Combinatorial Dominance Analysis by Yochai Twitto Keywords Combinatorial
Combinatorial Dominance Analysis by: Yochai Twitto Keywords: Combinatorial Optimization (CO) Approximation Algorithms (AA) Approximation Ratio (a. r) Combinatorial Dominance (CD) Domination number/ratio (domn, domr) DOM-good approximation DOM-easy problem 1
Overview n Background n n Combinatorial Dominance n n On approximations and approximation ratio. What is it ? Definitions & Notations. Problem: maximum Cut Summary 2
Overview n Background n n Combinatorial Dominance n n On approximations and approximation ratio. What is it ? Definitions & Notations. Problem: maximum Cut Summary 3
Background n NP complexity class. n AA and quality of approximations. n The classical approximation ratio analysis. n Example: Approximation for TSP. 4
NP n If P ≠ NP, then finding the optimum of NP-hard problem is difficult. If P = NP, P would encompass the NP and NP-Complete areas. 5
Approximations n n So we are satisfied with an approximate solution. OPT Near optimal Question: n How can we measure the solution quality ? Infeasible Solutions quality line 6
Solution Quality n n Most of the time, naturally derived from the problem definition. If not, it should be given as external information. 7
The classical Approximation Ratio (For maximization problem) n n Assume 0 ≤ β ≤ 1. A. r. ≥ β if n OPT Near optimal ½ OPT the solution quality is greater than β·OPT Infeasible Solutions quality line 8
Example: The Traveling Salesman Problem n n n Given a weighted complete graph G, find the optimal tour. We will assume the graph is metric. We will see: n n The MST approximation ratio analysis. 9
MST Approximation for TSP n n n Find a minimum spanning tree for G. DFS the tree. Make shortcuts. 10
MST Approx. ratio analysis n n Observation: If you remove an edge from a tour then you get a spanning tree! This means that n Tour cost more than a minimum spanning tree. 11
MST Approx. ratio analysis n Thus, DFSing the MST is of cost n n n No more than twice MST cost. I. e. no more than twice OPT. After shortcuts we get a tour with cost at most twice the optimum n Since the graph is metric. 12
Overview n Background n n Combinatorial Dominance n n On approximations and approximation ratio. What is it ? Definitions & Notations. Problem: maximum Cut Summary 13
Combinatorial Dominance n n What is a “combinatorial dominance guarantee” ? Why do we need such guarantees ? Example: the min partition problem Definitions and notations. 14
What is a “combinatorial dominance guarantee” ? n A letter of reference: n n “She is half as good as I am, but I am the best in the world…” “she finished first in my class of 75 students…” The former is akin to an approximation ratio. The latter to combinatorial dominance guarantee. 15
What is a “combinatorial dominance guarantee” ? (cont. ) n n We saw that MST provides a 2 -factor approximation. We can ask: n Is the returned solution guaranteed to be always in the top O(n) best solutions ? OPT top O(n) Near optimal Infeasible Solutions quality line 16
Why do we need that ? n Let us take another look at the MST approximation for TSP. All other edges of weight 1+ε (not shown) 17
Why do we need that ? n n The spanning tree here is a star. DFS + Shortcuts yields OPT = 6 + 4ε ≈ 6 OPT MST tour size: 10 In general: MST tour OPT: (n-2)(1+ε) + 2 MST: 2(n-2) + 2 18
Why do we need that ? n n But this is the worst possible tour! Such kind of analysis is called blackball analysis. Blackball instance 19
Corollary n n The approximation ratio analysis gives us only a partial insight of the performance of the algorithm. Dominance analysis makes the picture fuller. 20
Simple example of dominance analysis n The minimum partition problem. n Greedy-type algorithm. n Combinatorial dominance analysis of the algorithm. 21
Example: The minimum partition problem n Given is a set of n numbers n n V = { a 1, a 2, …, an} Find a bipartition (X, Y ) of the indices such that is minimal. 22
Greedy-type algorithm n n n Without loss of generality assume a 1 ≥ a 2 ≥ … ≥ a n. Initiate X = { }, Y = { }. For j = 1, …, n n n Add j to X if Otherwise add j to Y. , 23
