1. Cooperative TSP Amitai Armon, Adi Avidor and Oded Schwartz Tel-Aviv University, Israel

2. 2 Presentation Overview l Introducing Cooperative TSP problems »Cooperation modes »Minimization goals »Examples l Our results and related results l Outline of one of our PTAS algorithms

3. 3 Classical TSP A salesperson has to deliver goods to a set of customers, while minimizing the total travel. 2 1 2 1 3 2 2 1 3 1 5 3 5 4 5 2 3 4 3

4. 4 Cooperative TSP (cTSP) The customers cooperate with the salesperson, and can move to help in the deliveries. 2 1 2 1 3 2 2 1 3 1 5 3 5 4 5 2 3 4 3

5. 5 General Motivation In many realistic cases, all the participants are on the same side: The “customers” are actually other agents of the same organization. 2 1 2 1 3 2 2 1 3 1 5 3 5 4 5 2 3 4 3

6. 6 Cooperation Modes The agents (customers) may help in one of the following ways: 1. Purchase-cooperation: The agents can be instructed to move in order to receive the goods. 2. Sales-cooperation: The agents can be instructed to distribute the goods after receiving them. 3. Full-cooperation: Both 1 and 2. (We assume all the agents cooperate in the same mode)

7. 7 Minimization Goals The following goals are considered: Min-Sum : Minimizing the total distance traveled by all the participants. Min-Max : Minimizing the maximal distance traveled by any participant. Min-Makespan : Minimizing the time required for completing the deliveries.

8. 8 The Way Back: Two Versions l In the “Roundtrip” version, each participant must return to his/her starting-point at the end of the process. l In the “Path” version, we are only interested in what happens until all the deliveries are made.

9. 9 Example: Distributing a secret to spies l A secret should be distributed in person to all the spies as soon as possible. l Each spy is instructed where and when to receive it. l Spies who receive the secret can be instructed to help in further distributing it. Path Min-Makespan Full-Cooperation cTSP

10. 10 Example: Waking-up robots l Sleeping robots need to be awaken. l Each awake robot can help in waking-up the others (initially one robot is awake). l Each robot has limited battery, and thus should travel as little as possible. Path Min-Max Sales-Cooperation cTSP

11. 11 Example: Distributing Cash l A secured car distributes cash from the bank’s headquarters to its branches. l Each branch can be instructed to send a car to meet it and take its own part. l The bank wishes to minimize the total travel (or the total travel of cash-loaded cars). Roundtrip/Path Min-Sum Purchase-Cooperation cTSP

12. 12 The Metric Space We consider both: l A general metric space: a weighted undirected graph satisfying the triangle inequality. l A fixed-dimension Euclidean space (we refer to this problem as “Euclidean cTSP”).

13. 13 The Scope of this Work l We consider all the combinations of the problems mentioned: Cooperation-mode x Minimization-goal x Roundtrip/Path x Euclidean/graph (Except for versions which have already been considered, as described later). l We provide both algorithmic and hardness results.

14. 14 Presentation Overview l Introducing Cooperative TSP problems »Cooperation modes »Minimization goals »Examples l Our results and related results l Outline of one of our PTAS algorithms

15. 15 Previous Results Classical metric TSP: l In graphs: 3/2 approximation [Christofides 1976, Hoogeveen 1991] No approximation < 203/202 [Engebretsen and Karpinski 2001] l In fixed-dimension Euclidean space: PTAS [Arora 1998, Mitchell 1999] NP-hard [Papadimitriou 1977]

16. 16 Previous Results The Freeze-Tag problem [ABFMS, 2002] l This is exactly Path Min-Makespan Sales cTSP (waking-up robots ASAP). l In graphs: [Konemann, Levin, Sinha 2004] No approximation < 5/3 [ABFMS, 2002] l In fixed-dimension Euclidean space: PTAS [ABFMS, 2002] NP-hardness conjectured [ABFMS, 2002]

17. 17 Overview of Our Results l We provide constant-factor approximations for most of the problems, PTAS for some, and polynomial-time algorithms for a few. l We also provide lower-bounds on the approximability of most of the problems, some of them match our upper bounds.

18. 18 Our Results for cTSP in Graphs 2-  2 5/4-  Polynomial Min- Makespan 2-  2 3/2-  3 No FPTAS PTASMin-Max 203/202-  3/2 203/202-  3/2 203/202-  3/2Min-Sum RoundTrip 2-  2 5/3-  (2) Polynomial Min- Makespan 2-  4 3 No FPTAS PTASMin-Max APX-hard2+ln 3NP-hard2 2+ln 3Min-Sum PATH Inapprox.Approx.Inapprox.Approx.Inapprox.Approx. Full Cooperation Sales Cooperation Purchase Cooperation (1) by [KLS04] (2) by [ABFMS02]

19. 19 PTAS PolynomialMin-Makespan 23PTASMin-Max PTAS Min-Sum RoundTrip PTASPTAS 1 PolynomialMin-Makespan 43PTASMin-Max 2+  5/3+  PTASMin-Sum PATH Full Cooperation Sales Cooperation Purchase Cooperation Our Results for Euclidean cTSP (1) by [ABFMS02]

20. 20 Presentation Overview l Introducing Cooperative TSP problems »Cooperation modes »Minimization goals »Examples l Our results and related results l Outline of one of our PTAS algorithms

21. 21 Euclidean Path Min-Sum Purchase cTSP l Recall that in this problem: - Agents can approach the salesperson but cannot help in delivering the goods to others. - The total travel should be minimized. l We provide a PTAS for this problem by adapting Arora’s Euclidean-TSP technique. »The challenge is considering movements of many participants rather than one, without reaching a super- polynomial running-time.

