1. Jie Gao, Su Jia, Joseph S.B. Mitchell Stony Brook University
Exact and Approximation Algorithms for Time Window TSP and Time Window PC
Jie Gao, Su Jia, Joseph S.B. Mitchell
Stony Brook University

2. Problem Description Time Window Prize Collecting (TWPC):
Unit speed robot; must visit each site i during given time window, (ri , di).
Goal: max # sites visited (or total “prize”)
Time Window Travelling Salesman (TWTSP):
Robot with speed s; must visit each site i during given time window, (ri , di).
Goal: min distance robot travels to visit all sites (in TW)
Assume each TW has length at least one
(often called “TWTSP”)
(may not be feasible for small s)

3. Outline (1) Introduction (2) Dual QPTAS for 1D TWPC/TWTSP etc
(3) TWTSP in metric spaces

4. Some Prior Work

5. Literature Review of TWPC
Bar-Yehuda, Even, Shahar (2003): 1D TWPC has O(log n) approximation.
Bansal, Blum, Chawla, Meyerson (2004):
In metric space, O(log n) approximation if release time are 0;
O((log n)^2) for general time windows;
polytime bi-criteria approximation: visit

6. Literature Review of TWPC
Chekuri et al (2007):
In metric space, O(log L_{max}) approximation, assuming the shortest TW has length 1.
They also studied directed graph, wherewe need to pay an extra factor of O(poly(log OPT))
Frederickson (2012): Unit length time windows.
Azar et al (2015): Online version.
Special Case: When all time windows are the same: just ordinary PC and TSP.
Numerous research done in operations research literature. Not polytime.

7. Literature Review of TWPC/TSP
Tsitsiklis (1992):
TWTSP and TWPC are Strongly NP-complete in 1D. (Reduce to MSAT)
(A problem is said to be strongly NP-complete, if it remains so even when all of its numerical parameters are bounded by a polynomial in the length of the input)
Polytime (quadratic) solvable in 1D if all release time are 0. (Hint: DP)
Bockenhauer (2007) et al: There is no polytime constant factor approximation for TWTSP in metric space, unless P=NP.

8. Our Contributions (1D)
A framework for 1D TWTSP/PC: in time, we can compute a path collecting at least OPT prize (or at most OPT length), but visiting each point in relaxed time window
Assuming the max length is bounded by some poly(n), then this is a dual QPTAS.
Direct comparison with previous results: in time, we can compute a path collecting at least OPT prize but visiting each point in relaxed time window Compare with Bansal et al's result: visit

9. Our Contributions (1D)
The power of our method: it works for many 1D TSP type problem with TW (alternatively, 2D monotone TSP type problem).
In particular, dual QPTAS for 2D monotone TSPN problem, with arbitrary fatness/overlapping, assuming the max-min width ratio is poly(n). (relaxation: )

10. Our Contributions (metric space)
O(log n) apx for TWTSP with infinite speed.
(O(1), O(1)) dual apx for TWTSP with finite speed s when there are only 2 types of TW lengths.
(O(log L), O(log L)) dual apx for TWTSP with speed s, assuming that there is at least one vertex with time window [t,t+1], for each integer t<=L.
The apx factor depends only on the comlexity of time windows, regardless the number of points.

11. Outline (1) introduction and literature review
(2) dual QPTAS for 1D TWTSP etc (focus)
(3) TWTSP in metric space (brief)

12. Dyadic Interval
Interval [x,y] is said to be dyadic if (y-x) is a power of 2, and x is an integer multiple of its length.
e.g. [25,26], [64,96],...

13. Theorem. The TWTSP problem with infinite speed in 1D is polytime solvable for dyadic instances.
Hint: DP, Vertical Range Constraints (VRC).

14. Theorem. The TWTSP problem with infinite speed in 1D is polytime solvable for dyadic instances.
Hint: DP,
Encode north/southmost position visited

16. 1D TWPC, general instance, proof outline
(1) h-dyadic instances in O(n^O(h)) time
(2) general instance can be rounded to an h-dyadic instance by stretching each time window for at most (1+ ) times, where h= O(log L/ log (1+ )).
(1)+(2) => dual QPTAS for 1D TWTSP with infinite speed

17. the 'Inheriting Property'
Associate a partition to each dyadic interval
We say this family has 'the inheriting property' if the children intervals 'inherit' the partition points from their parents.
Very crucial for DP.

