1. On Generating All Shortest Paths and Minimal Cut-sets By: Beth Hayden, Daniel MacDonald July 20, 2005 Advisor: Dr. Endre Boros, RUTCOR

2. Problem Statement We are concerned with generating all short paths in a graph, and also with generating all minimal cut-sets in that graph such that all short paths are blocked.

3. Definitions A short path is defined as any path that is less than or equal to a certain threshold L A minimal cut-set is one that cuts all short paths. Furthermore, if any of the edges in the cut-set were to be put back into the original graph, a short s-t path would occur (this is what is meant by minimality).

4. An Example of a Minimal Cut-set L=3 [A] [B] s t s ts t

5. Independence Systems and our Problem: How do They Relate? Given a set E={1,2, …,n}, a family F of subsets I i E is independent when any subset of I i is also in the family of sets. An independent I set is maximal when there exists no I` I in the family of subsets.

6. Application of Independence Systems Define an independent set I in our case as a subset of a graph’s edge set E. I is independent if there exists no short path within I. Hence, an independent set I is maximal if when you add an additional edge to I, a short path now exists.

7. Maximal/Minimal: Complements The notion of maximal independent set in a graph as defined previously corresponds directly to the notion of minimal cutset that is part of our problem.

8. Maximal Independence – NP Complete? In a paper by Lawler, Lenstra and Rinnooy Kan (1980), a procedure known as the “I {j} problem” is shown to reduce the complexity of the general maximal independence set problem.

9. Another Definition Given E={1,2, …,n}, we say a family of independent sets I k is maximal through some k<n when all independent sets are maximal in the given subset K={1,2, …,k}.

10. The I {j} Problem The problem stated formally is: Given the family of maximally independent sets within j- 1, consider, for each I j-1 the addition of the next element j. Test for maximality for each independent set. Recall: we defined E={1,2, …,n}. This implies that labeling is an essential part of our procedure.

11. Labeling: Problems? t s s t

12. The Algorithm s t  Black edges are labeled arbitrarily.  Red edges are labeled according to a hierarchy based on the edge’s distance from the center vertex.

13. The Algorithm s t  Black edges are labeled arbitrarily.  Red edges are labeled according to a hierarchy based on the edge’s distance from the center vertex. 1 2 3 4 5 67 8 9 1011 12 13 14 15 16 17

14. s t  Pick threshold L = 4.  Begin with the empty set of red edges.  Add red edges in lexicographic order. The Algorithm: Getting Started

15. s t 1  Add next lexicographic red edge, j=1.  Do shortest path algorithm.  If there are no short paths, then the whole set of edges is an independent set I 1.

16. s t 1  Add next lexicographic red edge, j=2.  Do shortest path algorithm.  If there are no short paths, then the whole set of edges is an independent set I 2. 2

17. s t 1 2 3 4 5 67 8 9 1011 12 13 14  List short paths within : 12, l, n, 14 13, m, n, 14  Find all maximal independent sets: n l m

18. s t 1 2 3 4 5 67 8 9 1011 12 13 14  List short paths within : 12, l, o, 15 13, m, o, 15  Find all maximal independent sets:  Repeat for each I generated before, then continue to add red edges. l m 15 o

19. But… How can we generate all maximal independent sets for each in a systematic way?

20. Boolean Function: The Goal The goal of the Boolean function is to find the minimal cut-sets. We do this by representing each path as a disjunction of its edges, and take the “product” of each disjunction. When simplified, this gives us the minimal cut- sets.

21. The Boolean Function s t 10 11 14 n 15 16  Form a tree of the short paths, with each edge of a short path represented by a node of the tree o j k

22. The Boolean Function s t 10 11 14 n 15 16  Form a tree of the short paths, with each edge of a short path represented by a node of the tree: o j k 16 n o j 14 10 15 k 11

23. 16 n o j 14 10 15 11 The Boolean Function  4 short paths: k  To find minimal cuts: compute the conjunctive normal form of all the short paths:

24. Reducing the Complexity 16 n o j 14 10 15 11 k  First compute the conjunction of the branch farthest from the source node.  Use the results of the conjunction to form a single path from the branching node, with minimal cut represented by a node on the unified path.  Continue until the whole tree is one single path. 16 n o

25. n o j 14 10 15 11 16 n o k

26. Conclusions We have found that for a specific type of graph, a good algorithm for generating the minimal cut- sets in the “I {j}” problem exists. We have also found a good algorithm for generating the short paths.

27. Acknowledgments Lawler, Lenstra, Rinnooy Kan. Generating all Maximal Independent Sets: NP-Hardness and Polynomial-Time Algorithms, Society for Industrial and Applied Mathematics, 1980. Thanks to Dr. Boros, Kate Davidson, and Craig Bowles.