Traveling Salesman. Given G = (V,V V), find simple cycle C = (v 1,v 2,...,v n,v 1 ) such that n = V and (u,v) C d(u,v) is minimized.

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1 Sanders/van Stee: Approximations- und Online-Algorithmen 1 Traveling Salesman Given G = (V,V V), find simple cycle C = (v 1,v 2,...,v n,v 1 ) such that n = V and (u,v) C d(u,v) is minimized.

2 Sanders/van Stee: Approximations- und Online-Algorithmen 2 Wiederholung Approximationsalgorithmen Online-Algorithmen NP-harte Probleme Unvollständige Informationen suchen gute Lösung suchen gute Lösung Approximationsverhältnis Kompetitivitätsverhältnis Polynomialzeit evtl. exponentielle Zeit TSP, Rucksack, online TSP, Rucksack,... Lastverteilung,... Paging, Ski rental,...

3 Sanders/van Stee: Approximations- und Online-Algorithmen 3 Theorem 1. It is NP-hard to approximate the general TSP within any factor α. Proof. Reduction from Hamilton Cycle... Hamilton Cycle Problem: Given a graph decide whether it contains a simple cycle visiting all nodes

4 Sanders/van Stee: Approximations- und Online-Algorithmen 4 Proof We want to find a Hamilton Cycle in G = (V,E). Consider G = (V,V V) and the weight function d(u,v) = 1 if (u,v) E αn else Suppose G has a Hamilton cycle. Then there is a Hamilton cycle of weight n in G an α-approx. algorithm delivers one with weight αn If there is no Hamilton cycle in G, every Hamilton Cycle in G has weight αn+n 1> αn.

5 Sanders/van Stee: Approximations- und Online-Algorithmen 5 Proof (continued) Assume that there exists an α-approximation algorithm for TSP. Decision algorithm: Run α-approx TSP on G Solution has weight αn Hamilton path exists Else there is no Hamilton cycle. [e.g. Vazirani Theorem 3.6]

6 Sanders/van Stee: Approximations- und Online-Algorithmen 6 Metric TSP G is undirected and obeys the triangle inequality u,v,w V : d(u,w) d(u,v)+d(v,w) Metric completion Consider any connected undirected graph G = (V,E) with weight function c : E R +. Define d(u,v) := shortest path distance from u to v Example: (undirected) street graphs distance table

7 Sanders/van Stee: Approximations- und Online-Algorithmen 7 2-Approximation by MST Lemma 2. The total weight of an MST The total weight of any TSP tour Algorithm: T := MST(G) // weight(t) opt

8 Sanders/van Stee: Approximations- und Online-Algorithmen 8 2-Approximation by MST Lemma 2. The total weight of an MST The total weight of any TSP tour Algorithm: T := MST(G) T := T with every edge doubled // weight(t) opt // weight(t ) 2opt

9 Sanders/van Stee: Approximations- und Online-Algorithmen 9 2-Approximation by MST Lemma 2. The total weight of an MST The total weight of any TSP tour Algorithm: T := MST(G) T := T with every edge doubled T := EulerTour(T ) // weight(t) opt // weight(t ) 2opt // weight(t ) 2opt

10 Sanders/van Stee: Approximations- und Online-Algorithmen 10 2-Approximation by MST Lemma 2. The total weight of an MST The total weight of any TSP tour Algorithm: T := MST(G) T := T with every edge doubled T := EulerTour(T ) output removeduplicates(t ) // weight(t) opt // weight(t ) 2opt // weight(t ) 2opt // shortcutting Exercise: Implementation in timeo (m+nlogn) where m is number of edges before metric completion

11 Sanders/van Stee: Approximations- und Online-Algorithmen 11 Example LateX is soinput totally idiotic that weight: it requires a full 1line 2of white doubled text before it will MST put the figures right side up.

