In MAXSPACE-RDWV each ad A_i has a release date r_i >= 1, a deadline d_i >= r_i, a profit v_i that may not be related with s_i and lower and upper bounds w^min_i and w^max_i for frequency. In this problem, an ad may only appear in a slot B_j with r_i <= j <= d_i, and the goal is to find a feasible schedule that maximizes the sum of values of scheduled ads. This paper presents some algorithms based on meta-heuristics GRASP, VNS, Local Search, and Tabu Search for MAXSPACE and MAXSPACE-RDWV. We compare our proposed algorithms with Hybrid-GA proposed by Kumar et al. (2006). We also create a version of Hybrid-GA for MAXSPACE-RDWV and compare it with our meta-heuristics. Some meta-heuristics, such as VNS and GRASP+VNS, have better results than Hybrid-GA for both problems. In our heuristics, we apply a technique that alternates between maximizing and minimizing the fullness of slots to obtain better solutions. We also applied a data structure called BIT to the neighborhood computation in MAXSPACE-RDWV and showed that this enabled ours algorithms to run more iterations

Online Advertising Assignment Problems Considering Realistic Constraints

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 산업공학과, 2020. 8. 문일경.With a drastic increase in online communities, many companies have been paying attention to online advertising. The main advantages of online advertising are traceability, cost-effectiveness, reachability, and interactivity. The benefits facilitate the continuous popularity of online advertising. For Internet-based companies, a well-constructed online advertisement assignment increases their revenue. Hence, the managers need to develop their decision-making processes for assigning online advertisements on their website so that their revenue is maximized. In this dissertation, we consider online advertising assignment problems considering realistic constraints. There are three types of online advertising assignment problems: (i) Display ads problem in adversarial order, (ii) Display ads problem in probabilistic order, and (iii) Online banner advertisement scheduling for advertising effectiveness. Unlike previous assignment problems, the problems are pragmatic approaches that reflect realistic constraints and advertising effectiveness. Moreover, the algorithms the dissertation designs offer important insights into the online advertisement assignment problem. We give a brief explanation of the fundamental methodologies to solve the online advertising assignment problems in Chapter 1. At the end of this chapter, the contributions and outline of the dissertation are also presented. In Chapter 2, we propose the display ads problem in adversarial order. Deterministic algorithms with worst-case guarantees are designed, and the competitive ratios of them are presented. Upper bounds for the problem are also proved. We investigate the display ads problem in probabilistic order in Chapter 3. This chapter presents stochastic online algorithms with scenario-based stochastic programming and Benders decomposition for two probabilistic order models. In Chapter 4, an online banner advertisement scheduling model for advertising effectiveness is designed. We also present the solution methodologies used to obtain valid lower and upper bounds of the model efficiently. Chapter 5 offers conclusions and suggestion for future studies. The approaches to solving the problems are meaningful in both academic and industrial areas. We validate these approaches can solve the problems efficiently and effectively by conducting computational experiments. The models and solution methodologies are expected to be convenient and beneficial when managers at Internet-based companies place online advertisements on their websites.