22. 22 Euclidean Path Min-Sum Purchase cTSP l We define “portals-limited-solutions”. l Their optimum can be found in polynomial time, and it can be made arbitrarily close to the global optimum. l We next outline the algorithm for the planar case.

23. 23 Preliminaries l We assume w.l.o.g. that all the n participants are inside the bounding-square [0,L] 2, and that OPT ≥ L, where L is the smallest power of 2 s.t. L ≥ 4n 2. l We divide the bounding-square into four equal squares, “super-pixels of level 1”. l We repeatedly perform this division, until we reach super-pixels of level logL (and size 1), also called “pixels”. L L

24. 24 Portals l We mark each super-pixel boundary-line with m equidistant points, called “portals”. »m is the smallest power of 2 s.t. m ≥ 4 logL /  logn /  )) l A portal of a super-pixel of level i is also a portal of a super-pixel of level i+1. m m

25. 25 A portals-limited-solution: Requirements l A participant may cross a super-pixel boundary only at its portals. l Each participant first moves to its pixel’s center. l Deliveries are only made at pixel-centers. l If a pixel is visited by two or more participants, all the agents who visit it travel to its center and stop. l The salesperson may cross her/his own tour only at a portal. There are no other crossings.

26. 26 Tours in optimal portals-limited-solutions Straight segments, first connecting the starting point to the pixel-center, then connecting pixel-centers and portals.

27. 27 Tours in optimal portals-limited-solutions Straight segments, first connecting the starting point to the pixel-center, then connecting pixel-centers and portals.

28. 28 Optimal portals-limited-solution: Properties l Each portal is used by at most one participant. l If that participant is an agent, the portal is used once. l If that participant is the salesperson, we can assume that the portal is used at most once in each direction (by shortcutting).

29. 29 Portals in portals-limited-solutions Thus, for each portal p of a given super-pixel, there are six possibilities for its usage in an optimal solution: 1. The salesperson enters the super-pixel through p 2. The salesperson leaves the super-pixel through p 3. Both 1 and 2 4. One agent enters the super-pixel through p 5. One agent leaves the super-pixel through p 6. The portal is not used (In option 5 only this agent visits that super-pixel)

30. 30 Valid-lists in portals-limited-solutions Enumerating over at most 6 4m combinations for the usage of portals of a super-pixel takes poly(n 1/  ) time. l If such a combination may be a valid part of a solution, we call it a “valid-list”. For each valid-list there are poly(n 1/  ) possible pairings of entry and exit points of the salesperson in a pixel.

31. 31 The PTAS: Computing optimal costs for all the valid-lists l The computation is performed bottom-up. l For each of the pixels: - Enumerate over all the options for the pixel’s portals, and over all the possible pairings for each valid-list. - Compute the cost (total travel) of each of them. - Keep a table of the optimal costs (and the corresponding pairings) of each valid-list. (Note that there are O(n 4 ) pixels).

32. 32 The PTAS: Computing optimal costs for all the valid-lists l A valid-list of a super-pixel of level i fixes its entrance and exit points. l Enumerate over the 6 4m options for the 4m portals in the inner-boundaries. l Pick the cheapest option, using the tables of level i+1. Time: poly(n 1/  ) for each of the O(n 4 ) super-pixels.

33. 33 The Output of the Algorithm l Output: Value of level 0 table (the bounding-square), where no portal is used. l We can keep track of the table-entries of deeper- levels which led to this value, and thus have the solution itself. Overall running-time is poly(n 1/  ).

34. 34 Shifting the input relative to our grid l For ease of presentation, we haven’t mentioned so far that we start by shifting the input relative to the “gridlines”, similarly to Arora’s PTAS for TSP. l We enumerate over the L 2 =O(n 4 ) options for the best integer shifts (L in each dimension). l This reduces the error induced by moving crossings to portals. 2L

35. 35 Shifting the input relative to our grid l For ease of presentation, we haven’t mentioned so far that we start by shifting the input relative to the “gridlines”, similarly to Arora’s PTAS for TSP. l We enumerate over the L 2 =O(n 4 ) options for the best integer shifts (L in each dimension). l This reduces the error induced by moving crossings to portals. 2L

36. 36 General Outline of the proof l There is an optimal solution with a relatively small number of boundary crossings (about OPT). l The average error induced by moving a crossing to a portal is small (and so is the total error). l The other restrictions on a portals-limited-solution (e.g. moving to the pixels-centers) have a small effect.

37. 37 Summary and Open Problems l We introduced the Cooperative TSP (cTSP) family of problems, and considered each of them. l In this presentation we outlined a PTAS for Euclidean Path Min-Sum Purchase cTSP. l Several problems still have gaps between the lower and upper bounds. l Euclidean Path Min-Sum Sales cTSP is of special interest: Our current approximation ratio is 5/3, and a PTAS will imply a PTAS for 3-bounded MST, long-conjectured to exist.

38. 38 Cooperative TSP Thank You!