Arora s PTAS for the Euclidean TSP. R. Inkulu (Arora s PTAS for the Euclidean TSP) 1 / 23

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1 Arora s PTAS for the Euclidean TSP R. Inkulu (Arora s PTAS for the Euclidean TSP) 1 / 23

2 Problem description Given a set P of points in Euclidean plane, find a tour of minimum of cost that visits all the points of P. (Arora s PTAS for the Euclidean TSP) 2 / 23

3 Problem description Given a set P of points in Euclidean plane, find a tour of minimum of cost that visits all the points of P. known to be NP-hard objective of this talk: algo (PTAS) to compute an approximate tour whose expected length is (1 + ɛ)τ OPT in O(n( lg n ɛ )O( 1 ɛ ) ) time (Arora s PTAS for the Euclidean TSP) 2 / 23

4 Problem description Given a set P of points in Euclidean plane, find a tour of minimum of cost that visits all the points of P. known to be NP-hard objective of this talk: algo (PTAS) to compute an approximate tour whose expected length is (1 + ɛ)τ OPT in O(n( lg n ɛ )O( 1 ɛ ) ) time for the convenience, we let the following: 1 > ɛ > 1 n P is contained in [ 1 2, 1 ] [1, 1] 2 diam(p) 1 4 (Arora s PTAS for the Euclidean TSP) 2 / 23

5 Outline 1 A first attempt: Steiner points on the Hannan grid rectangle boundaries 2 Another attempt: portals placed on the quadtree square boundaries (Arora s PTAS for the Euclidean TSP) 3 / 23

6 Partition into subproblems consider rectangles from Hannan grid GR of points in P: grid obtained by drawing horizontal and vertical lines at every point p P (Arora s PTAS for the Euclidean TSP) 4 / 23

7 Solve subproblems check every feasible tour enter/exit order of the Steiner points placed along the edges of the Hannan grid rectangle (Arora s PTAS for the Euclidean TSP) 5 / 23

8 Assemble the subtours make sure that when subproblem solutions are merged, it should result in a tour (Arora s PTAS for the Euclidean TSP) 6 / 23

9 Dynamic programming over Hannan grid rectangles for every (sub)rectangle R GR, suppose an optimal tour is known to cross the boundary of R at most t times, then τ can be computed in O(n O(t) ) time: * for each rectangle try all possible subrectangles contained in it while memoizing the solutions * number of recursive calls are O(n 4 ( t i=2 (2( n 2) ) i i!)), which is O(n 4 n 2t t!) * work associated with each such call is O(n O(t) ), ignoring the work in the recursive calls (Arora s PTAS for the Euclidean TSP) 7 / 23

10 Dynamic programming over Hannan grid rectangles for every (sub)rectangle R GR, suppose an optimal tour is known to cross the boundary of R at most t times, then τ can be computed in O(n O(t) ) time: * for each rectangle try all possible subrectangles contained in it while memoizing the solutions * number of recursive calls are O(n 4 ( t i=2 (2( n 2) ) i i!)), which is O(n 4 n 2t t!) * work associated with each such call is O(n O(t) ), ignoring the work in the recursive calls but an optimal tour that crosses each rectangle at most t times may not always exist (Arora s PTAS for the Euclidean TSP) 7 / 23

11 Outline 1 A first attempt: Steiner points on the Hannan grid rectangle boundaries 2 Another attempt: portals placed on the quadtree square boundaries (Arora s PTAS for the Euclidean TSP) 8 / 23

12 Using quadtree q a b c d b1 b2 b3 b4 b21 b b23 b24 snapping (perturbing) each point in P to closest quadtree grid vertex and 1 having the smallest quadtree square of side length while the n 32 ɛ diameter of P being larger than 1/4: height of quadtree is O(lg n) letting no node of the quadtree is empty: number of nodes is O(n lg n) (Arora s PTAS for the Euclidean TSP) 9 / 23

13 Dynamic programming over squares of quadtree grid interaction between problems is according to the organization of quadtree nodes memoize the subproblem solutions base case: if a square has O(1/ɛ) points, then solve that subprobelm by brute force (Arora s PTAS for the Euclidean TSP) 10 / 23