온라인 커뮤니티의 급격한 성장에 따라, 많은 회사들이 온라인 광고에 관심을 기울이고 있다. 온라인 광고의 장점으로는 추적 가능성, 비용 효과성, 도달 가능성, 상호작용성 등이 있다. 온라인에 기반을 두는 회사들은 잘 짜여진 온라인 광고 할당결정에 관심을 두고 있고, 이는 광고 수익과 연관될 수 있다. 따라서 온라인 광고 관리자는 수익을 극대화 할 수 있는 온라인 광고 할당 의사 결정 프로세스를 개발하여야 한다. 본 논문에서는 현실적인 제약을 고려한 온라인 광고 할당 문제들을 제안한다. 본 논문에서 다루는 문제는 (1) adversarial 순서로 진행하는 디스플레이 애드문제, (2) probabilistic 순서로 진행하는 디스플레이 애드문제 그리고 (3) 광고효과를 위한 온라인 배너 광고 일정계획이다. 이전에 제안되었던 광고 할당 문제들과 달리, 본 논문에서 제안한 문제들은 현실적인 제약과 광고효과를 반영하는 실용적인 접근 방식이다. 또한 제안하는 알고리즘은 온라인 광고 할당 문제의 운영관리에 대한 통찰력을 제공한다. 1장에서는 온라인 광고 할당 문제에 대한 문제해결 방법론에 대해 간단히 소개한다. 더불어 연구의 기여와 개요도 제공된다. 2장에서는 adversarial 순서로 진행하는 디스플레이 애드문제를 제안한다. worst-case를 보장하는 결정론적 알고리즘을 설계하고, 이들의 competitive ratio를 증명한다. 더불어 문제의 상한도 입증된다. 3장에서는 probabilistic 순서로 진행하는 디스플레이 애드문제를 제안한다. 시나리오 기반의 확률론적 온라인 알고리즘과 Benders 분해방법을 혼합한 추계 온라인 알고리즘을 제시한다. 4장에서는 광고효과를 위한 온라인 배너 광고 일정계획을 설계한다. 또한, 모델의 유효한 상한과 하한을 효율적으로 얻는 데 사용되는 문제해결 방법론을 제안한다. 5장에서는 본 논문의 결론과 향후 연구를 위한 방향을 제공한다. 본 논문에서 제안하는 문제해결 방법론은 학술 및 산업 분야 모두 의미가 있다. 수치 실험을 통해 문제해결 접근 방식이 문제를 효율적이고 효과적으로 해결할 수 있음을 보인다. 이는 온라인 광고 관리자가 본 논문에서 제안하는 문제와 문제해결 방법론을 통해 온라인 광고 할당관련 의사결정을 진행하는 데 있어 도움이 될 것으로 기대한다.Chapter 1 Introduction 1 1.1 Display Ads Problem 3 1.1.1 Online Algorithm 4 1.2 Online Banner Advertisement Scheduling Problem 5 1.3 Research Motivations and Contributions 6 1.4 Outline of the Dissertation 9 Chapter 2 Online Advertising Assignment Problem in Adversarial Order 12 2.1 Problem Description and Literature Review 12 2.2 Display Ads Problem in Adversarial Order 15 2.3 Deterministic Algorithms for Adversarial Order 17 2.4 Upper Bounds of Deterministic Algorithms for Adversarial Order 22 2.5 Summary 28 Chapter 3 Online Advertising Assignment Problem in Probabilistic Order 30 3.1 Problem Description and Literature Review 30 3.2 Display Ads Problem in Probabilistic Order 33 3.3 Stochastic Online Algorithms for Probabilistic Order 34 3.3.1 Two-Stage Stochastic Programming 35 3.3.2 Known IID model 37 3.3.3 Random permutation model 41 3.3.4 Stochastic approach using primal-dual algorithm 45 3.4 Computational Experiments 48 3.4.1 Results for known IID model 55 3.4.2 Results for random permutation model 57 3.4.3 Managerial insights for Algorithm 3.1 59 3.5 Summary 60 Chapter 4 Online Banner Advertisement Scheduling for Advertising Effectiveness 61 4.1 Problem Description and Literature Review 61 4.2 Mathematical Model 68 4.2.1 Objective function 68 4.2.2 Notations and formulation 72 4.3 Solution Methodologies 74 4.3.1 Heuristic approach to finding valid lower and upper bounds 75 4.3.2 Hybrid tabu search 79 4.4 Computational Experiments 80 4.4.1 Results for problems with small data sets 82 4.4.2 Results for problems with large data sets 84 4.4.3 Results for problems with standard data 86 4.4.4 Managerial insights for the results 90 4.5 Summary 92 Chapter 5 Conclusions and Future Research 93 Appendices 97 A Initial Sequence of the Hybrid Tabu Search 98 B Procedure of the Hybrid Tabu Search 99 C Small Example of the Hybrid Tabu Search 101 D Linearization Technique of Bilinear Form in R2 104 Bibliography 106Docto

