在线次模块福利最大化:贪婪在随机顺序中击败1/2

Nitish Korula, V. Mirrokni, Morteza Zadimoghaddam
{"title":"在线次模块福利最大化:贪婪在随机顺序中击败1/2","authors":"Nitish Korula, V. Mirrokni, Morteza Zadimoghaddam","doi":"10.1145/2746539.2746626","DOIUrl":null,"url":null,"abstract":"In the Submodular Welfare Maximization (SWM) problem, the input consists of a set of n items, each of which must be allocated to one of m agents. Each agent l has a valuation function vl, where vl(S) denotes the welfare obtained by this agent if she receives the set of items S. The functions vl are all submodular; as is standard, we assume that they are monotone and vl(∅) = 0. The goal is to partition the items into m disjoint subsets S1, S2, ... Sm in order to maximize the social welfare, defined as ∑l = 1m vl(Sl). A simple greedy algorithm gives a 1/2-approximation to SWM in the offline setting, and this was the best known until Vondrak's recent (1-1/e)-approximation algorithm [34]. In this paper, we consider the online version of SWM. Here, items arrive one at a time in an online manner; when an item arrives, the algorithm must make an irrevocable decision about which agent to assign it to before seeing any subsequent items. This problem is motivated by applications to Internet advertising, where user ad impressions must be allocated to advertisers whose value is a submodular function of the set of users / impressions they receive. There are two natural models that differ in the order in which items arrive. In the fully adversarial setting, an adversary can construct an arbitrary / worst-case instance, as well as pick the order in which items arrive in order to minimize the algorithm's performance. In this setting, the 1/2-competitive greedy algorithm is the best possible. To improve on this, one must weaken the adversary slightly: In the random order model, the adversary can construct a worst-case set of items and valuations, but does not control the order in which the items arrive; instead, they are assumed to arrive in a random order. The random order model has been well studied for online SWM and various special cases, but the best known competitive ratio (even for several special cases) is 1/2 + 1/n [9,10], barely better than the ratio for the adversarial order. Obtaining a competitive ratio of 1/2 + Ω(1) for the random order model has been an important open problem for several years. We solve this open problem by demonstrating that the greedy algorithm has a competitive ratio of at least 0.505 for online SWM in the random order model. This is the first result showing a competitive ratio bounded above 1/2 in the random order model, even for special cases such as the weighted matching or budgeted allocation problems (without the so-called 'large capacity' assumptions). For special cases of submodular functions including weighted matching, weighted coverage functions and a broader class of \"second-order supermodular\" functions, we provide a different analysis that gives a competitive ratio of 0.51. We analyze the greedy algorithm using a factor-revealing linear program, bounding how the assignment of one item can decrease potential welfare from assigning future items. We also formulate a natural conjecture which, if true, would improve the competitive ratio of the greedy algorithm to at least 0.567. In addition to our new competitive ratios for online SWM, we make two further contributions: First, we define the classes of second-order modular, supermodular, and submodular functions, which are likely to be of independent interest in submodular optimization. Second, we obtain an improved competitive ratio via a technique we refer to as gain linearizing, which may be useful in other contexts (see [26]): Essentially, we linearize the submodular function by dividing the gain of an optimal solution into gain from individual elements, compare the gain when it assigns an element to the optimal solution's gain from the element, and, crucially, bound the extent to which assigning elements can affect the potential gain of other elements.","PeriodicalId":20566,"journal":{"name":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2015-06-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"69","resultStr":"{\"title\":\"Online Submodular Welfare Maximization: Greedy Beats 1/2 in Random Order\",\"authors\":\"Nitish Korula, V. Mirrokni, Morteza Zadimoghaddam\",\"doi\":\"10.1145/2746539.2746626\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In the Submodular Welfare Maximization (SWM) problem, the input consists of a set of n items, each of which must be allocated to one of m agents. Each agent l has a valuation function vl, where vl(S) denotes the welfare obtained by this agent if she receives the set of items