Combinatorial dominance analysis of the greedy-type algorithm n n n Observation: Any solution produced by the alg. satisfies. Assume (X ’, Y ’) is any solution for min partition for {a 2, a 3, …, an}. Now, add a 1 to Y ’ if , Otherwise add a 1 to X ’. 24
Combinatorial dominance analysis of the greedy-type algorithm (cont. ) n n Obtained solution: (X ’’, Y ’’) is a solution of the original problem. We have Conclusion: n The solution provided by the algorithm dominates at least 2 n-1 solutions. 25
Definitions & Notations n n Domination number: domn Domination ratio: domr DOM-good approximation DOM-easy problem 26
Domination Number: domn n Let P be a CO problem. Let A be an approximation for P. For an instance I of P, the domination number domn(I, A) of A on I is the number of feasible solutions of I that are not better than the solution found by A. 27
domn (example) n STSP on 5 vertices. n n There exist 12 tours If A returns a tour of length 7 then domn(I, A) = 8 4, 5, 5, 6, 7, 9, 9, 11, 12, 14 (tours lengths) 28
Domination Number: domn n Let P be a CO problem. Let A be an approximation for P. For any size n of P, the domination number domn(P, n, A) of an approximation A for P is the minimum of domn(I, A) over all instances I of P of size n. 29
Domination Ratio: domr n n Let P be a CO problem. Let A be an approximation for P. Denote by sol(I ) the number of all feasible solutions of I. For any size n of P, the domination ratio domn(P, n, A) of an approximation A for P is the minimum of domn(I, A) / sol(I ) taken over all instances I of P of size n. 30
DOM-good approximation n A is a DOM-good approximation algorithm for P, if n n It is a polynomial time complexity alg. There exists a polynomial p(n) in the size of P, such that n The domination ratio of A is at least 1/p(n) for any size n of P. 31
DOM-easy problem n n A CO problem is a DOM-easy problem if it admits a DOM-good approximation. Problems not having this property are DOM-hard. Corollary: Minimum Partition is DOM-easy. Furthermore, p(n) is a constant. 32
Overview n Background n n Combinatorial Dominance n n On approximations and approximation ratio. What is it ? Definitions & Notations. Problem: Maximum Cut Summary 33
Maximum Cut n n n The problem. Simple greedy algorithm. Combinatorial dominance of the algorithm. We’ll see… Maximum Cut is DOM-easy. 34
Problem: Maximum Cut n n Input: weighted complete graph G=(V, E, w) Find a bipartition (X, Y) of V maximizing the sum Denote n = |V|. Let W be the sum of weights of all edges. 35
Problem: Maximum Cut n n n Denote the average weight of a cut by Notice that. Next: n n We’ll see a simple algorithm which produces solutions that are always better than. We’ll show it is a DOM-good approximation for max. Cut. 36
Algorithm: greedy max. Cut n Algrorithm: n n n Initiate X = {}, Y = {} For each j = 1…n Add vj to X or Y so as to maximize its marginal value. Theorem: n n The above algorithm is a 2 -factor approximation for max. Cut. Moreover, it produces a cut of weight at least. 37
CD analysis n We will show that the number of cuts of weight at most is at least a polynomial part of all cuts n n Call them “bad” cuts Note that this is a general analysis technique. n Can be applied to another algs. /problems 38
CD analysis n A k-cut is a cut (X, Y) for which |X| = k. A fixed edge crosses k-cuts. n Hence the average weight of a k-cut is n 39
CD analysis n Let bk be the number of bad k-cuts. n n i. e. k-cuts of weight less than. Then 40
CD analysis n Solving for bk we get 41
CD analysis n Hence the number of bad cuts in G is at least (by De. Moivre-Laplace theorem) 42
CD analysis n n Thus, G has more than cuts. bad Corollary: Maximum Cut is DOM-easy. 43
Overview n Background n n Combinatorial Dominance n n On approximations and approximation ratio. What is it ? Definitions & Notations. Problem: maximum Cut Summary 44
Summary OPT Near optimal top O(n) Near optimal ½ OPT Infeasible Solutions quality line 45
Summary OPT MST tour 46
Summary n n Domination number: domn Domination ratio: domr DOM-good approximation DOM-easy problem 47
Summary all n b k c bla Domn(MST, TSP) = 1 n Minimum Partition is DOM-easy. Maximum Cut is DOM-easy. n Clique is DOM-hard unless P=NP. n 48
Combinatorial Dominance Analysis 49
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