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Polynomial-time approximation schemes for geometric NP-hard problems

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Polynomial-time approximation schemes for geometric NP-hard problems

Reto SpÃ¶hel

Reading Group, May 17, 2011

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or

On Euclidean vehicle routingwith allocation

Jan Remy, Reto SpÃ¶hel, Andreas WeiÃŸl

(appeared in WADS â€™07, CGTA â€™11)

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- The Traveling Salesman Problem (TSP)
- Input: edge-weighted graph G
- Output: Hamilton cycle in G with minimum edge-weight

- Motivation:
- Traveling salesman ;-)

- Complexity:
- NP-hard
- Admits no constant factor approximation (unless P=NP) [Sahni and Gonzalez 76]

- Metric TSP
- Input: edge-weighted graph G satisfying triangle inequality
- Output: Hamilton cycle in G with minimum edge-weight

- Motivation:
- real-world problems usually satisfy triangle inequality

- Complexity:
- still NP-hard
- admits 3/2-approximation [Christofides 76]
- admits no PTAS (unless P=NP) [Aroraet al. 98]

- Euclidean TSP
- Input: points P Â½R2
- Output: tour Â¼ through P with minimal length

- Complexity:
- still NP-hard [Papadimitriou 77]
- admits PTAS [Arora 96; Mitchell 96]

- Euclidean TSP
- Input: points P Â½R2
- Output: tour Â¼ through P with minimal length

- Complexity:
- still NP-hard [Papadimitriou 77]
- admits PTAS [Arora 96; Mitchell 96]
- â€¦even one with complexity O(n log n).

Arora (FOCS â€™97)

There is a randomized PTAS for Euclidean TSP with complexityn logO(1/Â²) n.

Rao, Smith (STOC â€™98)

There is a randomized PTAS for Euclidean TSP with complexityO(n log n).

- (Euclidean) VehicleRoutingwithAllocation (VRAP)
- Input: points P Â½R2 , constant Â¯Â¸ 1
- Output: tour Â¼ through subset T Âµ P minimizing

- Motivation:
- salesman does not visit all customers
- customers not visited go to next tourpoint, which is more expensive by a factor of Â¯.

- (Euclidean) VehicleRoutingwithAllocation (VRAP)
- Input: points P Â½R2 , constant Â¯Â¸ 1
- Output: tour Â¼ through subset T Âµ P minimizing

- Complexity:
- NP-hard, since setting Â¯Â¸ 2 yields Euclidean TSP
- as forEuclidean TSP, thereexists a quasilinear PTAS

Remy, S., WeiÃŸl (WADS â€™07)

There is a randomized PTAS for VRAP with complexityO(n log4n).

- Steiner VRAP
- Input: points P Â½R2 , constant Â¯Â¸ 1
- Output: subset T Âµ P, set of points S Â½R2(Steiner Points), tour Â¼ through T [ S minimizing

- Motivation:
- salesmanmay also stop en route to servecustomers

- Steiner VRAP
- Input: points P Â½R2 , constant Â¯Â¸ 1
- Output: subset T Âµ P, set of points S Â½R2(Steiner Points), tour Â¼ through T [ S minimizing â€¦

- Complexity:
- NP-hard
- admits PTAS
- â€¦even a quasilinear one

Armon, Avidor, Schwartz(ESA â€™06)

There is a randomized PTAS for Steiner VRAP with complexity nO(1/Â²).

Remy, S., WeiÃŸl (WADS â€™07)

There is a randomized PTAS for Steiner VRAP with complexity n logO(1/Â²) n.

- Finding a good solution for VRAP means
- finding a good set of tour points T Âµ 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, ESA â€™99]

- b) is Euclidean TSP.
- We use dynamic programming on â€˜patched short spannersâ€™, due to [Rao and Smith, STOC â€™98]

- To put both ideas into perspective, we start by explaining the basics of dynamic programming in quadtrees, as introduced in [Arora, FOCS â€™96] for Euclidean TSP

3

2

1

- We assume that the input points P
- have odd integer coordinates
- lieinside a square whose sidelength is
- a power of 2
- of order O(n/Â²)

- This is ok, since every (1+Â²/2)-approximation for the rescaled and shifted input Pâ€™ corresponds to a (1+Â²)-approximation for the original input P.

