2. Approximation Algorithms for NP-hard Combinatorial Problems Magnús M. Halldórsson Reykjavik University www.ru.is/~mmh/ewscs13/lec1.ppt

3. Post-doc & Ph.D. Positions available Part of research project titled “Algorithms for Wireless Scheduling”

4. Why? Motivation Instead of looking for the best solution, we may want to find a solution that is good enough – Instead of an optimal solution, we seek approximations Many computational problems are NP-hard Most

5. Why study approximation algorithms? Study valuable, general-purpose problem solving techniques Interesting combinatorics, from a CS viewpoint Related to various other paradigms: – Online algorithms – Streaming algorithms – Distributed algorithms

6. What I expect that you already know Discrete structures and problems – Graphs and networks Basic knowledge of algorithms Some experience with analysis of algorithms – E.g. why Dijkstra’s algorithm works Bonus: – Mathematical programming – Basic probability theory

7. Plan today Get comfortable with our main problems Overview the algorithmic strategies that we will examine Look at the most naive approximation algorithms

8. Fundamental problems There are millions of optimization problems We focus on a few fundamental problems: – The techniques developed can be relatively easily transferred to other problems The classic graph problems, w/o connectivity

9. Problems we shall consider Independent set Vertex cover Chromatic number Dominating set Domatic number Max Cut And sometimes weighted versions

10. Graphs and Notation n=9 m=13  =4  =2 d=26/9

11. Independent sets Find: An independent set S of vertices Objective: Maximize |S|

12. Independent sets Find: An independent set of vertices Maximize: Size

13. & corresponding interval graph Intervals

14. & corresponding interval graph Interval selection

15. & corresponding interval graph Interval selection

16. = independent set in the power graph G k k=2 distance-k-independent set in graph G (u,v)  E(G k )  dist G (u,v)  k

17. 16 Matching Input: Graph. Goal. Find maximum cardinality matching. C 1 5 2 A E 3 B D4

18. Vertex Cover S is a vertex cover if every edge has at least one endpoint in S Objective: Minimize |S|

19. Vertex Cover S is a vertex cover if every edge has at least one endpoint in S Objective: Minimize |S|

20. Map coloring

21. Austria and its neighbors ?

22. Graph Coloring

23. Chromatic Number  (G) = 3

24. Where to locate icecream stands? The Icecream Stands problem:Icecream Stands problem Every kid should have access to an icecream stand in the next street. How many stands are needed? A dominating set for a graph is a set of vertices whose neighbors, along with themselves, constitute all the vertices in the graph.

27. Domatic number Find: A partition (coloring) of the vertices such that each color is a dominating set Objective: Maximize the number of colors C A F E B

28. The MAX CUT problem Input. A graph. Feasible solution. A set S of vertices. Value of solution. Number of edges cut. Objective. Maximize.

29. Weighted versions In Weighted Independent Set: – Given a graph G with weights on vertices – Find an independent set with maximum total weight Weights on vertices: – Ind.set, Vertex cover, Dominating set Weights on edges – Max Cut

30. Algorithmic strategies Greedy Subgraph removal Local search Probabilistic method Local ratio Linear programming rounding Semi-definite programming

31. Core issue: Bounding OPT We want to compare „our“ solution, ALG, to OPT, the optimal solution – I.e., show that ALG   * OPT – But OPT is hard to get a handle on (NP-hard!) Instead, we compare ALG to an easier property that bounds OPT: –  (I)  OPT(I), and – ALG(I)   (I)

32. Greedy Previously known as „myopic“ In each round, the algorithm makes the action that is best according to some criteria

33. Subgraph removal An independent set can contain at most 1 vertex from a clique If we remove all big cliques, the problem may become easier. If OPT was large, it is still fairly large after removing the cliques

34. Local search Can I make small, „local“ changes to my solution, to get a better solution? – Ex. 1: „Shortcuts“ – Ex. 2: If I throw out 1 vertex, can I replace it with 2?