S. The functions vl are all submodular; as is standard, we assume that they are monotone and vl(∅) = 0. The goal is to partition the items into m disjoint subsets S1, S2, ... Sm in order to maximize the social welfare, defined as ∑l = 1m vl(Sl). A simple greedy algorithm gives a 1/2-approximation to SWM in the offline setting, and this was the best known until Vondrak's recent (1-1/e)-approximation algorithm [34]. In this paper, we consider the online version of SWM. Here, items arrive one at a time in an online manner; when an item arrives, the algorithm must make an irrevocable decision about which agent to assign it to before seeing any subsequent items. This problem is motivated by applications to Internet advertising, where user ad impressions must be allocated to advertisers whose value is a submodular function of the set of users / impressions they receive. There are two natural models that differ in the order in which items arrive. In the fully adversarial setting, an adversary can construct an arbitrary / worst-case instance, as well as pick the order in which items arrive in order to minimize the algorithm's performance. In this setting, the 1/2-competitive greedy algorithm is the best possible. To improve on this, one must weaken the adversary slightly: In the random order model, the adversary can construct a worst-case set of items and valuations, but does not control the order in which the items arrive; instead, they are assumed to arrive in a random order. The random order model has been well studied for online SWM and various special cases, but the best known competitive ratio (even for several special cases) is 1/2 + 1/n [9,10], barely better than the ratio for the adversarial order. Obtaining a competitive ratio of 1/2 + Ω(1) for the random order model has been an important open problem for several years. We solve this open problem by demonstrating that the greedy algorithm has a competitive ratio of at least 0.505 for online SWM in the random order model. This is the first result showing a competitive ratio bounded above 1/2 in the random order model, even for special cases such as the weighted matching or budgeted allocation problems (without the so-called 'large capacity' assumptions). For special cases of submodular functions including weighted matching, weighted coverage functions and a broader class of \\\"second-order supermodular\\\" functions, we provide a different analysis that gives a competitive ratio of 0.51. We analyze the greedy algorithm using a factor-revealing linear program, bounding how the assignment of one item can decrease potential welfare from assigning future items. We also formulate a natural conjecture which, if true, would improve the competitive ratio of the greedy algorithm to at least 0.567. In addition to our new competitive ratios for online SWM, we make two further contributions: First, we define the classes of second-order modular, supermodular, and submodular functions, which are likely to be of independent interest in submodular optimization. Second, we obtain an improved competitive ratio via a technique we refer to as gain linearizing, which may be useful in other contexts (see [26]): Essentially, we linearize the submodular function by dividing the gain of an optimal solution into gain from individual elements, compare the gain when it assigns an element to the optimal solution's gain from the element, and, crucially, bound the extent to which assigning elements can affect the potential gain of other elements.\",\"PeriodicalId\":20566,\"journal\":{\"name\":\"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2015-06-14\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"69\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1145/2746539.2746626\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the forty-seventh annual ACM symposium on Theory of Computing","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1145/2746539.2746626","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 69

摘要

在次模块福利最大化(Submodular Welfare Maximization, SWM)问题中,输入由一组n个项目组成,每个项目必须分配给m个代理中的一个。每个智能体l都有一个评估函数vl,其中vl(S)表示该智能体在接收项目S集合时所获得的福利。函数vl均为子模;作为标准,我们假定它们是单调的,且vl(∅)= 0。目标是将项目划分为m个不相交的子集S1, S2,…Sm以社会福利最大化,定义为∑l = 1m vl(Sl)。一个简单的贪心算法给出了离线环境下SWM的1/2近似,这是在Vondrak最近的(1-1/e)近似算法之前最著名的算法[34]。在本文中,我们考虑在线版本的SWM。在这里,物品以在线方式一次送达一件;当物品到达时,算法必须在看到任何后续物品之前,对将其分配给哪个代理做出不可撤销的决定。这个问题是由互联网广告的应用程序引起的,在互联网广告中,用户广告印象必须分配给广告商,广告商的价值是他们收到的用户/印象集的子模块函数。在物品到达的顺序上有两种不同的自然模型。在完全对抗的环境中,对手可以构建任意/最坏情况的实例,以及选择物品到达的顺序,以最小化算法的性能。在这种情况下,1/2竞争贪婪算法是最好的。为了改进这一点,我们必须稍微削弱对手:在随机顺序模型中,对手可以构建最坏情况下的物品和估值集合,但不能控制物品到达的顺序;相反,假设它们以随机顺序到达。随机顺序模型已经在在线SWM和各种特殊情况下得到了很好的研究,但最著名的竞争比(即使是在几个特殊情况下)是1/2 + 1/n[9,10],略好于对抗顺序的比率。获得1/2 + Ω(1)的随机排序模型的竞争比是多年来一个重要的开放问题。我们通过证明贪婪算法在随机顺序模型下对在线SWM具有至少0.505的竞争比来解决这个开放问题。这是在随机顺序模型中第一个显示竞争比率大于1/2的结果,即使对于特殊情况,如加权匹配或预算分配问题(没有所谓的“大容量”假设)也是如此。对于子模函数的特殊情况,包括加权匹配函数、加权覆盖函数和更广泛的一类“二阶超模”函数,我们提供了一个不同的分析,给出了0.51的竞争比。我们使用因子揭示线性规划分析贪婪算法,限定分配一个项目如何减少分配未来项目的潜在福利。我们还提出了一个自然猜想,如果成立,将贪婪算法的竞争比提高到至少0.567。除了我们在线SWM的新竞争比率之外,我们还做出了两个进一步的贡献:首先,我们定义了二阶模、超模和次模函数的类别,它们可能对次模优化有独立的兴趣。其次,我们通过一种称为增益线性化的技术获得了一个改进的竞争比,这在其他情况下可能很有用(见[26]):本质上,我们通过将最优解的增益划分为单个元素的增益来线性化子模块函数,在将一个元素分配给最优解的增益时比较增益,并且,至关重要的是,绑定分配元素对其他元素潜在增益的影响程度。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Online Submodular Welfare Maximization: Greedy Beats 1/2 in Random Order