12 Sanders/van Stee: Approximations- und Online-Algorithmen 12 Example input weight: 1 2 LateX is so totally idiotic that it requires a full line of white text before it will put the figures right side up. weight MST

13 Sanders/van Stee: Approximations- und Online-Algorithmen 13 Example input weight: 1 2 doubled MST LateX is so totally idiotic that it requires a full line of white text before it will put the figures right side up. 2 4 MST

14 Sanders/van Stee: Approximations- und Online-Algorithmen 14 Example input weight: 1 2 doubled MST LateX is so totally idiotic that it requires a full line of white text before it will put the figures right side up weight 10 MST Euler tour

15 Sanders/van Stee: Approximations- und Online-Algorithmen 15 Example input weight: 1 2 doubled MST output LateX is so totally idiotic that it requires a full line of white text before it will put the figures right side up weight 10 4 MST Euler tour

16 Sanders/van Stee: Approximations- und Online-Algorithmen 16 Example input weight: 1 2 doubled MST output LateX is so totally idiotic that it requires a full line of white text before it will put the figures right side up weight 10 4 MST Euler tour optimal weight: 6

17 Sanders/van Stee: Approximations- und Online-Algorithmen 17 Proof of Lemma 2 Lemma 2: The total weight of an MST The total weight of any TSP tour Proof: Let T denote the optimal TSP tour remove one edge from T now T is a spanning tree which is no lighter than the MST makes T lighter General Technique: Relaxation here: a TSP path is a special case of a spanning tree

18 Sanders/van Stee: Approximations- und Online-Algorithmen 18 More on TSP Practically better 2-approximations, e.g. lightest edge first Relatively simple yet unpractical 3/2-approximation PTAS for Euclidean TSP Guinea pig for just about any optimization heuristics Optimal solutions for practical instances. Rule of thumb: If it fits in memory it can be solved. [ lines of code is six digit number TSP-like applications are usually more complicated

19 Sanders/van Stee: Approximations- und Online-Algorithmen 19 Online TSP Metric space Algorithms move with speed at most 1 Requests appear over time Future requests are unknown Minimize finishing time (makespan)

20 Sanders/van Stee: Approximations- und Online-Algorithmen 20 Online TSP What is the worst that can happen to an online algorithm? Algorithm is at location X Request occurs somewhere very far away from it, at Y Optimal solution is to serve it immediately No further requests arrive Algorithm still needs to move to Y: high competitive ratio

21 Sanders/van Stee: Approximations- und Online-Algorithmen 21 Online TSP However... The optimal solution must have had enough time to travel to Y before the request arrives It started at the origin, like the online algorithm Idea: do not move too far from the origin Close enough = within a factor of time elapsed

22 Sanders/van Stee: Approximations- und Online-Algorithmen 22 Algorithm Return Home (RH) Whenever a new request arrives, return to O at full speed In O, calculate optimal tour for requests that appeared so far Follow this tour at maximum speed such that distance to O is at most ( 2 1)t at time t, for all t

23 Sanders/van Stee: Approximations- und Online-Algorithmen 23 Theorem 3. Return Home has a competitive ratio of 2+1. Let t be the time at which the last request arrives. Clearly opt t. If RH does not slow down after time t, it needs time at most t +( 2 1)t + opt ( 2+1)opt let the last request for which RH slows down be a distance x from the origin. RH serves it at time x/( 2 1) = ( 2+1)x. RH serves remainder of tour (T x ) at full speed. We have opt = x+t x and RH is ready at time ( 2+1)x+T x ( 2+1)opt

24 Sanders/van Stee: Approximations- und Online-Algorithmen 24 Theorem 3. Return Home has a competitive ratio of 2+1. Let t be the time at which the last request arrives. Clearly opt t. If RH does not slow down after time t, it needs time at most t +( 2 1)t + opt ( 2+1)opt) Else, let the last request for which RH slows down be a distance x from the origin. RH serves it at time x/( 2 1) = ( 2+1)x. RH serves remainder of tour (T x ) at full speed. We have opt x+t x and RH is ready at time ( 2+1)x+T x ( 2+1)opt

25 Sanders/van Stee: Approximations- und Online-Algorithmen 25 Theorem 3. Return Home has a competitive ratio of 2+1. Let t be the time at which the last request arrives. Clearly opt t. If RH does not slow down after time t, it needs time at most t +( 2 1)t + opt ( 2+1)opt) Else, let the last request for which RH slows down be a distance x from the origin. RH serves it at time x/( 2 1) = ( 2+1)x. RH serves remainder of tour (T x ) at full speed. We have opt = x+t x and RH is ready at time ( 2+1)x+T x ( 2+1)opt

