{"id":324947,"date":"2016-11-20T20:28:38","date_gmt":"2016-11-21T04:28:38","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/?post_type=msr-research-item&p=324947"},"modified":"2018-10-16T20:54:34","modified_gmt":"2018-10-17T03:54:34","slug":"online-non-clairvoyant-scheduling-simultaneously-minimize-convex-functions","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/online-non-clairvoyant-scheduling-simultaneously-minimize-convex-functions\/","title":{"rendered":"Online Non-clairvoyant Scheduling to Simultaneously Minimize All Convex Functions"},"content":{"rendered":"

We consider scheduling jobs online to minimize the objective \u2211\u2009 i<\/em>\u2009\u2208\u2009[n<\/em>]<\/sub> w<\/em> i<\/em><\/sub>g<\/em>(C<\/em> i<\/em> <\/sub>\u2009\u2212\u2009r<\/em> i<\/em> <\/sub>), where w<\/em> i<\/em> <\/sub>is the weight of job i<\/em>, r<\/em> i<\/em> <\/sub>is its release time, C<\/em> i<\/em> <\/sub>is its completion time and g<\/em> is any non-decreasing convex function. Previously, it was known that the clairvoyant algorithm Highest-Density-First (HDF) is (2\u2009+\u2009\u03b5<\/em>)-speed O<\/em>(1)-competitive for this objective on a single machine for any fixed 0\u2009<\u2009\u03b5<\/em>\u2009<\u20091 [1]. We show the first non-trivial results for this problem when\u00a0g<\/em> is not concave and the algorithm must be non-clairvoyant<\/em>. More specifically, our results include:<\/p>\n

\n
\n
    \n
  • \n

    A (2\u2009+\u2009\u03b5<\/em>)-speed O<\/em>(1)-competitive non-clairovyant algorithm on a single machine for all non-decreasing convex\u00a0g<\/em>, matching the performance of HDF for any fixed 0\u2009<\u2009\u03b5<\/em>\u2009<\u20091.<\/p>\n<\/li>\n

  • \n

    A (3\u2009+\u2009\u03b5<\/em>)-speed O<\/em>(1)-competitive non-clairovyant algorithm on multiple identical machines for all non-decreasing convex\u00a0g<\/em> for any fixed 0\u2009<\u2009\u03b5<\/em>\u2009<\u20091.<\/p>\n<\/li>\n<\/ul>\n<\/div>\n<\/div>\n

    Our positive result on multiple machines is the first non-trivial one even when the algorithm is clairvoyant. Interestingly, all performance guarantees above hold for all non-decreasing convex functions g<\/em> simultaneously<\/em>. We supplement our positive results by showing any algorithm that is oblivious to\u00a0g<\/em> is not\u00a0O<\/em>(1)-competitive with speed less than\u00a02 on a single machine. Further, any non-clairvoyent algorithm that knows the function\u00a0g<\/em> cannot be\u00a0O<\/em>(1)-competitive with speed less than\u00a02<\/mn><\/msqrt><\/math>\">2<\/span><\/span>\u2013\u221a<\/span><\/span><\/span>2<\/span><\/span><\/span> on a single machine or speed less than\u00a02<\/mn>−<\/mo>1<\/mn>m<\/mi><\/mfrac><\/math>\">2<\/span>\u2212<\/span>1<\/span>m<\/span><\/span><\/span><\/span>2\u22121m<\/span><\/span><\/span> on\u00a0m<\/em> identical machines.<\/p>\n","protected":false},"excerpt":{"rendered":"

    We consider scheduling jobs online to minimize the objective \u2211\u2009 i\u2009\u2208\u2009[n] w ig(C i \u2009\u2212\u2009r i ), where w i is the weight of job i, r i is its release time, C i is its completion time and g is any non-decreasing convex function. Previously, it was known that the clairvoyant algorithm Highest-Density-First (HDF) […]<\/p>\n","protected":false},"featured_media":0,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr-author-ordering":null,"msr_publishername":"Springer Berlin 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