In the Submodular Welfare Maximization (SWM) problem, the input consists of a set of n items, each of which must be allocated to one of m agents. Each agent l has a valuation function vl, where vl(S) denotes the welfare obtained by this agent if she receives the set of items S. The functions vl are all submodular; as is standard, we assume that they are monotone and vl(∅) = 0. The goal is to partition the items into m disjoint subsets S1, S2, ... Sm in order to maximize the social welfare, defined as ∑l = 1m vl(Sl). A simple greedy algorithm gives a 1/2-approximation to SWM in the offline setting, and this was the best known until Vondrak's recent (1-1/e)-approximation algorithm [34]. In this paper, we consider the online version of SWM. Here, items arrive one at a time in an online manner; when an item arrives, the algorithm must make an irrevocable decision about which agent to assign it to before seeing any subsequent items. This problem is motivated by applications to Internet advertising, where user ad impressions must be allocated to advertisers whose value is a submodular function of the set of users / impressions they receive. There are two natural models that differ in the order in which items arrive. In the fully adversarial setting, an adversary can construct an arbitrary / worst-case instance, as well as pick the order in which items arrive in order to minimize the algorithm's performance. In this setting, the 1/2-competitive greedy algorithm is the best possible. To improve on this, one must weaken the adversary slightly: In the random order model, the adversary can construct a worst-case set of items and valuations, but does not control the order in which the items arrive; instead, they are assumed to arrive in a random order. The random order model has been well studied for online SWM and various special cases, but the best known competitive ratio (even for several special cases) is 1/2 + 1/n [9,10], barely better than the ratio for the adversarial order. Obtaining a competitive ratio of 1/2 + Ω(1) for the random order model has been an important open problem for several years. We solve this open problem by demonstrating that the greedy algorithm has a competitive ratio of at least 0.505 for online SWM in the random order model. This is the first result showing a competitive ratio bounded above 1/2 in the random order model, even for special cases such as the weighted matching or budgeted allocation problems (without the so-called 'large capacity' assumptions). For special cases of submodular functions including weighted matching, weighted coverage functions and a broader class of "second-order supermodular" functions, we provide a different analysis that gives a competitive ratio of 0.51. We analyze the greedy algorithm using a factor-revealing linear program, bounding how the assignment of one item can decrease potential welfare from assigning future items. We also formulate a natural conjecture which, if true, would improve the competitive ratio of the greedy algorithm to at least 0.567. In addition to our new competitive ratios for online SWM, we make two further contributions: First, we define the classes of second-order modular, supermodular, and submodular functions, which are likely to be of independent interest in submodular optimization. Second, we obtain an improved competitive ratio via a technique we refer to as gain linearizing, which may be useful in other contexts (see [26]): Essentially, we linearize the submodular function by dividing the gain of an optimal solution into gain from individual elements, compare the gain when it assigns an element to the optimal solution's gain from the element, and, crucially, bound the extent to which assigning elements can affect the potential gain of other elements.
求助全文
通过发布文献求助,成功后即可免费获取论文全文。 去求助
来源期刊
自引率
0.00%
发文量
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整,为了账户安全,请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
确定
请完成安全验证×
copy
已复制链接
快去分享给好友吧!
我知道了
右上角分享
点击右上角分享
0
联系我们:info@booksci.cn Book学术提供免费学术资源搜索服务,方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号:481959085
Book学术官方微信