26 Sanders/van Stee: Approximations- und Online-Algorithmen 26 Remarks about RH Uses exponential time to calculate optimal tour Nevertheless, leaves O immediately after arriving there Theoretical result More reasonable: use some approximation algorithm in O Competitive ratio increases Time for serving requests should be much longer than time needed to calculate approximate tour

27 Sanders/van Stee: Approximations- und Online-Algorithmen 27 Steiner Trees [C. F. Gauss 18??] Given G = (V,E), with positive edge weights cost : E R + V = R F, i.e., Required vertices and Steiner vertices find a minimum cost tree T E that connects all required vertices weight: 1 2 u,v R : T contains a u-v path THE network design problem

28 Sanders/van Stee: Approximations- und Online-Algorithmen 28 Metric Steiner Trees Find Steiner tree in complete graph with triangle inequality u,v,w V : d(u,w) d(u,v)+d(v,w) Easier? No!

29 Sanders/van Stee: Approximations- und Online-Algorithmen 29 Approximation Factor Preserving Reduction Steiner Tree of G? Metric Steiner Tree of G? Complete the graph G. u,v V : cost(u,v) := shortest path distance between u and v weight: 1 2 we only add edges. Hence, opt(g ) opt(g). Now consider any Steiner tree I G. We construct a Steiner tree I G with cost(i) cost(i ): replace edges paths remove edges from cycles

30 Sanders/van Stee: Approximations- und Online-Algorithmen 30 Examples From Metric Steiner Tree to Steiner Tree weight: 1 2

31 Sanders/van Stee: Approximations- und Online-Algorithmen 31 2-Approximation by MST G weight: 1 2 Given metric graph G = (R F,E) Find MST T of subgraph G R induced by R T* Theorem 4. cost(t) 2opt Proof: consider optimal solution T (cost(t ) = opt)) double edges of T find Euler tour B (cost(b) = 2opt) use shortcuts to obtain Hamilton cycle H (cost(h) cost(b) = 2opt). drop heaviest edge. Now H is a spanning tree of G R cost(mst) cost(h) cost(b) = 2opt

32 Sanders/van Stee: Approximations- und Online-Algorithmen 32 2-Approximation by MST G weight: 1 2 Given metric graph G = (R F,E) Find MST T of subgraph G R induced by R T* Theorem 4. cost(t) 2opt Proof: consider optimal solution T (cost(t ) = opt)) double edges of T find Euler tour B (cost(b) = 2opt) use shortcuts to obtain Hamilton cycle H (cost(h) cost(b) = 2opt). drop heaviest edge. Now H is a spanning tree of G R cost(mst) cost(h) cost(b) = 2opt

33 Sanders/van Stee: Approximations- und Online-Algorithmen 33 2-Approximation by MST G weight: 1 2 Given metric graph G = (R F,E) Find MST T of subgraph G R induced by R T* Theorem 4. cost(t) 2opt Proof: consider optimal solution T (cost(t ) = opt)) double edges of T find Euler tour B (cost(b) = 2opt) use shortcuts to obtain Hamilton cycle H B (cost(h) cost(b) = 2opt). drop heaviest edge. Now H is a spanning tree of G R cost(mst) cost(h) cost(b) = 2opt

34 Sanders/van Stee: Approximations- und Online-Algorithmen 34 2-Approximation by MST G weight: 1 2 Given metric graph G = (R F,E) Find MST T of subgraph G R induced by R T* Theorem 4. cost(t) 2opt Proof: consider optimal solution T (cost(t ) = opt)) double edges of T find Euler tour B (cost(b) = 2opt) use shortcuts to obtain Hamilton cycle H (cost(h) cost(b) = 2opt). drop heaviest edge. Now H is a spanning tree of G R H cost(mst) cost(h) cost(b) = 2opt B

35 Sanders/van Stee: Approximations- und Online-Algorithmen 35 Tight Example weight: G T* MST cost(t ) = 4,cost(MST) = 6. More general: R = n cost(t ) = n, cost(mst) = (2 ε)(n 1)

36 Sanders/van Stee: Approximations- und Online-Algorithmen 36 Fast Implementation Theorem 5. MST(G R ) can be computed in timeo (m+nlogn) where m, n refer to the graph before metric completion. [K. Mehlhorn, A Faster Approximation Algorithm for the Steiner Problem in Graphs, Information Processing Letters 27(1988) ]