P

- We assume that the input points P
- have odd integer coordinates
- lieinside a square whose sidelength is
- a power of 2
- of order O(n/Â²)

- This is ok, since every (1+Â²/2)-approximation for the rescaled and shifted input Pâ€™ corresponds to a (1+Â²)-approximation for the original input P.

Pâ€™

- We assume that the input points P
- have odd integer coordinates
- lieinside a square whose sidelength is
- a power of 2
- of order O(n/Â²)

- This is ok, since every (1+Â²/2)-approximation for the rescaled and shifted input Pâ€™ corresponds to a (1+Â²)-approximation for the original input P.

Pâ€™

- We assume that the input points P
- have odd integer coordinates
- lieinside a square whose sidelength is
- a power of 2
- of order O(n/Â²)

P

1

- Choose origin of coordinate system (= center of large square) randomly.
- thisistheonlysource of randomnessin all algorithms

- Split large square recursively into 4 smaller squares until squares have sidelength 2
- Since bounding square has sidelength O(n), resulting tree has O(n2) nodes (squares) and depth O(log n)

- Truncatedquadtree: stop subdivision at empty squares
- remaining tree has O(n log n) nodes

- Place O(log n/Â²) many equidistant points (â€˜portalsâ€™) on the boundary of each square.
- Imposerestriction: Salesman may enter/leave a square only via its portals.

Lemma (Arora)

In expectation, detouring all edges of the optimal salesmantour via the nearest portal increases its length only by a factor of 1+Â².

- Place O(log n/Â²) many equidistant points (â€˜portalsâ€™) on the boundary of each square.
- Imposerestriction: Salesman may enter/leave a square only via its portals.

- Intuition: fortwofixedpoints:
- good

Lemma (Arora)

In expectation, detouring all edges of the optimal salesmantour via the nearest portal increases its length only by a factor of 1+Â².

- Place O(log n/Â²) many equidistant points (â€˜portalsâ€™) on the boundary of each square.
- Imposerestriction: Salesman may enter/leave a square only via its portals.

- Intuition: fortwofixedpoints:
- bad
- butunlikely!

Lemma (Arora)

In expectation, detouring all edges of the optimal salesmantour via the nearest portal increases its length only by a factor of 1+Â².

- Place O(log n/Â²) many equidistant points (â€˜portalsâ€™) 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 (Arora)

- For a given square Q, guess which portals are used bysalesman 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

- Working in a non-truncated quadtree, we have to consider O(n2) squares. For each of these we have to consider2O(logn/Â²) = nO(1/Â²) configurations, and the estimate for each configuration can be calculated in time nO(1/Â²).
- We obtain a PTAS with running time
O(n2) Â¢nO(1/Â²)Â¢nO(1/Â²)= nO(1/Â²)

- This is essentially the technique used in the PTAS for Steiner VRAP by Armonet al.

- We obtain a PTAS with running time

Arora(FOCS â€™96)

Armon, Avidor, Schwartz(ESA â€™06)

There is a randomized PTAS for Euclidean TSP with complexity nO(1/Â²).

There is a randomized PTAS for Steiner VRAP with complexity nO(1/Â²).

- Working in a non-truncated quadtree, we have to consider O(n2) squares. For each of these we have to consider2O(logn/Â²) = nO(1/Â²) configurations, and the estimate for each configuration can be calculated in time nO(1/Â²).
- We obtain a PTAS with running time
O(n2) Â¢nO(1/Â²)Â¢nO(1/Â²)= nO(1/Â²)

- to achieve quasilinear time, we can only use polylogarithmic time per square. In particular, we can only consider polylogarithmically many configurations per square.

- We obtain a PTAS with running time

Arora(FOCS â€™96)

There is a randomized PTAS for Euclidean TSP with complexity nO(1/Â²).

Patching Lemma (Arora)

- Idea: proceedbottom-upthroughquadtree and modifyeachsquarewithtoomanycrossingsbyintroducing line segments parallel to sides.

