IEOR E4008: Computational Discrete Optimization Yuri Faenza IEOR - PowerPoint PPT Presentation

IEOR E4008: Computational Discrete Optimization Yuri Faenza – IEOR Department Jan 23th, 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

Logistics Instructor: Yuri Faenza Assistant Professor @ IEOR from 2016 Research area: Discrete Optimization Schedule: MW, 10:10-11:25 Room: 303 Mudd O ffi ce Hours: M, 5:30-7pm or by appointment – Mudd 334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

Website Course website on Courseworks: ◮ Slides, lecture notes, links to further material,... ◮ I’ll (try to) upload the slides of each lecture before class starts, so you can print them if you want to. Piazza: Online discussions on topics of the class. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

Grading 30%: Assignments, roughly one every two weeks. 10%: Class participation. will list I projects some - 60%: Project. ~ idea with for you - can come up c project , talk to and me . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

Examples of final projects ◮ How much would we save if all taxis were replaced by car sharing? ◮ How to solve discrete optimization problems with algorithms inspired by statistical physics and genetics. ◮ How to summarize the content of documents using machine learning and submodular functions. ◮ How can we visit all streets in a neighborhood as quickly as possible? ◮ How can we build index funds using algorithms on graphs? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

Examples of assignments In an image, each pixel is represented by a 3-dimensional vector in the RGB space, i.e. each component represents the amount of one of the three primary colors (red, green blue) that appears in the pixel. Each component has value between 0 and 256. Adapt one of the algorithms seen in class to the following problem. You are given an image as a set of pixels in the RGB space, and an integer k . Reduce the size of the picture by representing it using at most k colors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

Image compression via clustering: an example Original image, 302KB 2 colors, 13KB 16 colors, 44KB 32 colors, 73KB Yuri Faenza, Columbia University IEORE4004: Optimization Models and Methods

What is this course about? Discrete optimization: choose the solution of maximum profit (or minimum cost) from a discrete family. Some discrete optimization problems you have probably seen in basic classes: ◮ Many problems on graphs (e.g. shortest path). " ÷ : in . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

What is this course about? Discrete optimization: choose the solution of maximum profit (or minimum cost) from a discrete family. Some discrete optimization problems you have probably seen in basic classes: ◮ Many problems on graphs (e.g. shortest path). ◮ Integer Programming. min cx f Ax ≤ b ↳ Z n x ∈ Linear programming integer program I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

Why a class focused on Discrete Optimization? Many applications: ... machine learning network design production planning DNA assembling biomedicine In general, very hard to solve ⇒ not a unique approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

Example: online ads Bis 25$ ◮ Where does the 97% of Google revenue come from 1 ? Advertising! ,w ) di 70$ car we -_ Google Adwords: w ) $ s tire cci = - - , 3$ ◮ Company i chooses: ,w ) cci = race = ◮ a monthly budget B i ; ◮ for each word w a cost c ( i , w ) that the company is willing to pay to have its ad shown when word w is searched on Google. ◮ When w is searched, Google chooses a set of ads to display. If the user clicks on the ad of company i , Google gets c ( i , w ) from the company (unless the company has finished its budget B i ). 1 Source: http://www.wordstream.com/blog/ws/2011/07/18/most-expensive-keywords-google- . . . . . . . . . . . . . . . . . . . . adwords?dt=1 . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

Which word produces the biggest revenue 2 ? Which is the most expensive 3 ? 2 Source: https://searchenginewatch.com/2016/05/31/the-most-expensive-100-google- adwords-keywords-in-the-us/ 3 Source: http://www.wordstream.com/blog/ws/2011/07/18/most-expensive-keywords-google- . . . . . . . . . . . . . . . . . . . . adwords?dt=1 . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

The Adwords problem TO Y OTA WORDS 4 8 CAR 3 BUDGET 3 8 RACE 1 G- H Who should we assign each word to? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

The Adwords problem TOYOTA WORDS f [ ③ ④ 4 I 2X CAR 8 O CAR 2 RACE x ) ① 3 STRATEGY I - :/GRCED Bugg , , µ agg , won q , , , . TO THE HIGHEST BIDDER f) 3 WHO STILL HAS 7) G 8 SOME BUDGET LEFT RACE [ I 1 I GM TOYOTA G can → → Who should we assign each word to? TOYOTA 9 CAR → → I → a- ti RACE → I c- n a - RACE → - 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

The Adwords problem TOYOTA WORDS I ⑦ L 4 2x CAR 8 yy3 CAR TSU 2x RACE ) £ 3 BUDGET ① 3 TOYOTA CAR -79 → 8 RACE Gn can 3 → → [ I 1 [ RACE TOYOTA gu 3 → → Who should we assign each word to? → RACE → g- n y - II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

The Adwords problem 4 8 3 3 8 1 Who should we assign each word to? What if arrivals are random? Dr. Balasubramanian Sivan, Google Research. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

The Stable Marriage problem A stable if marriage is § w m : , to his partner prefers m w hav x w m " a , above is m ,W as called " blocking pain , a - . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

The Stable Marriage problem a blocking is my W1 pain I not a stable matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

The Stable Marriage problem NO BLOCKING PAIR ¥ MATCHING STABLE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization

The Stable Marriage problem Applications include: Allocations of students to high schools in NYC, of doctors to hospitals, of kidney donors to patients, ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yuri Faenza, Columbia University IEOR E4008: Computational Discrete Optimization