On the Configuration LP for Maximum Budgeted Allocation

We study the Maximum Budgeted Allocation problem, i.e., the problem of selling a set of $m$ indivisible goods to $n$ players, each with a separate budget, such that we maximize the collected revenue. Since the natural assignment LP is known to have an integrality gap of $\frac{3}{4}$, which matches the best known approximation algorithms, our main focus is to improve our understanding of the stronger configuration LP relaxation. In this direction, we prove that the integrality gap of the configuration LP is strictly better than $\frac{3}{4}$, and provide corresponding polynomial time roundings, in the following restrictions of the problem: (i) the Restricted Budgeted Allocation problem, in which all the players have the same budget and every item has the same value for any player it can be sold to, and (ii) the graph MBA problem, in which an item can be assigned to at most 2 players. Finally, we improve the best known upper bound on the integrality gap for the general case from $\frac{5}{6}$ to $2\sqrt{2}-2\approx 0.828$ and also prove hardness of approximation results for both cases.Comment: 29 pages, 4 figures. To appear in the 17th Conference on Integer Programming and Combinatorial Optimization (IPCO), 201

The Preemptive Resource Allocation Problem

We revisit a classical scheduling model to incorporate modern trends in data center networks and cloud services. Addressing some key challenges in the allocation of shared resources to user requests (jobs) in such settings, we consider the following variants of the classic resource allocation problem (RAP). The input to our problems is a set J of jobs and a set M of homogeneous hosts, each has an available amount of some resource. A job is associated with a release time, a due date, a weight and a given length, as well as its resource requirement. A feasible schedule is an allocation of the resource to a subset of the jobs, satisfying the job release times/due dates as well as the resource constraints. A crucial distinction between classic RAP and our problems is that we allow preemption and migration of jobs, motivated by virtualization techniques. We consider two natural objectives: throughput maximization (MaxT), which seeks a maximum weight subset of the jobs that can be feasibly scheduled on the hosts in M, and resource minimization (MinR), that is finding the minimum number of (homogeneous) hosts needed to feasibly schedule all jobs. Both problems are known to be NP-hard. We first present an Omega(1)-approximation algorithm for MaxT instances where time-windows form a laminar family of intervals. We then extend the algorithm to handle instances with arbitrary time-windows, assuming there is sufficient slack for each job to be completed. For MinR we study a more general setting with d resources and derive an O(log d)-approximation for any fixed d >= 1, under the assumption that time-windows are not too small. This assumption can be removed leading to a slightly worse ratio of O(log d log^* T), where T is the maximum due date of any job

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

Optimal advertising campaign generation for multiple brands using MOGA

The paper proposes a new modified multiobjective genetic algorithm (MOGA) for the problem of optimal television (TV) advertising campaign generation for multiple brands. This NP-hard combinatorial optimization problem with numerous constraints is one of the key issues for an advertising agency when producing the optimal TV mediaplan. The classical approach to the solution of this problem is the greedy heuristic, which relies on the strength of the preceding commercial breaks when selecting the next break to add to the campaign. While the greedy heuristic is capable of generating only a group of solutions that are closely related in the objective space, the proposed modified MOGA produces a Pareto-optimal set of chromosomes that: 1) outperform the greedy heuristic and 2) let the mediaplanner choose from a variety of uniformly distributed tradeoff solutions. To achieve these results, the special problem-specific solution encoding, genetic operators, and original local optimization routine were developed for the algorithm. These techniques allow the algorithm to manipulate with only feasible individuals, thus, significantly improving its performance that is complicated by the problem constraints. The efficiency of the developed optimization method is verified using the real data sets from the Canadian advertising industry

Buyback Problem - Approximate matroid intersection with cancellation costs

In the buyback problem, an algorithm observes a sequence of bids and must decide whether to accept each bid at the moment it arrives, subject to some constraints on the set of accepted bids. Decisions to reject bids are irrevocable, whereas decisions to accept bids may be canceled at a cost that is a fixed fraction of the bid value. Previous to our work, deterministic and randomized algorithms were known when the constraint is a matroid constraint. We extend this and give a deterministic algorithm for the case when the constraint is an intersection of $k$ matroid constraints. We further prove a matching lower bound on the competitive ratio for this problem and extend our results to arbitrary downward closed set systems. This problem has applications to banner advertisement, semi-streaming, routing, load balancing and other problems where preemption or cancellation of previous allocations is allowed

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

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