The optimal solution can be modified such that it crosses the boundary of every square at most O(1/Â²) many times.In expectation, this increases the length of the tour only by a factor of 1+Â².

- The total length of thenew line segmentsisat most 3x
- ïƒ¨ modification on lowlevels of thequadtreearecheap.

x

Patching Lemma (Arora)

- i.e., thereis an expectednearly-optimalportal-respectingsalesman tour whichforeverysquareusesonly O(1/Â²) many of the O(log n) portals.

The optimal solution can be modified such that it crosses the boundary of every square at most O(1/Â²) many times.In expectation, this increases the length of the tour only by a factor of 1+Â².

Patching Lemma (Arora)

- We only have to consider logO(1/Â²) n configurations per square.
- Working in a truncatedquadtree, we obtain a PTAS with running time
O(n log n) Â¢logO(1/Â²) n Â¢logO(1/Â²) n = n logO(1/Â²) n

- Working in a truncatedquadtree, we obtain a PTAS with running time

The optimal solution can be modified such that it crosses the boundary of every square at most O(1/Â²) many times.In expectation, this increases the length of the tour only by a factor of 1+Â².

Arora (FOCS â€™97)

There is a randomized PTAS for Euclidean TSP with complexityn logO(1/Â²) n.

Patching Lemma (Arora)

- Combiningtheextendedpatchinglemmawithstandardquadtreetechniquesforfacilitylocationproblems[Arora, Raghavan, Rao, STOC â€™98], weobtain

Lemma

The Patching Lemma extends to Steiner VRAP.

Remy, S., WeiÃŸl (WADS â€™07)

There is a randomized PTAS for Steiner VRAP with complexity n logO(1/Â²) n.

2

- Patchingrevisited:
- In Aroraâ€™s technique, the â€˜patchingâ€™ is not part of the algorithm â€“ we simply know a nearly-optimal patched solution exists, and try to find it by dynamic programming.
- Rao and Smith (STOC â€™98) improved Aroraâ€™s running time by making the â€˜patchingâ€™ part of the algorithm.

- A (1+Â²)-spanner Son P is a straight-line graph on P such that for every two points the shortest path in S is at most (1+Â²) time their Euclidean distance.
- A â€˜shortâ€™(1+Â²)-spannercanbecomputed in time O(n log n) [Gudmundsson, Levcopoulos, Narasimhan, SWAT â€™00]

- Clearly, given such a spanner S there is a nearly-optimal salesman tour that only uses edges of S.

Patching Lemma (Rao and Smith)

- Idea: proceedbottom-upthroughquadtree and modifyeachsquarewithtoomanycrossingsbyintroducing line segments parallel to sides.

- A short spanner S can be modified such that every square of the quadtree is crossed by at most O(1/Â²) relevant edges.In expectation, this increases the length of an optimal tour on the graph only by a factor of 1+Â².

- The total length of thenew line segmentsisat most 2x
- ïƒ¨ modification on lowlevels of thequadtreearecheap.

x

- A betteralgorithmforEuclidean TSP:
- Computeshortspanner S on P
- Patch S, call the new graph Sâ€™
- Dynamic programming in quadtree, but instead of portals use edges of Sâ€™.

- We now only have to consider constantly many configurations per square!
- We obtain a PTAS with running time
O(n log n) Â¢O(1) Â¢O(1)= O(n log n)

- We obtain a PTAS with running time

Rao, Smith (STOC â€™98)

There is a randomized PTAS for Euclidean TSP with complexityO(n log n).

- I promised a running time ofO(n log4 n) for (non-Steiner) VRAP.
- The techniquesdiscussed so fartakecareoftheEuclidean TSP partoftheproblem, and itremainstodiscussthefacilitylocationpart.
- I will presentthekeyideasdirectlyforthe VRAP setting.

3

- To improvetherunning time forfacilitylocationproblems, Kolliopoulos and Rao (ESA â€™99) introducedtheadaptive dissectiontechnique:
- thequadtreeisreplacedby a morecomplicatedstructure, a zoom tree
- thestructure of the 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 T Âµ P

- Key Advantage: constantlymanyportalsper rectanglesuffice!
- Everythingdiscussed so far (fortheâ€˜TSP partâ€™) needstobeadaptedfromthequadtreesettingtothe zoom treesetting!