37 Sanders/van Stee: Approximations- und Online-Algorithmen 37 The Graph Voronoi Diagram Voronoi diagram of a set of points R in the plane: Assign each point in the plane to its closest point in R. Here, assign each node v V to its closest required node s(v) R. Use shortest path distance d. break ties arbitrarily weight: 1 2

38 Sanders/van Stee: Approximations- und Online-Algorithmen 38 Computing the Graph Voronoi Diagram Solve single source shortest path problem from s 0 in G 0 := ({s 0 } R F,E {(s 0,v) : v R}) where the new edges have zero weight. s0 TimeO (m+nlogn) (Dijkstra s Algorihm using Fibonacci heaps)

39 Sanders/van Stee: Approximations- und Online-Algorithmen 39 (Yet another) Auxiliary Graph G 1 := (R,E 1 ) where (u,v) E 1 if (u,v ) E such that u is assigned to u and v is assigned to v in the graph Voronoi diagram of G. cost (u,v):= d(u,u )+cost(u,v )+d(v,v). Can be computed in linear time from shortest path tree in G 0

40 Sanders/van Stee: Approximations- und Online-Algorithmen 40 Finishing Up build G 1 from G //O (m+nlogn) G 2 := MST(G 1 ) //O (m+nlogn) G 3 := G 2 with edges replaced by shortest paths in G //O (m+n) G 4 := MST(G 3 ) // omittable while v F :v is a leaf of G 4 do // [Floren 91] remove the edge leading to v in G 4 return G 4

41 Sanders/van Stee: Approximations- und Online-Algorithmen 41 Lemma 6. MST(G 1 ) = MST(G R ) Why this is not obvious: 3 We assume wlog that all the MSTs are unique (no equal edge weights)

42 Sanders/van Stee: Approximations- und Online-Algorithmen 42 Lemma: MST(G 1 ) = MST(G R ) Proof: Assume G 2 = MST(G 1 ) MST(G R ) Case A: (s,t) E 2 but (s,t) E 1 Case B: (s,t) E 2 with cost (s,t) > d(s,t) 3

43 Sanders/van Stee: Approximations- und Online-Algorithmen 43 Proof Let s = v 0,...v i,v i+1,...,v k = t denote shortest s t path. If s(v i ) s(v i+1 ) we have (s(v i ),s(v i+1 )) E 1. d(s(v i ),s(v i+1 )) cost (s(v i ),s(v i+1 )) d(s(v i ),v i )+cost(v i,v i+1 )+d(v i+1,s(v i+1 )) d(s,v i )+cost(v i,v i+1 )+d(v i+1,t) = d(s,t) < cost (s,t) (case B only) remove (s,t) from E 2 G 2 gets disconnected i : s(v i ),s(v i+1 ) are in different components. E 2 \ {(s,t)} {(s(v i ),s(v i+1 ))} different MST s(v ) i s(v ) i+1 s t

44 Sanders/van Stee: Approximations- und Online-Algorithmen 44 More on Steiner trees Complicated Approximation down to 1.55 [Robins and Zelikovsky 00] Optimal solutions for large practical instances. [PhD Polzin,Daneshmand, 2003, Dortmund, Mannheim, MPII-SB] Many applications: multicasting in networks, VLSI design(?), phylogeny reconstruction

45 Sanders/van Stee: Approximations- und Online-Algorithmen 45 Directed Steiner Trees Theorem 7. It is hard to approximate the directed Steiner tree problem within a factor ln R. Proof by approximation preserving reduction from the set covering problem

46 Sanders/van Stee: Approximations- und Online-Algorithmen 46 The Set Covering Problem Given universe U, subsetss = {S 1,...,S k }, cost function c :S N. Find minimum costs S such that [ S = U S S Theorem 8. It is hard to approximate the set covering problem within a factor ln U. [Feige 98]

47 Sanders/van Stee: Approximations- und Online-Algorithmen 47 Approximation Preserving Reduction: Directed Steiner Tree from Set Covering r S U V = {r} S U E = {(r,s) : S S } cost c(s) {(S,u) : S S,u S} cost 0.