{"title":"超越局部搜索的路径:随机不动点计算的紧界","authors":"X. Chen, S. Teng","doi":"10.1109/FOCS.2007.14","DOIUrl":null,"url":null,"abstract":"In 1983, Akhus proved that randomization can speedup local search. For example, it reduces the query complexity of local search over grid [1 : n]d from ominus(nd-1) to 0(d1/2nd/2). It remains open whether randomisation helps fixed-point computation. Inspired by the recent advances on the complexity of equilibrium computation, we solve this open problem by giving an asymptotically tight bound of (Omega(n))d-1 on the randomized query complexity for computing a fixed point of a discrete Brouwer function over grid [1 : n]d. Our result can be extended to the black-box query model for Sperner's I&mma in any dimension. It also yields a tight bound for the computation of d-dimensional approximate Brouwer fixed points as defined by Scarf and by Hirsch, Papadimitriou, and Vavasis. Since the randomized query complexity of global optimization over [1 : n]d is ominus(nd), the randomized query model over [ 1 : n]d strictly separates these three important search problems: Global optimization is harder than fixed-point computation, and fixed-point computation is harder than local search. Our result indeed demonstrates that randomization does not help much in fixed-point computation in the black-box query model. Our randomized lower bound matches the deterministic complexity of this problem, which is ominus(nd-1).","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"17","resultStr":"{\"title\":\"Paths Beyond Local Search: A Tight Bound for Randomized Fixed-Point Computation\",\"authors\":\"X. Chen, S. Teng\",\"doi\":\"10.1109/FOCS.2007.14\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In 1983, Akhus proved that randomization can speedup local search. For example, it reduces the query complexity of local search over grid [1 : n]d from ominus(nd-1) to 0(d1/2nd/2). It remains open whether randomisation helps fixed-point computation. Inspired by the recent advances on the complexity of equilibrium computation, we solve this open problem by giving an asymptotically tight bound of (Omega(n))d-1 on the randomized query complexity for computing a fixed point of a discrete Brouwer function over grid [1 : n]d. Our result can be extended to the black-box query model for Sperner's I&mma in any dimension. It also yields a tight bound for the computation of d-dimensional approximate Brouwer fixed points as defined by Scarf and by Hirsch, Papadimitriou, and Vavasis. Since the randomized query complexity of global optimization over [1 : n]d is ominus(nd), the randomized query model over [ 1 : n]d strictly separates these three important search problems: Global optimization is harder than fixed-point computation, and fixed-point computation is harder than local search. Our result indeed demonstrates that randomization does not help much in fixed-point computation in the black-box query model. Our randomized lower bound matches the deterministic complexity of this problem, which is ominus(nd-1).\",\"PeriodicalId\":197431,\"journal\":{\"name\":\"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"1900-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"17\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1109/FOCS.2007.14\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1109/FOCS.2007.14","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Paths Beyond Local Search: A Tight Bound for Randomized Fixed-Point Computation
In 1983, Akhus proved that randomization can speedup local search. For example, it reduces the query complexity of local search over grid [1 : n]d from ominus(nd-1) to 0(d1/2nd/2). It remains open whether randomisation helps fixed-point computation. Inspired by the recent advances on the complexity of equilibrium computation, we solve this open problem by giving an asymptotically tight bound of (Omega(n))d-1 on the randomized query complexity for computing a fixed point of a discrete Brouwer function over grid [1 : n]d. Our result can be extended to the black-box query model for Sperner's I&mma in any dimension. It also yields a tight bound for the computation of d-dimensional approximate Brouwer fixed points as defined by Scarf and by Hirsch, Papadimitriou, and Vavasis. Since the randomized query complexity of global optimization over [1 : n]d is ominus(nd), the randomized query model over [ 1 : n]d strictly separates these three important search problems: Global optimization is harder than fixed-point computation, and fixed-point computation is harder than local search. Our result indeed demonstrates that randomization does not help much in fixed-point computation in the black-box query model. Our randomized lower bound matches the deterministic complexity of this problem, which is ominus(nd-1).