On Euclidean Vehicle Routing With Allocation

1. On Euclidean Vehicle RoutingWith Allocation Reto Spöhel, ETH Zürich Joint work with Jan Remy and Andreas Weißl TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

2. Euclidean TSP • Euclidean TSP • Input: points P ½R2 • Output: tour ¼ through P with minimum length • Complexity: • NP-hard [Papadimitriou 77]

3. Euclidean TSP • Euclidean TSP • Input: points P ½R2 • Output: tour ¼ through P with minimal length • Complexity: • NP-hard [Papadimitriou 77] • admits PTAS [Arora 96; Mitchell 96] • …even one with complexity O(n log n). Arora (1997) There is a randomized PTAS for Euclidean TSP with complexityn logO(1/²) n. Rao, Smith (1998) There is a randomized PTAS for Euclidean TSP with complexityO(n log n).

4. VRAP • (Euclidean) VehicleRoutingwithAllocation (VRAP) • Input: points P ½R2 , constant ¯¸ 1 • Output: tour ¼ through subset T µ P minimizing • Motivation: • salesman visits not all customers • customers not visited go to next tourpoint, which is more expensive by a factor of ¯.

5. VRAP • (Euclidean) VehicleRoutingwithAllocation (VRAP) • Input: points P ½R2 , constant ¯¸ 1 • Output: tour ¼ through subset T µ P minimizing • Complexity: • NP-hard, since setting ¯¸ 2 yields Euclidean TSP • as forEuclidean TSP, thereexists a quasilinear PTAS Remy, S., Weissl (2007) There is a randomized PTAS for VRAP with complexityO(n log5n).

6. Steiner VRAP • Steiner VRAP • Input: points P ½R2 , constant ¯¸ 1 • Output: subset T µ P, set of points S ½R2(Steiner Points), tour ¼ through T [ S minimizing • Motivation: • salesmanmay also stop en route to servecustomers

7. Steiner VRAP • Steiner VRAP • Input: points P ½R2 , constant ¯¸ 1 • Output: subset T µ P, set of points S ½R2(Steiner Points), tour ¼ through T [ S minimizing … • Complexity: • NP-hard • admits PTAS • even a quasilinear one Armon, Avidor, Schwartz(ESA ´06) There is a randomized PTAS for Steiner VRAP with complexity nO(1/²). Remy, S., Weissl (2007) There is a randomized PTAS for Steiner VRAP with complexity n logO(1/²) n.

8. Our techniques • Finding a good solution for (non-Steiner) VRAP means • finding a good set of tourpointsT µ P • finding a good tour on this set T simultaneously. • a) is essentially a facility location problem. • We use the adaptive dissection technique, due to [Kolliopoulos and Rao 99] • b) is Euclidean TSP. • We use dynamic programming on sparse Euclidean spanners, due to [Rao and Smith 98] • In the remainder of the talk, we • outline the basics of dynamic programming in quadtrees, as introduced in [Arora 96] for Euclidean TSP, and • sketch the key ideas of the adaptive dissection technique.

9. Quadtrees • Choose origin of coordinate system (= center of large square) randomly • thisistheonlysource of randomnessin all algorithms!

10. Quadtrees • Split large square recursively into 4 smaller squares until squares have sidelength 2 • Since bounding square has sidelength O(n), resulting tree has depth O(log n)

11. Quadtrees • Truncatequadtree: stop subdivision at empty squares. • remaining tree has O(n log n) nodes

12. Portal-respecting solutions • Place O(log n/²) many equidistant points (‘portals’) on the boundary of each square. • Imposerestriction: Salesman may enter/leave a square only via its portals. Lemma In expectation, detouring all edges of the optimal salesmantour via the nearest portal increases its length only by a factor of 1+².

13. Portal-respecting solutions • Place O(log n/²) many equidistant points (‘portals’) on the boundary of each square. • Imposerestriction: Salesman may enter/leave a square only via its portals. • Intuition: fortwofixedpoints: • good Lemma In expectation, detouring all edges of the optimal salesmantour via the nearest portal increases its length only by a factor of 1+².

14. Portal-respecting solutions • Place O(log n/²) many equidistant points (‘portals’) on the boundary of each square. • Imposerestriction: Salesman may enter/leave a square only via its portals. • Intuition: fortwofixedpoints: • bad • butunlikely! Lemma In expectation, detouring all edges of the optimal salesmantour via the nearest portal increases its length only by a factor of 1+².

