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1 Combinatorial Dominance Analysis 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 by: Yochai Twitto

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2 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. maximum Cut Problem: maximum Cut Summary

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3 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. maximum Cut Problem: maximum Cut Summary

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4 Background NP complexity class. AA and quality of approximations. The classical approximation ratio analysis. Example: Approximation for TSP.

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5 NP 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.

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6 Approximations So we are satisfied with an approximate solution. Question: How can we measure the solution quality ? Solutions quality line OPT Infeasible Near optimal

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7 Solution Quality Most of the time, naturally derived from the problem definition. If not, it should be given as external information.

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8 The classical Approximation Ratio (For maximization problem) Assume 0 ≤ β ≤ 1. A.r. ≥ β if the solution quality is greater than β·OPT Solutions quality line OPT Infeasible Near optimal ½ OPT

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9 Example: The Traveling Salesman Problem Given a weighted complete graph G, find the optimal tour. We will assume the graph is metric. We will see: The MST approximation. MST approximation ratio analysis.

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10 MST Approximation for TSP Find a minimum spanning tree for G. DFS the tree. Make shortcuts.

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11 MST Approx. ratio analysis Observation: If you remove an edge from a tour then you get a spanning tree! This means that Tour cost more than a minimum spanning tree.

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12 MST Approx. ratio analysis Thus, DFSing the MST is of cost 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 Since the graph is metric.

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13 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. maximum Cut Problem: maximum Cut Summary

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14 Combinatorial Dominance What is a “ combinatorial dominance guarantee ” ? Why do we need such guarantees ? min partition problem Example: the min partition problem. Definitions and notations.

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15 What is a “ combinatorial dominance guarantee ” ? A letter of reference: “ 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.

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16 What is a “ combinatorial dominance guarantee ” ? (cont.) We saw that MST provides a 2-factor approximation. We can ask: Is the returned solution guaranteed to be always in the top O(n) best solutions ? Solutions quality line OPT Infeasible Near optimal top O(n)

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17 Why do we need that ? Let us take another look Let us take another look at the MST approximation for TSP. All other edges of weight 1+ε (not shown)

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18 Why do we need that ? The spanning tree here is a star. DFS + Shortcuts yields OPT = 6 + 4ε ≈ 6 MST tour size: 10 In general: OPT: (n-2)(1+ε) + 2 MST: 2(n-2) + 2 OPT MST tour

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19 Why do we need that ? But this is the worst possible tour! Such kind of analysis is called blackball analysis. Blackball instance

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20 Corollary The approximation ratio analysis gives us only a partial insight of the performance of the algorithm. Dominance analysis makes the picture fuller.

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21 Simple example of dominance analysis The minimum partition problem. Greedy-type algorithm. Combinatorial dominance analysis of the algorithm.

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22 Example: The minimum partition problem Given is a set of n numbers V = { a 1, a 2, …, a n } Find a bipartition (X,Y ) of the indices such that is minimal.

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23 Greedy-type algorithm Without loss of generality assume a 1 ≥ a 2 ≥ … ≥ a n. Initiate X = { }, Y = { }. For j = 1, …, n Add j to X if, Otherwise add j to Y.

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24 Combinatorial dominance analysis of the greedy-type algorithm Observation: Any solution produced by the alg. satisfies. Assume (X ’,Y ’ ) is any solution for min partition for {a 2, a 3, …, a n }. Now, add a 1 to Y ’ if, Otherwise add a 1 to X ’.

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25 Combinatorial dominance analysis of the greedy-type algorithm (cont.) Obtained solution: (X ’’,Y ’’ ). (X ’’, Y ’’ ) is a solution of the original problem. We have Conclusion: The solution provided by the algorithm dominates at least 2 n-1 solutions.

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26 Definitions & Notations Domination number: domn Domination ratio: domr DOM -good approximation DOM -easy problem

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27 Domination Number: domn 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.

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28 domn (example) STSP on 5 vertices. 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, 11, 12, 14, 14 (tours lengths)

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29 Domination Number: domn Let P be a CO problem. Let A be an approximation for P. minimum 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.

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30 Domination Ratio: domr Let P be a CO problem. Let A be an approximation for P. sol( I ) Denote by sol( I ) the number of all feasible solutions of I. minimum 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.

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31 DOM -good approximation A is a DOM -good approximation algorithm for P, if It is a polynomial time complexity alg. There exists a polynomial p(n) in the size of P, such that The domination ratio of A is at least 1/p(n) for any size n of P.

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32 DOM -easy problem 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.

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33 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. Maximum Cut Problem: Maximum Cut Summary

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34 Maximum Cut The problem. Simple greedy algorithm. Combinatorial dominance of the algorithm. We ’ ll see … Maximum Cut is DOM -easy.

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35 Problem: Maximum Cut 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.

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36 Problem: Maximum Cut Denote the average weight of a cut by Notice that. Next: We ’ ll see a simple algorithm which produces solutions that are always better than. We ’ ll show it is a DOM -good approximation for maxCut.

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37 Algorithm: greedy maxCut Algrorithm: Initiate X = {}, Y = {} For each j = 1 … n Add v j to X or Y so as to maximize its marginal value. Theorem: The above algorithm is a 2-factor approximation for maxCut. Moreover, it produces a cut of weight at least.

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38 CD analysis We will show that the number of cuts of weight at most is at least a polynomial part of all cuts Call them “ bad ” cuts Note that this is a general analysis technique. Can be applied to another algs./problems

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39 CD analysis A k-cut is a cut (X, Y) for which |X| = k. A fixed edge crosses k-cuts. Hence the average weight of a k-cut is

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40 CD analysis Let b k be the number of bad k-cuts. i.e. k-cuts of weight less than. Then

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41 CD analysis Solving for b k we get

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42 CD analysis Hence the number of bad cuts in G is at least (by DeMoivre-Laplace theorem)

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43 CD analysis Thus, G has more than bad cuts. Corollary: Maximum Cut is DOM -easy.

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44 Overview Background On approximations and approximation ratio. Combinatorial Dominance What is it ? Definitions & Notations. maximum Cut Problem: maximum Cut Summary

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45 Summary Solutions quality line OPT Infeasible Near optimal ½ OPT Solutions quality line OPT Infeasible Near optimal top O(n)

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46 Summary MST tour OPT

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47 Summary Domination number: domn Domination ratio: domr DOM -good approximation DOM -easy problem

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48 Summary Domn(MST, TSP) = 1 Minimum Partition is DOM -easy. Maximum Cut is DOM -easy. Clique is DOM -hard unless P=NP. blackball

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49 Combinatorial Dominance Analysis

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