Combinatorial Dominance Analysis

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

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

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

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.

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

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

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

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.

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

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.

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.

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

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.

OPT Near optimal top O(n) Infeasible Solutions quality line 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 ?

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

OPT MST tour 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

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

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

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

Example:The minimum partition problem • Given is a set of n numbers • V = { a1, a2, …, an} • Find a bipartition (X,Y ) of the indices such that is minimal.

Greedy-type algorithm • Without loss of generality assume a1 ≥ a2 ≥ … ≥ an . • Initiate X = { }, Y = { } . • For j = 1, …, n • Add j to X if , • Otherwise add j to Y .

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 {a2, a3, …, an}. • Now, add a1 to Y’ if , Otherwise add a1 to X ’.

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 2n-1 solutions.

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

Domination Number: domn • Let Pbe a CO problem. • Let A be an approximation for P . • For an instance I of P, the domination numberdomn(I, A) of A on I is the number of feasible solutions of I that are not better than the solution found by A.

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)

Domination Number: domn • Let Pbe a CO problem. • Let A be an approximation for P . • For any size n of P, the domination numberdomn(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.

Domination Ratio: domr • Let Pbe 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 ratiodomn(P, n, A) of an approximation A for P is the minimumof domn(I, A) / sol(I ) taken over all instances I of P of size n.

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.

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.

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

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

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.

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.

Algorithm: greedy maxCut • Algrorithm: • Initiate X = {}, Y = {} • For each j = 1…n Add vj 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 .

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

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

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

CD analysis • Solving for bk we get

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

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

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

OPT Summary MST tour

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

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

Combinatorial Dominance Analysis The End

Combinatorial Dominance Analysis