18. Dyadic Midpoint

19. h-dyadic Instance
recall definition of inheriting property, W(I), h-partition

20. Thm. For h-dyadic instance, the 1D TWTSP problem with infinite speed can be solved in O(n^O(h)) time.
Mimic dyadic case.
Encode the north/southmost position that P visits in each subinterval (i.e. vertical ranges)

21. Level of intervals:. unit interval is on level 0;
Level of intervals: unit interval is on level 0; the longest interval is on level logL,etc..

22. So far: h-dyadic instance can be solved in O(n^O(h)) time
Natural Idea for general case: Stretch the intervals to the partition points.

23. Reduce to: find a family of partitions, s. t
Reduce to: find a family of partitions, s.t. (1) dense enough for each dyadic interval (epsilon-dense) (2) inheriting property (3) “small” number of partition points Such family of partitions exists! Proof complete

24. -dense Partition
example.

25. the Existence of -dense Partition''
Why can we only get dual QPTAS (instead of dual PTAS)? Because of the number of each partition.
What's Challenging: these partitions are not isolated from each other!
Tradoff between the relaxation of TW and number of partition points

26. Now we have found a family of partitions, s. t
Now we have found a family of partitions, s.t. (1) dense enough for each dyadic interval (epsilon-dense) (2) inheriting property (3) reasonable number of partition points Proof Complete!

27. the proof of dual QPTAS for 1D TWTSP is complete.
Next, TWPC

28. Foundation of DP: Polynomial Boundary Complexity
Intuitions:
Assume dyadic
define the green boxes
the 'no-benefit' lemma: we benefit nothing from leaving or entering the small boxes for more than once, or zigzaging outside the box.

29. Structure of OPT what to encode for each subinterval:
the vertical range constraints (x's, i.e. the dashed black lines)
the north/southmost segment fully contained in I that P visits (y's, i.e. green boxes)
the time that the path leaves/enters the small boxes, and the time P starts and stops moving. (tau's)
source, sink

30. summary of this section

31. Direct Comparison with Previous Work

32. Dual QPTAS for Monotone TSPN in 2D

33. high level idea PTAS vs dual PTAS LP:
simplex: “improve the value, maintain feasibility”
dual simplex: “improve feasibility, maintain optimality condition”

34. Outline (1) introduction and literature review
(2) dual QPTAS for 1D TWPC/TWTSP etc
(3) TWTSP in metric space

35. TWTSP in Metric Space: Simple Case

36. Thank You! Open Problems:
Q1: there is PTAS or dual PTAS for 1D TWTSP/TWPC?
Q2: better dual apx factor?
Q3: a generalization of TWPC: orienteering with changing prize
Q4: disconnected TW
Q5: Euclidean space: how can we make use of geometry?

37. the Cut-Tree Algorithm for TWTSP (simple case)
red nodes: unit length TW
black nodes: [0,L]
Algo:
1. Find MST on all points
2. Cut into disjoint subtrees (called blocks), each of size O(s)
3. Build unweighted bipartite graph H and find a perfect delta-matching
4. For each red node, traverse the blue blocks assigned to it, and connect the reds.

38. the Cut-Tree Algorithm for TWTSP (simple case)
1. Find MST on all points
2. Cut into disjoint subtrees (called blocks), each of size O(s)
3. Build unweighted bipartite graph H and find a perfect delta-matching
4. For each red node, traverse the blue blocks assigned to it, and connect the reds.

39. the “incompressible lemma”
Given an MST, say T
Pick any sollection of disjoint subtrees
Build MST, say T' on V'
compare T' with the sum of the subtrees you chose

41. H satisfies the generalized Hall's matching condition
Intuition: If a set of blocks does not have enough number of red nodes that are within distance s to them (i.e. red neighbors in H), then there is no feasible solution. (Equivalently, if there is a feasible solution, then ...)
why is this intuition correct?
This is just the Hall's marriage condition!

42. proof sketch:
speed:
existence of feasible solution
=> generalized Hall's marriage condition
=> it is possible to “assign” blue blocks to red nodes such that each red node is assigned at most const # of blocks
=> the excess of speed is within const times (note that each block has size O(s).)
distance:
each blue block is traversed once, and the union of the blocks is a subset of MST(all nodes), which is a lower bound for d*(s) for any s.

43. General Time Windows
(O(log L),O(log L) ) apx, assuming that for each i, there is at least one node with TW [i,i+1].