1. 1 A New Calculational Law for Combinatorial Optimization Problems Akimasa Morihata PhD student of IPL (Takeichi/Hu Lab.), the University of Tokyo IFIP WG2.1 @ Kyoto

2. 2 Maximum Segment Sum (Maximum segment sum problem) Given a list of numbers, find the segment that has the maximum weight-sum.

3. 3 Resource Constrained Shortest Path Problem (Resource constrained shortest path problem) Given an edge-weighed graph and a resource function, find the shortest path between given two nodes such that. s t 5 1 1 3 1 -3 4 -2 4 2 2

4. 4 Our Contribution Proposing a new calculational law for combinatorial optimization problems –Generic –Easy to use Suitable for automation

5. 5 Calculational Laws for Combinatorial Optimization Problems Greedy theorems (Bird and de Moor, Curtis) –Generic but not automatic Maximum marking problems (Sasano et al.: ICFP 2000) –Automatic but specific Derivation of the result of Sasamo.et.al (Bird, JFP 2001) –Based on the thinning law (» greedy theorem)

6. 6 Notations map : union: filter : means minimals (not “least elements”) where is a preorder

7. 7 Formalizing Combinatorial Optimization Problems Combinatorial optimization problems » each solution is given by a sequence of decisions –a decision: –enumeration of all solutions: Greedy Theorem (Bird and de Moor, Curtis) : If, then.

8. 8 Example: Maximum Segment Sum (MSS as a Maximum Marking Problem) Compute the Maximum marking problem on lists where marking should be accepted by the automaton: M M N N N M: marked N: not marked

9. 9 Problem: Greedy Theorem is HARD to use How do we find an appropriate order?

10. 10 A New Calculational Law Theorem: If and then where. Monotonic Fusible

11. 11 Why Correct? Lemma: if Lemma: if and only if Lemma:

12. 12 Deriving an Algorithm for MSS M and N is monotonic for · ? ) Yes! (trivial) is fusible? ) Yes! (it’s an automaton!)

13. 13 Next Example: Shortest Path Problem (Shortest path problem) Given an edge-weighed graph, find the shortest path from to.

14. 14 Derivation of Bellman-Ford Algorithm is monotonic for · ? ) Yes! (trivial) is fusible for ! (you can easily confirm it by a small calculation)

15. 15 Final Example: Resource Constrained Shortest Path (Resource constrained shortest path problem) Given an edge-weighed graph and a resource function, find the shortest path between given two nodes such that. s t 5 1 1 3 1 -3 4 -2 4 2 2

16. 16 Deriving DP Algorithm for Resource Constrained Shortest Path is fusible for ) is fusible for

17. 17 Conclusion We propose a new caluclational law for deriving dynamic programming algorithms –Fusion » Dynamic programming –Generic –Suitable for automation

18. 18 Future Work General recursion schema? Giving a DSL for dynamic programming –Generate efficient program automatically Derivation of greedy algorithms