Prof. Swarat Chaudhuri

1. COMP 482: Design and Analysis of Algorithms Prof. Swarat Chaudhuri Spring 2013 Lecture 20

2. Recap: Project Selection can be positive or negative • Projects with prerequisites. • Set P of possible projects. Project v has associated revenue pv. • some projects generate money: create interactive e-commerce interface, redesign web page • others cost money: upgrade computers, get site license • Set of prerequisites E. If (v, w)  E, can't do project v and unless also do project w. • A subset of projects A  P is feasible if the prerequisite of every project in A also belongs to A. • Project selection. Choose a feasible subset of projects to maximize revenue.

3. Recap: Project Selection: Prerequisite Graph • Prerequisite graph. • Include an edge from v to w if can't do v without also doing w. • {v, w, x} is feasible subset of projects. • {v, x} is infeasible subset of projects. w w v x v x feasible infeasible

4. 7.10 Image Segmentation

5. Image Segmentation • Image segmentation. • Central problem in image processing. • Divide image into coherent regions. • Ex: Three people standing in front of complex background scene. Identify each person as a coherent object.

6. Image Segmentation • Foreground / background segmentation. • Label each pixel in picture as belonging toforeground or background. • V = set of pixels, E = pairs of neighboring pixels. • ai  0 is likelihood pixel i in foreground. • bi  0 is likelihood pixel i in background. • pij  0 is separation penalty for labeling one of iand j as foreground, and the other as background. • Goals. • Accuracy: if ai > bi in isolation, prefer to label i in foreground. • Smoothness: if many neighbors of i are labeled foreground, we should be inclined to label i as foreground. • Find partition (A, B) that maximizes: foreground background

7. Image Segmentation • Properties of the problem. • Maximization. • No source or sink. • Undirected graph. • Turn into minimization problem. • Maximizingis equivalent to minimizing • or alternatively

8. Q1 • …Can you solve this problem using max flow or min-cut?

9. Image Segmentation • Formulate as min cut problem. • G' = (V', E'). • Add source to correspond to foreground;add sink to correspond to background • Use two anti-parallel edges instead ofundirected edge. pij pij pij aj pij i j s t bi G'

10. Image Segmentation • Consider min cut (A, B) in G'. • A = foreground. • Precisely the quantity we want to minimize. if i and j on different sides, pij counted exactly once aj pij i j s t bi A G'

11. 7.12 Baseball Elimination

12. Baseball Elimination • Which teams have a chance of finishing the season with most wins? • Montreal eliminated since it can finish with at most 80 wins, but Atlanta already has 83. • wi + ri < wj team i eliminated. • Only reason sports writers appear to be aware of. • Sufficient, but not necessary! Teami Winswi Lossesli To playri Against = rij Atl Phi NY Mon Atlanta 83 71 8 - 1 6 1 Philly 80 79 3 1 - 0 2 New York 78 78 6 6 0 - 0 Montreal 77 82 3 1 2 0 -

13. Baseball Elimination • Which teams have a chance of finishing the season with most wins? • Philly can win 83, but still eliminated . . . • If Atlanta loses a game, then some other team wins one. • Remark. Answer depends not just on how many games already won and left to play, but also on whom they're against. Teami Winswi Lossesli To playri Against = rij Atl Phi NY Mon Atlanta 83 71 8 - 1 6 1 Philly 80 79 3 1 - 0 2 New York 78 78 6 6 0 - 0 Montreal 77 82 3 1 2 0 -

14. Baseball Elimination • Baseball elimination problem. • Set of teams S. • Distinguished team s  S. • Team x has won wx games already. • Teams x and y play each other rxy additional times. • Is there any outcome of the remaining games in which team s finishes with the most (or tied for the most) wins?

15. Baseball Elimination: Max Flow Formulation • Can team 3 finish with most wins? • Assume team 3 wins all remaining games  w3 + r3wins. • Divvy remaining games so that all teams have  w3 + r3 wins. 1-2 1 team 4 can stillwin this manymore games 1-4 games left 2  1-5 2-4  w3 + r3 - w4 s t 4 r24 = 7 2-5 5 4-5 game nodes team nodes

16. Baseball Elimination: Max Flow Formulation • Theorem. Team 3 is not eliminated iff max flow saturates all edges leaving source. • Integrality theorem  each remaining game between x and y added to number of wins for team x or team y. • Capacity on (x, t) edges ensure no team wins too many games. 1-2 1 team 4 can stillwin this manymore games 1-4 games left 2  1-5 2-4  w3 + r3 - w4 s t 4 r24 = 7 2-5 5 4-5 game nodes team nodes

17. NP-completeness and computational intractability

18. Algorithm Design Patterns and Anti-Patterns • Algorithm design patterns. Ex. • Greed. O(n log n) interval scheduling. • Divide-and-conquer. O(n log n) FFT. • Dynamic programming. O(n2) edit distance. • Duality. O(n3) bipartite matching. • Reductions. • Local search. • Randomization. • Algorithm design anti-patterns. • NP-completeness. O(nk) algorithm unlikely. • PSPACE-completeness. O(nk) certification algorithm unlikely. • Undecidability. No algorithm possible.