14 Dynamic programming over squares of quadtree grid interaction between problems is according to the organization of quadtree nodes memoize the subproblem solutions base case: if a square has O(1/ɛ) points, then solve that subprobelm by brute force but how to limit the interaction between any two subproblems? (Arora s PTAS for the Euclidean TSP) 10 / 23

15 Portals along boundaries of quadtree squares introduce equi-spaced m = O( lg n ɛ ) portals (Steiner points) along each side of each quadtree square together with four at the corners of the same 1 1 choosing m + 1 as a power of two, each portal on the sides of a level i 1 square are at the same location as a portal on the side of some level i square contained in the level i 1 square (Arora s PTAS for the Euclidean TSP) 11 / 23

16 Requisite characteristics of a constrained tour tour is permitted to cross an edge e of a quadree square only via portals on e (Arora s PTAS for the Euclidean TSP) 12 / 23

17 Requisite characteristics of a constrained tour tour is permitted to cross an edge e of a quadree square only via portals on e no portal can be used more than twice (if a tour uses a portal more than twice, it can be modified to use it at most twice) (Arora s PTAS for the Euclidean TSP) 12 / 23

18 Requisite characteristics of a constrained tour tour is permitted to cross an edge e of a quadree square only via portals on e no portal can be used more than twice (if a tour uses a portal more than twice, it can be modified to use it at most twice) tour can use at most r = O( 1 ɛ ) portals corresponding to any side of any quadtree square i.e., tour need to be r-light w.r.t. any quadtree square edge (Arora s PTAS for the Euclidean TSP) 12 / 23

Requisite characteristics of a constrained tour tour is permitted to cross an edge e of a quadree square only via portals on e no portal can be used more than twice (if a tour uses a portal more than

19 Requisite characteristics of a constrained tour tour is permitted to cross an edge e of a quadree square only via portals on e no portal can be used more than twice (if a tour uses a portal more than twice, it can be modified to use it at most twice) tour can use at most r = O( 1 ɛ ) portals corresponding to any side of any quadtree square i.e., tour need to be r-light w.r.t. any quadtree square edge is it possible to have such a tour while being a good apprx to an optimal tour? (Arora s PTAS for the Euclidean TSP) 12 / 23

20 Time complexity number of ways the above said tour interacts with any quadtree square is 8r ) i=0 i! i.e., ( 1 ɛ lg n)o( 1 ɛ ) ) ( 2(4m+4) i since a subproblem at each quadtree node (ignoring the recursive subproblems) can be solved in (( 1 ɛ lg n)o( 1 ɛ ) )) O(1) time, the overall running time is n( 1 ɛ lg n)o( 1 ɛ ) (Arora s PTAS for the Euclidean TSP) 13 / 23

21 Sliding grid Randomly choose the unit grid origin from [0, 0] [ 1 2, 1 2 ] helps in giving expected error bounds (see the following slides) (Arora s PTAS for the Euclidean TSP) 14 / 23

22 Expected error due to snapping points in P to grid vertices for each snapped point, the expected error in walking to its true location and back is n( 2 n 32 ) ɛ 2 OPT (since π OPT 1 4 ) ɛ (Arora s PTAS for the Euclidean TSP) 15 / 23

23 The other possible errors Let Q be set of points due to snapping set P of points to grid vertices. In the following, we compare the cost of optimal tour of Q with the constrained tour that we are building for Q. To recapitulate, the errors are due to the following: limiting the tour to be r-light snapping the tour intersection with edges of quadtree squares to portals (Arora s PTAS for the Euclidean TSP) 16 / 23

24 Patch the path with respect to a line segment q 1 s p 1 π π q 1 s p 1 q 2 p 2 q 2 p 2 q 3 p 3 q 3 p 3 q 4 p 4 find an Eulerian tour in the graph replace a closed curve π that crosses a line segment s at least three times with a closed curve π that crosses s at most twice so that π π + ( i being odd p ip i+1 + q i q i+1 ) + 2 s π + 4 s as a corollary, viewing the portal is of zero lengthed segment, any optimal solution need to use any portal at most twice (Arora s PTAS for the Euclidean TSP) 17 / 23

