{"id":184292,"date":"2005-01-19T00:00:00","date_gmt":"2009-10-31T13:34:53","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/the-benefit-of-adaptivity-in-stochastic-optimization\/"},"modified":"2016-09-09T10:01:02","modified_gmt":"2016-09-09T17:01:02","slug":"the-benefit-of-adaptivity-in-stochastic-optimization","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/the-benefit-of-adaptivity-in-stochastic-optimization\/","title":{"rendered":"The Benefit of Adaptivity in Stochastic Optimization"},"content":{"rendered":"<div class=\"asset-content\">\n<p>Consider the Stochastic Knapsack problem where items have deterministic values but random sizes. The motivation for this problem is in the area of stochastic scheduling where a sequence of jobs should be scheduled on a machine within a limited amount of time. The running times of jobs are considered random and independent. A priori, only some information on their probability distributions is available. When a job has been scheduled and completed, its precise running time is revealed and this information can be used in subsequent decisions. An &#8220;adaptive policy&#8221; is a decision strategy which chooses a job to be processed based on the remaining time available. Alternatively, we consider &#8220;non-adaptive policies&#8221; choosing a fixed ordering of jobs beforehand. A natural question is how much advantage we can gain by being adaptive. We measure the &#8220;adaptivity gap&#8221; for a given instance as the ratio of the expected values achieved by the optimal adaptive vs. non-adaptive policy. We are also interested in the algorithmic questions of finding a good adaptive or non-adaptive policy efficiently.<\/p>\n<p>In FOCS 2004, we showed that the adaptivity gap cannot be larger than a constant factor of 7. I will present a recent improvement of the upper bound to 4; i.e., a non-adaptive policy which achieves at least 1\/4 of the adaptive optimum. The proof involves a new analysis of adaptive policies which leads to a polymatroid optimization problem. Using this technique, we can also find an adaptive policy in polynomial time, which approximates the adaptive optimum to within a factor of 3+epsilon.<\/p>\n<p>Next, I will address multidimensional generalizations of the Stochastic Knapsack problem where items with random vector sizes are to be packed in a unit hypercube (&#8220;Stochastic Packing&#8221;). I will show that for &#8220;Stochastic Set Packing&#8221; (with random 0\/1 vectors), the adaptivity gap can be as large as d<sup>1\/2<\/sup> and it is always bounded by O(d<sup>1\/2<\/sup>). Although even deterministic Set Packing has a known inapproximability factor of d<sup>1\/2-epsilon<\/sup>, we can approximate the adaptive optimum to within O(d<sup>1\/2<\/sup>) using a non-adaptive policy. For general Stochastic Packing, we can only achieve an O(d)-approximation. However, we show that this is also nearly optimal, by improving the inapproximability factor for Packing Integer Programs to d<sup>1-epsilon<\/sup>.<\/p>\n<p>This is joint work with Brian Dean and Michel Goemans.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider the Stochastic Knapsack problem where items have deterministic values but random sizes. The motivation for this problem is in the area of stochastic scheduling where a sequence of jobs should be scheduled on a machine within a limited amount of time. The running times of jobs are considered random and independent. A priori, only [&hellip;]<\/p>\n","protected":false},"featured_media":195385,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-184292","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/-xpOkkj6rJI","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/184292","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/184292\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/195385"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=184292"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=184292"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=184292"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=184292"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=184292"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=184292"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=184292"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=184292"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=184292"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=184292"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}