A Recursive Greedy Algorithm for Walks in Directed Graphs

1. Discussion based on :- A Recursive Greedy Algorithm for Walks in Directed Graphs

2. Conclusions !! • Polynomial time algorithms for SOP and SOP-TW that have a poly-logarithmic approximation ratio. • An O(log2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). • an O(log2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups • This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than Ω(log1− OPT), even in undirected graphs.

3. Orienteering (SOP) • Given: G(V,A,l) ; s-t; B; • Our goal is to find an s-t walk P of length at most B, to maximize reward collected. • Reward Function: • Submodular + Monotone

4. Time Windowed (TW) • Release and deadline • Timed sequence of nodes • Time proportional to length of arc • Stalling is allowed

5. Assumptions/ Priliminaries • Asymmetric triangle inequality • f is an integer valued submodular function • Given a submodular function f on V and a subset X ⊆ V we define a new submodularfunction fXon V as fX(S) = f(S ∪ X) − f(X). • Let f be a monotone submodular set function on V . Then for any A ⊆ B ⊆ V , fA(S) ≥ fB(S) for all S ∈ 2V. • Polynomially Bounded Rewards:

6. RG

7. Analysis The running time of RG(s, t,B,X, i) is O((2nB)i · Tf ) where Tf is the maximum time to compute f on a given set. To obtain a logarithmic approximation, the algorithm takes O((2nB)log k) time

8. RG-QP

9. Analysis • For the submodular orienteering problem (SOP) there is an algorithm with running time (n logB) O(log n) that yields an O(log OPT) approximation. The running time of RG-QP(s, t,B,X, i) is O((2 + nAlogB)i · Tf ) where Tf is the maximum time to compute f on a given set.

10. RG-QP-TW

11. Analysis For the submodular orienteering problem with time windows (SOP-TW), there is an algorithm with running time (n logB)O(log n) that provides an O(log OPT) approximation where B is an upper bound on the tour length.

12. Improved Approximation Ratios • Instead of k/2 divide the path P* into h steps • Depth of each recursion will be O(log k/ log h) • (n LogB) ^O(h log k/ log h) : the time

13. Analysis • Approximation ratio O(log OPT/ log h) while increasing the running time to (n logB)O(h log n/ log h).

14. Applications • Orienteering with Multiple Disjoint Time Windows • Assume equal number of windows for each node • Special case of SOP-TW (how not given ; probably by copying the nodes) • By using appropriate windows, we can say the result for any arbitrary time varying profit function for each node. • Running time will be quasi-poly in nLogB

15. Applications • Rooted k-TSP in Directed Graphs • Using the algorithm for SOP with a budget of B, we can find a tour of length B that contains Ω(k/ log k) nodes. • after O(log2 k) iterations, the algorithm will cover k nodes.

16. Applications • Group Steiner and Covering Steiner Problems • Group Steiner : one from each grp. • Covering Steiner: at least di from ithgrp. • Find SOP stitch multiple SOP

17. Applications Putting together the tours yields tree of length O(log2Sumidi)B that is a feasible solution. We can use binary search to find a B that is within a constant factor of OPT and hence we obtain an O(log2Sumidi) approximation. When specialized to the group Steiner problem the ratio becomes O(log2 k) where k is the number of groups.

18. Eureka • The above discussion implies that an α-approximation for SOP in undirected graphs implies an O(α log k) approximation for the group Steiner problem in undirected graphs. • Halperin and Krauthgamerhave shown that the group Steiner problem is hard to approximate to within an Ω(log2−ek) factor unless NP has quasi-polynomial time Las-Vegas algorithms. • The submodular orienteering problem (SOP) in undirected graphs is hard to approximate to within a factor of Ω(log1−eOPT) unless NP ⊆ ZTIME(npolylog(n)).

19. Conclusions !! • Polynomial time algorithms for SOP and SOP-TW that have a poly-logarithmic approximation ratio. • An O(log2 k) approximation for the k-TSP problem in directed graphs (satisfying asymmetric triangle inequality). • an O(log2 k) approximation (in quasi-poly time) for the group Steiner problem in undirected graphs where k is the number of groups • This connection to group Steiner trees also enables us to prove that the problem we consider is hard to approximate to a ratio better than Ω(log1− OPT), even in undirected graphs.