25 Patching edges of quadtree while considering quadtree nodes in bottom-to-top, patch the tour w.r.t. an edge e of a quadtree rectangle if the current tour intersects e more than r times so that it intersects e at most twice after patching 2 level number of root is zero (Arora s PTAS for the Euclidean TSP) 18 / 23

26 Patching edges of quadtree while considering quadtree nodes in bottom-to-top, patch the tour w.r.t. an edge e of a quadtree rectangle if the current tour intersects e more than r times so that it intersects e at most twice after patching expected number of intersections of a polygonal curve s with the grid at level 2 i (denoted with G i ), is 2 i+1 s prob argument is due to sliding grid (1) (homework from a talk!) 2 level number of root is zero (Arora s PTAS for the Euclidean TSP) 18 / 23

27 Patching edges of quadtree while considering quadtree nodes in bottom-to-top, patch the tour w.r.t. an edge e of a quadtree rectangle if the current tour intersects e more than r times so that it intersects e at most twice after patching expected number of intersections of a polygonal curve s with the grid at level 2 i (denoted with G i ), is 2 i+1 s prob argument is due to sliding grid (1) (homework from a talk!) given that a point p belongs to open edges of quadtree grid at level i, then with probability 1 2, point p belongs to open edges of quadtree grid at level i 1 2 level number of root is zero (Arora s PTAS for the Euclidean TSP) 18 / 23

28 Patching edges of quadtree while considering quadtree nodes in bottom-to-top, patch the tour w.r.t. an edge e of a quadtree rectangle if the current tour intersects e more than r times so that it intersects e at most twice after patching expected number of intersections of a polygonal curve s with the grid at level 2 i (denoted with G i ), is 2 i+1 s prob argument is due to sliding grid (1) (homework from a talk!) given that a point p belongs to open edges of quadtree grid at level i, then with probability 1 2, point p belongs to open edges of quadtree grid at level i 1 due to patching lemma, error in patching any one edge of G i is 4 2 i (as the length of any edge of G i is 1/2 i ) 2 level number of root is zero (Arora s PTAS for the Euclidean TSP) 18 / 23

29 Expected error due to patching the path at every level from bottom-to-top F i : number of patching operations performed at i th level; Y i : number of times the current tour crosses the edges of G i just before considering G i for patching; n i : number of times the tour crosses G i after patching; H: height of quadtree E[err] (Arora s PTAS for the Euclidean TSP) 19 / 23

30 Expected error due to patching the path at every level from bottom-to-top F i : number of patching operations performed at i th level; Y i : number of times the current tour crosses the edges of G i just before considering G i for patching; n i : number of times the tour crosses G i after patching; H: height of quadtree E[err] = E[ H i=1 4F i 2 i ] (Arora s PTAS for the Euclidean TSP) 19 / 23

31 Expected error due to patching the path at every level from bottom-to-top F i : number of patching operations performed at i th level; Y i : number of times the current tour crosses the edges of G i just before considering G i for patching; n i : number of times the tour crosses G i after patching; H: height of quadtree E[err] = E[ H i=1 4F i 2 i ] E[2F 1 + H i=2 4F i 2 i ] (Arora s PTAS for the Euclidean TSP) 19 / 23

32 Expected error due to patching the path at every level from bottom-to-top F i : number of patching operations performed at i th level; Y i : number of times the current tour crosses the edges of G i just before considering G i for patching; n i : number of times the tour crosses G i after patching; H: height of quadtree E[err] = E[ H i=1 4F i 2 i ] E[2F 1 + H i=2 4F i 2 i ] E[2F 1 ] + 4 r 2 H i=2 (since E[Y i 1 ] = E[ ni 2 E[Y i ] 2E[Y i 1 ] 2 i ] E[ Yi (r 2)Fi 2 ] i.e., E[F i ] 1 r 2 (E[Y i] 2E[Y i 1 ])) (Arora s PTAS for the Euclidean TSP) 19 / 23

