Introduction Some lower and upper bounds PTAS PTAS for the k-Tour Cover Problem on the Euclidean Plane for Moderately Large Values of k Anna Adamaszek, Artur Czumaj, Andrzej Lingas DIMAP, University of Warwick, UK Lund University, Sweden ISAAC 18.12.2009 Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS k-tour problem in R 2 input: a set of n points P and a special point O called the origin (in R 2 ) O Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS k-tour problem in R 2 input: a set of n points P and a special point O called the origin (in R 2 ) O k-tour cover: a set of tours (cycles), each including the origin and at most k points from P, which covers all points from P Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS k-tour problem in R 2 input: a set of n points P and a special point O called the origin (in R 2 ) O k-tour cover: a set of tours (cycles), each including the origin and at most k points from P, which covers all points from P goal: find a k-tour cover with minimum total length Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS k-tour problem in R 2 input: a set of n points P and a special point O called the origin (in R 2 ) O k-tour cover: a set of tours (cycles), each including the origin and at most k points from P, which covers all points from P goal: find a k-tour cover with minimum total length example: depot and clients Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Results In the metric space: APX-hard for all k ≥ 3 Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Results In the metric space: APX-hard for all k ≥ 3 In the Euclidean plane: PTAS for k = O(log n/ log log n) [Asano et al., 1997] PTAS for k = Ω(n) [Asano et al., 1997] quasi-polynomial time approximation scheme (QPTAS); O(1/ε) n [Das and Mathieu, 2008] running time: nlog Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Results In the metric space: APX-hard for all k ≥ 3 In the Euclidean plane: PTAS for k = O(log n/ log log n) [Asano et al., 1997] PTAS for k = Ω(n) [Asano et al., 1997] quasi-polynomial time approximation scheme (QPTAS); O(1/ε) n [Das and Mathieu, 2008] running time: nlog Our result: δ PTAS for k ≤ 2log n , where δ = δ(ε) in particular for k = polylog(n) Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Lemma TSP ≤ OPT From a solution to the k-tour cover problem we can easily get a solution to the TSP problem using only shortcutting. O Anna Adamaszek, Artur Czumaj, Andrzej Lingas ⇒ O PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Lemma 2 P p∈P d(p, O) ≤ OPT k · Let T be a tour from a k-tour cover. c(T ) — cost of the tour T p For any p on T : c(T ) ≥ 2 · d(p, O). Tour T visits at mostP k points, therefore c(T ) ≥ k2 · p∈T d(p, O). O Now it is enough to sum up the results for all tours. Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Lemma OPT ≤ TSP + 2 k · P p∈P d(p, O) Suppose that k|n. Let T be the optimal TSP tour for our points. Remove point O from T . We can divide the tour into paths, such that each path visits exactly k consecutive points from P. We add edges to the origin to transform paths into tours. P Average cost of such a tour cover — TSP + k2 · p∈P d(p, O) O O Anna Adamaszek, Artur Czumaj, Andrzej Lingas O PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS We have shown the following inequalities: max{TSP, 2 X 2 X · d(p, O)} ≤ OPT ≤ TSP + · d(p, O) k k p∈P p∈P Using a simple 2-approximation algorithm for the TSP we can construct in polynomial time a k-tour cover with cost at most 2 P 2 · TSP + k · p∈P d(p, O). This gives us a 3-approximation algorithm. Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Removing close points L - maximum distance from a point in P to the origin We can cover all points at a distance at most Lε/n from the origin by 1-tours. L Lε/n O The additional cost generated is at most n · 2Lε/n = 2Lε ≤ εOPT . Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Locations Idea: OPT ≥ 2 k · P p∈P d(p, O) moving each point p ∈ P by a distance at most can change the total cost by at most ε · OPT Anna Adamaszek, Artur Czumaj, Andrzej Lingas ε k · d(p, O) PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Locations Idea: OPT ≥ 2 k · P p∈P d(p, O) moving each point p ∈ P by a distance at most can change the total cost by at most ε · OPT ε k · d(p, O) Circles and rays: smallest radius: Lε/n radius growing by a factor (1 + kε ) distance between rays ≈ radius O location: intersection of a circle and a ray T = Θ(k 2 ε−2 log(n/ε)) locations Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Moving points to nearest location O O Such an operation changes the cost by at most ε · OPT . Advantage: all points are in T = Θ(k 2 ε−2 log(n/ε)) locations. Remark: for the rest of the talk we will consider the modified problem (all points are in locations). Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Trivial and nontrivial tours We call a tour trivial if it visits only a single location; a tour is nontrivial otherwise. Theorem There is an optimal solution in which each location is visited by at most T nontrivial tours (T — number of locations). Idea: we take the optimal k-tour cover as long as we have a pair of nontrivial tours which have two locations in common we will perform an operation which eliminates such a pair without increasing the cost if each pair of nontrivial tours has at most one location in common — each location is visited by at most T nontrivial tours Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Trivial and nontrivial tours Suppose that we have tours T1 and T2 which visit locations p and q. p q We will swap the points from locations p and q between the tours. We can do it in such a way that after our operation one of the tours will visit only one of the locations p, q. This operation does not increase the cost of the solution. O Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Decreasing the number of points We already know that there is an optimal solution in which: each location is visited by at most T nontrivial tours in each location there are at most T · k points visited by nontrivial tours Corollary Using trivial tours we can reduce the problem to one with at most T 2 · k = (k/ε · log n)O(1) points. Remark: we can improve this result and reduce the problem to a collection of independent problems, each with (k/ε)O(1) points. Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Polynomial time approximation scheme There is a quasi-polynomial time approximation scheme (QPTAS) for k-tour cover problem on the Euclidean plane. Its running time O(1/ε) n. is nlog We can run this algorithm on the modified problem instance with (k/ε · log n)O(1) points. We will achieve (1 + ε)-approximation. δ Running time will be polynomial for k ≤ 2log n , where δ = δ(ε). Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ... Introduction Some lower and upper bounds PTAS Summary Our approximation algorithm: Removing close points Moving points to nearest location Decreasing the number of points by using trivial tours Applying QPTAS Open problem: Does there exist a polynomial time approximation scheme for arbitrary k? Anna Adamaszek, Artur Czumaj, Andrzej Lingas PTAS for the k-Tour Cover Problem on the Euclidean Plane ...