Approximation Algorithms Lecture 2 10/19/10 1 (Metric) Steiner Tree Problem: Steiner Tree. Given an undirected graph G=(V,E) with non-negative edge + costs c : E Q whose vertex set is partitioned into required and →Steiner vertices, find a minimum cost tree in G that contains all required vertices. In the metric Steiner Tree Problem, G is the complete graph and edge costs satisfy the triangle inequality, i.e. c(u, w) c(u, v)+c(v, w) ≤ for all u, v, w V . ∈ 2 (Metric) Steiner Tree Steiner required 3 (Metric) Steiner Tree Steiner required 4 Metric Steiner Tree Let Π 1 , Π 2 be minimization problems. An approximation factor preserving reduction from Π 1 to Π 2 consists of two poly-time computable functions f,g , such that • for any instance I 1 of Π 1 , I 2 = f ( I 1 ) is an instance of Π2 such that opt ( I ) opt ( I ) , and Π2 2 ≤ Π1 1 • for any solution t of I 2 , s = g ( I 1 ,t ) is a solution of I 1 such that obj (I ,s) obj (I ,t). Π1 1 ≤ Π2 2 Proposition 1 There is an approximation factor preserving reduction from the Steiner Tree Problem to the metric Steiner Tree Problem. 5 Metric Steiner Tree Algorithm 3: Metric Steiner Tree via MST. Given a complete graph G=(V,E) with metric edge costs, let R be its set of required vertices. Output a spanning tree of minimum cost (MST) on vertices R. Theorem 2 Algorithm 3 is a factor 2 approximation algorithm for the metric Steiner Tree Problem. 6 Metric Steiner Tree 7 Metric Steiner Tree Algorithm 3: Metric Steiner Tree via MST. Given a complete graph G=(V,E) with metric edge costs, let R be its set of required vertices. Output a spanning tree of minimum cost (MST) on vertices R. Theorem 2 Algorithm 3 is a factor 2 approximation algorithm for the metric Steiner Tree Problem. Corollary 1 Together with the approximation preserving reduction from the previous slide we obtain a factor 2 approximation algorithm for the Steiner Tree Problem. 8 Steiner Tree An example that our analysis is tight: Steiner required ( n ) c(u, v)=1 × c(u, v)=2 Every MST on the required vertices has cost 2(n-1). Using the Steiner vertex yields a tree of cost n. 9 Traveling Salesman Problem Problem: Traveling Salesman Problem (TSP). Given a complete graph G=(V,E) with non-negative edge + costs c : E Q for all, find a minimum cost cycle visiting every vertex→ exactly once. In the metric TSP, edge costs are required to satisfy the triangle inequality. Theorem 3 For any poly-time computable function α ( n ) , TSP cannot be approximated within a factor of α ( n ) , unless P=NP. 10 Metric TSP Algorithm 4: Metric TSP. 1. Find an MST of G, say T. 2. Compute a minimum cost perfect matching, say M, on the set of odd-degree vertices of T. Add M to T to obtain an Eulerian graph. 3. Find an Euler tour, say Q, of this graph. 4. Output the tour that visits the vertices of G in order of their first appearance in Q. Theorem 4 Algorithm 4 is a factor 3/2 approximation algorithm for the metric TSP. 11 Metric TSP Lemma 2 Let V V , such that V is even, and let M be a minimum ⊆ | | cost perfect matching on V . Then cost( M ) opt / 2 . ≤ An example that our analysis is tight: 1 ... 1 1 1 1 1 1 1 ... 1 1 1 1 n/2 Thick edges indicate the MST found in step 1. 12 Multiway Cut Problem: Multiway Cut. Given an undirected graph G=(V,E) with non-negative edge + weights w : E Q and a set S = s 1 ,...,s k V of terminals, a multiway→ cut is a set of edges{ whose removal} ⊆ disconnects the terminals from each other. The Multiway Cut Problem asks for such a set of minimum total weight. 13 Multiway Cut Problem: Multiway Cut. Given an undirected graph G=(V,E) with non-negative edge + weights w : E Q and a set S = s 1 ,...,s k V of terminals, a multiway→ cut is a set of edges{ whose removal} ⊆ disconnects the terminals from each other. The Multiway Cut Problem asks for such a set of minimum total weight. 14 Multiway Cut For k=2, i.e. given just 2 terminals, this is the standard Min-Cut problem. Recall that the Min-Cut problem can be solved in polynomial time by computing a maximum flow between the terminals (“Min-Cut / Max-Flow Theorem”). The Multiway Cut Problem is NP-hard for any fixed k 3. ≥ 15 Multiway Cut We define an isolating cut for s i to be a set of edges whose removal disconnects s from all s ,j = i . i j Minimum weight isolating cuts can be found in poly time: 1. Merge all terminals in S s into a single vertex s ∗ . \{ i} 2. Compute a min-cut separating s i and s ∗ . si merge terminals into supernode 16 Multiway Cut We define an isolating cut for s i to be a set of edges whose removal disconnects s from all s ,j = i . i j Minimum weight isolating cuts can be found in poly time: 1. Merge all terminals in S s into a single vertex s ∗ . \{ i} 2. Compute a min-cut separating s i and s ∗ . si compute min-cut 17 Multiway Cut We define an isolating cut for s i to be a set of edges whose removal disconnects s from all s ,j = i . i j Minimum weight isolating cuts can be found in poly time: 1. Merge all terminals in S s into a single vertex s ∗ . \{ i} 2. Compute a min-cut separating s i and s ∗ . si 18 Multiway Cut Algorithm 5: Multiway Cut. 1. For each i=1,...,k, compute a minimum weight isolating cut for s i , say C i . 2. Discard the heaviest of these cuts and output the union of the rest, say C. Theorem 5 Algorithm 5 is a factor 2-2/k approximation algorithm for the Multiway Cut Problem. 19.