35. Probabilistic method A „random“ solution is often pretty good – Sometimes close to best possible – Sometimes it needs minor changes Basic facts from probability: – Product rule – Union bound – Linearity of expectation

36. LP rounding Linear programming is a powerful hammer – The most general purpose solving method All of our problems can be written as Integer Programming problems – X_i = 1, if v_i is in the independent set, – X_i = 0, if v_i is not in the set LP: Allow any value in the range [0,1] – Solvable in polynomial time „Rounding“: Use LP-value to choose the binary 0/1 value

37. Semi-definite programming „Vector programming“ assigns each vertex a vector in n-dimensional space – LP value = 1-dimensional vector Use nearness or farness to decide on rounding

38. Performance ratio An algorithm A is  -approximate if, for every input instance I, A(I)    OPT(I) (maximization) or A(I)    OPT(I) (minimization)

39. VC - Approximation Algorithm C   E’  E while E’   – do let (u,v) be an arbitrary edge of E’ – C  C  {u,v} – remove from E’ every edge incident to either u or v. return C.

40. Demo Compare this cover to the one from the example one from the example

41. Polynomial Time C   E’  E while E’   do – let (u,v) be an arbitrary edge of E’ – C  C  {u,v} – remove from E’ every edge incident to either u or v return C O(n 2 ) O(1)O(n) O(n 2 )

42. Correctness The set of vertices our algorithm returns is clearly a vertex-cover, since we iterate until every edge is covered.

43. How Good an Approximation is it? Observe the set of edges our algorithm chooses  any VC contains 1 in each our VC contains both, hence at most twice as large no common vertices!

44. Another algorithm Given a graph G=(V,E), a subset of the edges M  E is a matching if no two edges of M share an endpoint. A matching is maximal if no more edges can be added The previous solution is a maximal matching! Algorithm 1.2 (Cardinality vertex cover)  Find a maximal matching in G and output the set of matched vertices

45. Linking ALG to OPT (relating the approximation to the optimal) Approximate solution Optimal solution ? Maximal matching

46. Lower bounding OPT OPT : an optimal solution (vertex cover) (also its size) ALG : solution found by our algorithm M : maximal matching We argued a lower bound for OPT in terms of M: OPT ≥ |M| Thus, maximal matching is the combinatorial structure that is relevant here. Performance analysis follows from this: – Since, ALG = 2 |M|, the performance ratio is at most 2

47. Can we improve this approximation? Can we improve the analysis for this algorithm? Can we make this algorithm more clever? – What if we carefully choose the maximal matching? Can we find a better algorithm?

48. Tight example #1 But, this is only one small graph. We want a family of graphs, of infinite size OPT picks 1 vertex ALG selected 2

49. Tight example #2 Complete bipartite graph K n,n (here n=4) Any maximal matching contains all the 2n vertices So the algorithm solution always contains 2n verts. A matching and the algorithm solution Optimal solution contains only n vertices An optimal solution

50. “This was a too simple solution” It is simple, but not stupid. Many seemingly smarter heuristics can give a far worse performance, in the worst case. It is easy to complicate heuristics by adding more special cases or details. For example, if you just picked the edge whose endpoints had highest degree, it doesn’t improve the worst case. It just makes it harder to analyze.

51. Naive approximations Bound ALG and OPT, independently Forms the baseline, with which we compare

52. Naïve approximation of Dominating Set ALG  n OPT  n/(  +1)

53. Naïve approximation of Domatic Partition ALG  1 OPT   +1

54. Naïve approximation of Coloring ALG   +1 OPT  2

55. Naïve approximation of Independent Set ALG  n/(  +1) OPT  n

56. Acknowledgment Much of the presentations are adapted from slides on the web In this talk, material on Vertex Cover was taken from: – www.cs.tau.ac.il/~safra/Complexity/appalgo.ppt

57. Problem #1: Min-degree greedy What is the most natural algorithm for the Independent Set problem? „Min-degree greedy“: – 1. Find a vertex of minimum degree; 2. delete its neighbors; 3. repeat (until the graph is empty) A) How good/bad is it on general graphs? – Construct a bad instance for that algorithm – As a function of n B) Does it do better than  +1 ?.

58. Problem #2: Max Cut What (easy) approximations can we give for Max Cut?