Essays on optimization and incentive contracts

Includes bibliographical references (p. 167-176).Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2007.(cont.) In the second part of the thesis, we focus on the design and analysis of simple, possibly non-coordinating contracts in a single-supplier, multi-retailer supply chain where retailers make both pricing and inventory decisions. Specifically, we introduce a buy-back menu contract to improve supply chain efficiency, and compare two systems, one in which the retailers compete against each other, and another in which the retailers coordinate their decisions to maximize total expected retailer profit. In a linear additive demand setting, we show that for either retailer configuration, the proposed buy-back menu guarantees the supplier, and hence the supply chain, at least 50% of the optimal global supply chain profit. In particular, in a coordinated retailers system, the contract guarantees the supply chain at least 75% of the optimal global supply chain profit. We also analyze the impact of retail price caps on supply chain performance in this setting.In this thesis, we study important facets of two problems in methodological and applied operations research. In the first part of the thesis, motivated by optimization problems that arise in the context of Internet advertising, we explore the performance of the greedy algorithm in solving submodular set function maximization problems over various constraint structures. Most classic results about the greedy algorithm assume the existence of an optimal polynomial-time incremental oracle that identifies in any iteration, an element of maximum incremental value to the solution at hand. In the presence of only an approximate incremental oracle, we generalize the performance bounds of the greedy algorithm in maximizing nondecreasing submodular functions over special classes of matroids and independence systems. Subsequently, we unify and improve on various results in the literature for problems that are specific instances of maximizing nondecreasing submodular functions in the presence of an approximate incremental oracle. We also propose a randomized algorithm that improves upon the previous best-known 2-approximation result for the problem of maximizing a submodular function over a partition matroid.by Pranava Raja Goundan.Ph.D

Heuristics and approximation algorithms for scheduling advertisements problem

Orientadores: Rafael Crivellari Saliba Schouery, Lehilton Lelis Chaves PedrosaDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: No problema MAXSPACE, dado um conjunto de propagandas A, desejamos dispor um subconjunto A' de A em K slots B_1, ..., B_K de tamanho L. Cada propaganda A_i em A tem um tamanho s_i e uma frequência w_i. Uma disposição é viável se o tamanho total das propagandas em qualquer slot é no máximo L e cada A_i em A' aparece em exatamente w_i slots e no máximo uma vez por slot. O objetivo é encontrar uma disposição viável que maximiza a soma dos espaços ocupados por todos os slots. Introduzimos algumas variantes para esse problema, como o MAXSPACE-R, o MAXSPACE-RD e o MAXSPACE-RDWV. Consideramos o problema MAXSPACE-RD com número de slots constante e apresentamos um esquema de aproximação polinomial, isto é, para qualquer ? > 0, damos um algoritmo de tempo polinomial que devolve uma solução com valor pelo menos (1-?)OPT, em que OPT é o valor de um solução ótima. Também apresentamos uma 1/9-aproximação para o MAXSPACE-R com K polinomialmente limitado. Abordamos os problemas MAXSPACE e MAXSPACE-RDWV do ponto de vista de heurísticas, utilizando as meta-heurísticas Greedy Randomized Adaptive Search Procedure, Variable Neighborhood Search, Variable Neighborhood Descent, Busca Tabu e Busca Local. Os resultados computacionais das heurísticas implementadas foram comparados com o algoritmo genético híbrido proposto por Kumar et alAbstract: In the MAXSPACE problem, given a set of ads A, one wants to place a subset A' of A into K slots B_1, ..., B_K of size L. Each advertisement A_i in A has a size s_i and a frequency w_i. A schedule is feasible if the total size of ads in any slot is at most L, and each ad A_i in A' appears in exactly w_i slots with at most one copy per slot. The objective is to find a feasible schedule which maximizes the sum of the space occupied by all slots. We introduce some generalizations for this problem, such as MAXSPACE-R, MAXSPACE-RD, and MAXSPACE-RDWV. We consider the MAXSPACE-RD with a constant number of slots and present a polynomial-time approximation scheme, i.e., for any ? > 0, we give a polynomial-time algorithm which returns a solution with value at least (1-?)OPT, where OPT is the optimal value. We also present a 1/9-approximation for MAXSPACE-R with a polynomial number of slots. We implement the metaheuristics Greedy Randomized Adaptive Search Procedure, Variable Neighborhood Search, Variable Neighborhood Descent, Tabu Search and Local Search for MAXSPACE and MAXSPACE-RDWV. The computational results of these heuristics were compared with the results of the hybrid genetic algorithm proposed by Kumar et alMestradoCiência da ComputaçãoMestre em Ciência da Computação2017/21297-2FAPES