15. Portal-respecting solutions • Place O(log n/²) many equidistant points (‘portals’) on the boundary of each square. • Imposerestriction: Salesman may enter/leave a square only via its portals. • i.e., thereis an expectednearly-optimalportal-respectingsalesman tour. • Wetry to find the best portal-respectingsalesman tour bydynamicprogramming in thequadtree. Lemma In expectation, detouring all edges of the optimal salesmantour via the nearest portal increases its length only by a factor of 1+².

16. Dynamic programming in quadtrees • For a given square Q, guess which portals are used by salesman tour, and enumerate all possible configurations C. • For each configuration C, calculate estimate for the length of a good tour inside Q, subject to the restrictions given by C: • If Q is a leaf of the quadtree, by brute force. • If Q is an inner node of the quadtree, by recursing to its four children. C

17. Running time Patching Lemma (Arora) • Looking for such a patched solution, it suffices to consider logO(1/²) n configurations per square. • Working in a truncated quadtree, we obtain a PTAS with running time O(n log n) ¢logO(1/²) n ¢logO(1/²) n = n logO(1/²) n The optimal solution can be modified such that it crosses the boundary of every square at most O(1/²) many times.In expectation, this increases the length of the tour only by a factor of 1+². Arora (1997) There is a randomized PTAS for Euclidean TSP with complexityn logO(1/²) n.

18. Running time Patching Lemma (Arora) • Combiningtheextendedpatchinglemmawithstandardquadtreetechniquesforfacilitylocationproblems[Arora, Raghavan, Rao 98], weobtain The optimal solution can be modified such that it crosses the boundary of every square at most O(1/²) many times.In expectation, this increases the length of the tour only by a factor of 1+². Lemma The Patching Lemma extends to Steiner VRAP. Remy, S., Weissl (2007) There is a randomized PTAS for Steiner VRAP with complexity n logO(1/²) n.

19. Advancedtechniques • These techniques also easilyyield an nlogO(1/²) n-PTASfor (non-Steiner) VRAP. Improvingthis to O(n log5 n) requirestwoadvancedtechniques. • Euclidean TSP: usesparseEuclideanspanners[Rao and Smith 98; Gudmundsson, Levcopoulos, and Narasimhan 00] • Facilitylocation: adaptive dissection[Kolliopoulos and Rao 99]

20. Adaptive dissection • Quadtreeisreplacedbyzoom tree • Thestructure of a zoom treechangeswiththelocation of thefacilities(in ourcase, the tour points T). • Guessingthelocation of the tour pointsisdonebyguessinghow to best recurse. • wehave to do dynamicprogramming in larger structure, whichisessentiallytheunion of the zoom treesfor all possiblechoices of Tµ P • Key Advantage: constantlymanyportalsper rectanglesuffice!

21. The zoom tree • The zoom treealternatesbetweensplitstepsand zoom steps. • splitstepsworkverysimilar to recursion in quadtree. • zoom stepslook as follows: we zoom on bounding box of tourpoints( + somesafetymargin)

22. Howdoesthishelp? • Twoconceptualadvantages: • On one hand, directlyzooming on thetourpointsskipslevels in between, whichmightintroduce large errors in thequadtreetechnique. • On theother hand, in theresultingnearly-optimalsolution, a point isnot necessarilyallocated to itsnearesttourpoint, butpossibly to a different nearby point. added flexibility in analysis. • Theneteffectisthatweonlyhave to considerconstantlymanyconfigurationsper rectangle.

23. Running time • Running time isdominatedby zoom steps • Weconsiderrectangles of boundedaspect ratiowithsides on suitablegridscontainingat least one point. •  ThereareonlyO(n log2 n) pairs of rectangleswhichcorrespond to zoom steps • For each such pair, the zoom stepcanbeperformed in timeO(log3 n). • Thisrequiresallocatingnon-tourpoints in batchesusingrangesearchingtechniques. • Weobtain a running time of O(n log2 n) ¢O(1) ¢O(log3n) = O(n log5 n) Remy, S., Weissl (2007) There is a randomized PTAS for VRAP with complexityO(n log5n).

24. Concludingremarks • Bothalgorithmsextend to higherdimensions d, yieldingrunningtimes of O(n logd+3 n) and O(n logC(d,²) n) respectively. • Bothalgorithmscanbederandomizedbyenumerating all possiblerandomshifts of thequadtree (zoom tree), at thecost of an extra factor O(nd).

25. Summary • VRAP is a combination of Euclidean TSP and a facilitylocationproblem. • Thestate-of-the-arttechniquesforEuclidean TSP and facilitylocationcanbecombinedinto a O(n log5 n)-PTAS forVRAP. • For Steiner VRAP, bynowwell-establishedstandardtechniquesyield a n logO(1/²) n-PTAS.

26. Questions?