33 Expected error due to patching the path at every level from bottom-to-top F i : number of patching operations performed at i th level; Y i : number of times the current tour crosses the edges of G i just before considering G i for patching; n i : number of times the tour crosses G i after patching; H: height of quadtree E[err] = E[ H i=1 4F i 2 i ] E[2F 1 + H i=2 4F i 2 i ] E[2F 1 ] + 4 r 2 H i=2 E[2 Y 1 (since E[Y i 1 ] = E[ ni 2 r 2 ] + 4 r 2 ( E[Y H] E[Y i ] 2E[Y i 1 ] 2 i E[Y 1] 2 H 2 ) (since F 1 Y 1 /r Y 1 /(r 2)) ] E[ Yi (r 2)Fi 2 ] i.e., E[F i ] 1 r 2 (E[Y i] 2E[Y i 1 ])) (Arora s PTAS for the Euclidean TSP) 19 / 23

34 Expected error due to patching the path at every level from bottom-to-top F i : number of patching operations performed at i th level; Y i : number of times the current tour crosses the edges of G i just before considering G i for patching; n i : number of times the tour crosses G i after patching; H: height of quadtree E[err] = E[ H i=1 4F i 2 i ] E[2F 1 + H i=2 4F i 2 i ] E[2F 1 ] + 4 r 2 H i=2 E[2 Y 1 (since E[Y i 1 ] = E[ ni 2 r 2 ] + 4 r 2 ( E[Y H] E[Y i ] 2E[Y i 1 ] 2 i E[Y 1] 2 H 2 ) (since F 1 Y 1 /r Y 1 /(r 2)) = 4 r 2 ( E[Y 1] 2 + E[Y H] 2 H E[Y 1] 2 ) ] E[ Yi (r 2)Fi 2 ] i.e., E[F i ] 1 r 2 (E[Y i] 2E[Y i 1 ])) (Arora s PTAS for the Euclidean TSP) 19 / 23

35 Expected error due to patching the path at every level from bottom-to-top F i : number of patching operations performed at i th level; Y i : number of times the current tour crosses the edges of G i just before considering G i for patching; n i : number of times the tour crosses G i after patching; H: height of quadtree E[err] = E[ H i=1 4F i 2 i ] E[2F 1 + H i=2 4F i 2 i ] E[2F 1 ] + 4 r 2 H i=2 E[2 Y 1 (since E[Y i 1 ] = E[ ni 2 r 2 ] + 4 r 2 ( E[Y H] = 4 r 2 ( E[Y 1] 2 + E[Y H] 4 r 2 E[Y i ] 2E[Y i 1 ] 2 i E[Y 1] 2 H 2 ) (since F 1 Y 1 /r Y 1 /(r 2)) E[Y 1] 2 H 2 ) E[Y H ] 2 H ] E[ Yi (r 2)Fi 2 ] i.e., E[F i ] 1 r 2 (E[Y i] 2E[Y i 1 ])) (Arora s PTAS for the Euclidean TSP) 19 / 23

36 Expected error due to patching the path at every level from bottom-to-top F i : number of patching operations performed at i th level; Y i : number of times the current tour crosses the edges of G i just before considering G i for patching; n i : number of times the tour crosses G i after patching; H: height of quadtree E[err] = E[ H i=1 4F i 2 i ] E[2F 1 + H i=2 4F i 2 i ] E[2F 1 ] + 4 r 2 H i=2 (since E[Y i 1 ] = E[ ni 2 r 2 ] + 4 r 2 ( E[Y H] E[Y i ] 2E[Y i 1 ] 2 i E[2 Y 1 E[Y 1] 2 H 2 ) (since F 1 Y 1 /r Y 1 /(r 2)) = 4 r 2 ( E[Y 1] 2 + E[Y H] E[Y 1] 2 H 2 ) 4 E[Y H ] r 2 2 H = 4 2 H+1 OPT r 2 2 H ] E[ Yi (r 2)Fi 2 ] i.e., E[F i ] 1 r 2 (E[Y i] 2E[Y i 1 ])) (Arora s PTAS for the Euclidean TSP) 19 / 23

