Improving Steiner Trees in Network Theory and VLSI Projects

1. Projects Network Theory VLSI PSM 1. Network 1. Steiner trees survivability - run time (Purav doing it)improvement (B) 2. Arbitrary - quality improvement (A) (theoretical) 2. Zero-skew trees problems - run time (B) (problems from - theory (quality impr.) (A) conferences, 3. Floor planning metric problems - computational with graphs ..) geometry 3. Implement fast MST O(E+V logV) (Karger algorithm)

2. Approximation Algorithm Steiner trees in graphs G = (V,E,cost), cost:E  R+ S  V set of terminals if S = V  MST algorithm exactly if S  V  problem is NP-hard MST - heuristic for the Steiner tree problem 1. Construct graph G’= (S,E’,cost’) (for any pair of points define distance=cost of the shortest path between them) there is an algorithm (Mehlhorn) for this problem with O(E+VlogV) 2. Find T=MST(G’) O(n3) Run Dijkstra from one point

3. MST - heuristic for the Steiner tree problem 3. Construct H = T* T*={G-paths corresponding to edges at T} (T* may have some cycles, remove arbitrary edges from cycles) 4. Toutput = MST(H) a a  G’ G c b b c cycles cannot appear, but duplicates are possible

4. Approximation ratio of MSTH Theorem: Approximation ratio of MST heuristic  2 Proof: 2 OPT = Tour  Shortcut Tour  MST  Toutput (+ cost of the longest edge in the green tour) terminals Steiner points

5. Approximation ratio of MSTH Proof that approximation ratio = 2:   > 0  I  STP such that: MST (I)  (2- ) OPT(I) OPT(Ik) = k MST( Ik ) = 2(k-1) supIk (2k-2)/k = 2 1 2 k terminals 2 distance between any 2 terminals=2

6. Traveling Salesperson Problem (TSP) -first problem proved to be NP-hard Given: complete graph G=(V,E,cost) Find: minimum cost tour which visits all nodes traveling salesperson problem - if we have -inequality in G  MST-heuristic = 2 approximation

7. Traveling Salesperson Problem (TSP) 1. Find T = MST(G) 2. H = T+T (traverse this tree visiting each node twice) 3. Tour  Shortcut (H) output tour (heuristic) make a shortcut OPT  MST Approx  2 MST 2 OPT

8. Eulerian Graphs Eulerian graph - you can traverse all edges visiting each only once (Euler, 18th cent., problem: Köningsberg’s bridges) graph Eulerian  all degrees are even Theorem: Graph G is Eulerian  D(G) is bipartite Proof: homework Theorem: In every graph the number of odd degree nodes is even. Proof: Homework red = G black = D(G) - dual of G

9. 1.5 Approximation for TSP Christophides (in 1976) - better heuristic, approx. ratio 1.5 Matching problem Given G = (V,E,cost) |V|=even matching - no 2 edges have common point (there is exact algorithm with run time O(v3)that will find minim. cost) 3 matchings bipartite graph

10. 1.5 Approximation for TSP Algorithm: 1. Find MST 2. Find minimum weight matching M of odd- degree nodes 3. Make Tour M  MST Proof that approx. is 1.5: we know : MST  OPT need to prove: M  0.5 OPT then  Tour  1.5 OPT ?

11. 1.5 Approximation for TSP Proof that M  0.5 OPT optimal  shortcut = M1 +M2  2*M shortcut for odd nodes (shorter because of triang.ineq.) optimal other mathing M2 one matching M1

12. Minimum Vertex Cover • NP - hard problem Given: G = (V,E,cost) cost: V  R+ Find: C  V cost(C)  min such thateach edge in E has at least one end point in C C 2 approximation : - pick an edge e - delete both endpoints of e - repeat until no edges are left

13. Minimum Vertex Cover Let Ce be output of edge-deletion heuristic. Then |Ce|  2 OPT. Proof: Let M be the set of chosen edges in the heuristic Ce = 2|M| OPT  |M| |M| = OPT (M)  OPT(G) we proved  |Ce|  2 OPT ? here all edges of G need to be covered number of edges in M only some edges of G to cover