19. 8.1 Polynomial-Time Reductions

20. Classify Problems According to Computational Requirements • Q. Which problems will we be able to solve in practice? • A working definition. [Cobham 1964, Edmonds 1965, Rabin 1966] Those with polynomial-time algorithms. Yes Probably no Shortest path Longest path Matching 3D-matching Min cut Max cut 2-SAT 3-SAT Planar 4-color Planar 3-color Bipartite vertex cover Vertex cover Primality testing Factoring

21. Classify Problems • Desiderata. Classify problems according to those that can be solved in polynomial-time and those that cannot. • Provably requires exponential-time. • Given a Turing machine, does it halt in at most k steps? • Given a board position in an n-by-n generalization of chess, can black guarantee a win? • Frustrating news. Huge number of fundamental problems have defied classification for decades. • This chapter. Show that these fundamental problems are "computationally equivalent" and appear to be different manifestations of one really hard problem.

22. Polynomial-Time Reduction • Desiderata'. Suppose we could solve X in polynomial-time. What else could we solve in polynomial time? • Reduction. Problem X polynomial reduces to problem Y if arbitrary instances of problem X can be solved using: • Polynomial number of standard computational steps, plus • Polynomial number of calls to oracle that solves problem Y. Notation. X  P Y. Remarks. • We pay for time to write down instances sent to black box  instances of Y must be of polynomial size. • Note: Cook reducibility. don't confuse with reduces from computational model supplemented by special pieceof hardware that solves instances of Y in a single step in contrast to Karp reductions

23. Polynomial-Time Reduction • Purpose. Classify problems according to relative difficulty. • Design algorithms. If X  P Y and Y can be solved in polynomial-time, then X can also be solved in polynomial time. • Establish intractability. If X  P Y and X cannot be solved in polynomial-time, then Y cannot be solved in polynomial time. • Establish equivalence. If X  P Y and Y  P X, we use notation X  P Y. up to cost of reduction

24. Reduction By Simple Equivalence Basic reduction strategies. Reduction by simple equivalence. Reduction from special case to general case. Reduction by encoding with gadgets.

25. Independent Set • INDEPENDENT SET: Given a graph G = (V, E) and an integer k, is there a subset of vertices S  V such that |S|  k, and for each edge at most one of its endpoints is in S? • Ex. Is there an independent set of size  6? Yes. • Ex. Is there an independent set of size  7? No. independent set

26. Vertex Cover • VERTEX COVER: Given a graph G = (V, E) and an integer k, is there a subset of vertices S  V such that |S|  k, and for each edge, at least one of its endpoints is in S? • Ex. Is there a vertex cover of size  4? Yes. • Ex. Is there a vertex cover of size  3? No. vertex cover

27. Vertex Cover and Independent Set • Claim. VERTEX-COVERPINDEPENDENT-SET. • Pf. We show S is an independent set iff V  S is a vertex cover. independent set vertex cover

28. Vertex Cover and Independent Set • Claim. VERTEX-COVERPINDEPENDENT-SET. • Pf. We show S is an independent set iff V  S is a vertex cover. •  • Let S be any independent set. • Consider an arbitrary edge (u, v). • S independent  u  S or v  S  u  V  S or v  V  S. • Thus, V  S covers (u, v). •  • Let V  S be any vertex cover. • Consider two nodes u  S and v  S. • Observe that (u, v)  E since V  S is a vertex cover. • Thus, no two nodes in S are joined by an edge  S independent set. ▪

29. Reduction from Special Case to General Case Basic reduction strategies. Reduction by simple equivalence. Reduction from special case to general case. Reduction by encoding with gadgets.

30. Set Cover • SET COVER: Given a set U of elements, a collection S1, S2, . . . , Sm of subsets of U, and an integer k, does there exist a collection of  k of these sets whose union is equal to U? • Sample application. • m available pieces of software. • Set U of n capabilities that we would like our system to have. • The ith piece of software provides the set Si U of capabilities. • Goal: achieve all n capabilities using fewest pieces of software. • Ex: U = { 1, 2, 3, 4, 5, 6, 7 }k = 2 S1 = {3, 7} S4 = {2, 4} S2 = {3, 4, 5, 6} S5 = {5}S3 = {1} S6 = {1, 2, 6, 7}

31. Vertex Cover Reduces to Set Cover • Claim. VERTEX-COVER PSET-COVER. • Pf. Given a VERTEX-COVER instance G = (V, E), k, we construct a set cover instance whose size equals the size of the vertex cover instance. • Construction. • Create SET-COVER instance: • k = k, U = E, Sv = {e  E : e incident to v } • Set-cover of size  k iff vertex cover of size  k. ▪ SET COVER U = { 1, 2, 3, 4, 5, 6, 7 }k = 2 Sa = {3, 7} Sb = {2, 4} Sc = {3, 4, 5, 6} Sd = {5}Se = {1} Sf= {1, 2, 6, 7} VERTEX COVER a b e7 e4 e2 e3 f c e6 e5 e1 k = 2 d e

32. Q2: Hitting set • HITTING SET: Given a set U of elements, a collection S1, S2, . . . , Sm of subsets of U, and an integer k, does there exist a subset of U of size  k such that U overlaps with each of the sets S1, S2, . . . , Sm? • Show that SET COVER polynomial reduces to HITTING SET.