37 Expected error due to patching the path at every level from bottom-to-top F i : number of patching operations performed at i th level; Y i : number of times the current tour crosses the edges of G i just before considering G i for patching; n i : number of times the tour crosses G i after patching; H: height of quadtree E[err] = E[ H i=1 4F i 2 i ] E[2F 1 + H i=2 4F i 2 i ] E[2F 1 ] + 4 r 2 H i=2 (since E[Y i 1 ] = E[ ni 2 r 2 ] + 4 r 2 ( E[Y H] E[Y i ] 2E[Y i 1 ] 2 i E[2 Y 1 E[Y 1] 2 H 2 ) (since F 1 Y 1 /r Y 1 /(r 2)) = 4 r 2 ( E[Y 1] 2 + E[Y H] E[Y 1] 2 H 2 ) 4 E[Y H ] r 2 2 H = 4 2 H+1 OPT r 2 2 H ] E[ Yi (r 2)Fi 2 ] i.e., E[F i ] 1 r 2 (E[Y i] 2E[Y i 1 ])) = 8 r 2 OPT (Arora s PTAS for the Euclidean TSP) 19 / 23

38 Expected error due to snapping tour to portals after applying patching lemma to grid squares at all levels, the tour intersects with any edge of any grid square at most twice (Arora s PTAS for the Euclidean TSP) 20 / 23

39 Expected error due to snapping tour to portals after applying patching lemma to grid squares at all levels, the tour intersects with any edge of any grid square at most twice in G i, maximum error in snapping tour to portals of G i occurs when there are two such intersections, and it is 2( 1/2i 2(m+1) ) (Arora s PTAS for the Euclidean TSP) 20 / 23

40 Expected error due to snapping tour to portals after applying patching lemma to grid squares at all levels, the tour intersects with any edge of any grid square at most twice in G i, maximum error in snapping tour to portals of G i occurs when there are two such intersections, and it is 2( 1/2i 2(m+1) ) let Z i be the number of intersections of τ OPT with G i due to (1), E[Z i ] = 2 i+1 OPT E[err] = E[ lg n Z i i=1 2 i (m+1) ] 2 lg n m+1 OPT (Arora s PTAS for the Euclidean TSP) 20 / 23

41 Overall apprx factor (1 + ɛ 2 )( lg n r 2 )(1 + m+1 )OPT by choosing r = O(1/ɛ) and m = O( lg n ɛ ), the apprx factor is upper bounded by (1 + ɛ) in expectation (Arora s PTAS for the Euclidean TSP) 21 / 23

42 Take-home lessons PTAS Las Vegas algo approximation in expectation dynamic programming Steiner points along grid edges rounding input points locally patching a path and, more importantly, the shifting grid hence, the Godel award for the result! (Arora s PTAS for the Euclidean TSP) 22 / 23

43 References Geometric Approximation Algorithms by Sariel Har-Peled, American Mathematical Society, First Edition, S. Arora, Polynomial time approximation schemes for Euclidean TSP and other geometric problems, Journal of ACM (Arora s PTAS for the Euclidean TSP) 23 / 23

44 References Geometric Approximation Algorithms by Sariel Har-Peled, American Mathematical Society, First Edition, S. Arora, Polynomial time approximation schemes for Euclidean TSP and other geometric problems, Journal of ACM Parallel/further significant work - J. S. B. Mitchell, Guillotine subdivisions approximate polygonal subdivisions: A simple new method for the geometric TSP, k-mst, and related problems, SIAM Journal on Computing Satish B. Rao and Warren D. Smith, Approximating geometric graphs via spanners and banyans, STOC (Arora s PTAS for the Euclidean TSP) 23 / 23

Algorithms for Euclidean TSP

Algorithms for Euclidean TSP This week, paper [2] by Arora. See the slides for figures. See also http://www.cs.princeton.edu/~arora/pubs/arorageo.ps Algorithms for Introduction This lecture is about the polynomial time approximation