PDF-Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints Rishabh

washingtonedu Jeff Bilmes Department of Electrical Engineering University of Washington bilmesuwashingtonedu Abstract We investigate two new optimization problems minimizing a submodular function subject to a submodular lower bound constraint submod. e EE364A Chance Constrained Optimization brPage 7br Portfolio optimization example gives portfolio allocation is fractional position in asset must satisfy 1 8712 C convex portfolio constraint set portfolio return say in percent is where 8764 N p Bayesian Submodular Models. Josip . Djolonga. joint work with Andreas Krause. Motivation. inference with higher order potentials. MAP Computation . ✓. Inference? . ✘. We provide a method for inference in such models. A Mini-Survey. Chandra . Chekuri. Univ. of Illinois, Urbana-Champaign. Submodular Set Functions. A function . f. : 2. N. . . . R . is submodular if. . f(A. ) + . f(B. ) ≥ . f(A. . B. ) + . Input:. Output:. Objective: . a number W and a set of n items, the i-th item has a weight w. i. and a cost c. i. a subset of items with total weight . ·. W. and. Given positive integers v. i. and w. i. for . i = 1, 2, ..., n.. and positive integers. K . and. W.. Does there exist a subset . S . of. {1, 2, ..., n}. such that:. A special case: S. UBSET. A dynamic approach. Knapsack Problem. Given a sack, able to hold K kg. Given a list of objects. Each has a weight and a value. Try to pack the object in the sack so that the total value is maximized. 1. Tsvi. . Kopelowitz. Knapsack. Given: a set S of n objects with weights and values, and a weight bound:. w. 1. , w. 2. , …, w. n. , B (weights, weight bound).. v. 1. , v. 2. , …, v. n. (values - profit).. 1. Merkle-Hellman Knapsack. Public Key Systems . 2. Merkle-Hellman Knapsack. One of first public key systems. Lecture 5 – Part 1. M. Pawan Kumar. pawan.kumar@ecp.fr. Slides available online http://. cvn.ecp.fr. /personnel/. pawan. /. Submodular. Functions. Examples. Outline. Submodular. Function. Set S. Function f over power set of S. and. Given positive integers v. i. and . w. i. for . i = 1, 2, ..., n.. and positive integers. K . and. W.. Does there exist a subset . S . of. {1, 2, ..., n}. such that:. A special case: S. UBSET. A dynamic approach. Knapsack Problem. Given a sack, able to hold . W. . kg. Given a list of objects. Each has a weight and a value. Try to pack the object in the sack so that the total value is maximized. 1. Dynamic Programming: . 0/1 Knapsack. Presentation for use with the textbook, . Algorithm Design and Applications. , by M. T. Goodrich and R. Tamassia, Wiley, 2015. Dynamic Programming. 2. The 0/1 Knapsack Problem. . for Binary . Pairwise. Energies. Lena . Gorelick. . joint work with. . O. . Veksler. I. Ben . Ayed. A. Delong. . Y. . Boykov. Example of Simple Binary Energy. 2. Potts Model. Binary . Pairwise. Outline. Knapsack revisited: How to output the optimal solution, and how to prove correctness?. Longest Common Subsequence. Maximum Independent Set on Trees.. Example 2 Knapsack Problem. There is a knapsack that can hold items of total weight at most .