- The zoom treealternatesbetweensplitstepsand zoom steps.
- splitstepsworkverysimilar to recursion in quadtree.
- zoom stepslook as follows:

we zoom on bounding box of tour points(+ somesafetymargin),

thesides of thisrectanglelieon a suitablegrid.

ïƒ¨ for a fixedset of tour points, thestructure of theresulting zoom treedepends on therandomchoice of thecoordinateorigin.

- Twoconceptualadvantages:
- On one hand, directlyzooming on the tour pointsskipslevels 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.ïƒ¨ added flexibility in analysis.
- The neteffectisthatweonlyhave to considerconstantlymanyconfigurationsper rectangle.

- Running time isdominatedby zoom steps
- Weconsiderrectangles of boundedaspect ratiowithsides on suitablegridscontainingat least one point.
- ïƒ¨ ThereareonlyO(n log2 n) pairs of rectangleswhichcorrespond to zoom steps

- For each such pair, the zoom stepcanbeperformed in timeO(log2 n).
- Thisrequiresallocatingnon-tourpoints in batchesusingrangesearchingtechniques.

- Weobtain a running time of
O(n log2 n) Â¢O(1) Â¢O(log2n) = O(n log4 n)

- Weconsiderrectangles of boundedaspect ratiowithsides on suitablegridscontainingat least one point.

Remy, S., WeiÃŸl (WADS â€™07)

There is a randomized PTAS for VRAP with complexityO(n log4n).

- Aroraâ€™s PTAS forEuclidean TSP and our PTAS forSteiner VRAP extend to anyfixeddimension d,yielding a running time of O(n logC(d,Â²) n).
- Here , i.e., therunning time isdoubly exponential in d.
- Main difficulty: thepatchingbecomesmorecomplicated, sincetheâ€˜sidesâ€™ of the hyper-â€˜squaresâ€™arenow (dâ€“1)-dimensional hypercubes.

- Aroraâ€™s PTAS forEuclidean TSP and our PTAS forSteiner VRAP extend to anyfixeddimension d,yielding a running time of O(n logC(d,Â²) n).
- The Rao-Smith PTAS forEuclidean TSP extends to anyfixeddimension d, still having a running time of O(n log n).
- (butwiththeimplicitconstantdependingbadly on d)

- Our PTAS forVRAPextends to anyfixeddimension d, yielding a running time of O(n logd+2 n).
- Therangesearchingadds an extra log-factor per dimension.

- All algorithmscanbederandomizedbyenumerating all possiblerandomshifts of thequadtree (zoom tree), at thecost of an extra factor O(nd).

- VRAP is a combination of Euclidean TSP and a facilitylocationproblem.
- Thetwostate-of-the-arttechniques
- Dynamicprogramming on â€˜patchedshortspannersâ€™ (Rao and Smith, STOC â€™98) for Euclidean TSP
- Adaptive dissection(Kolliopoulos and Rao, ESA â€™99) forfacilitylocation
canbecombinedinto a O(n log4 n)-PTAS forVRAP.

Thankyou!Questions?

- Considertwofixedpointsu,v2 P, and a fixed line gwhichis on theeven-integergrid and intersectsuv.
- Ifgispart of thelevel in whichthesquareshavesidelength s, itintroduces an error of at mosts/m at thislevel.
- Theprobabilitythatgispart of thelevel in whichthesquareshavesidelength s is2/s.
- Thatis, thegrid line gintroduces an expectederrorof at most
2/s Â¢ s/m = 2/m

at eachlevel, whichisO(log n/m) in total.

- Sincethereare at mostd(u,v) even-integergridlinesintersectinguv, the total expectederror in the distance fromu to vis
O(log n/m) Â¢ d(u,v) = O(Â²)Â¢ d(u,v) .

- increasenumber of portalsbyconstantfactor to beatconstanthiding in O ïƒ¨ in expectation, tour lengthincreasesby a factor of 1+Â².