On Generating All Shortest Paths and Minimal Cutsets

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On Generating All Shortest Paths and Minimal
Cut-sets

• By Beth Hayden, Daniel MacDonald
• July 20, 2005
• Advisor Dr. Endre Boros, RUTCOR

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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.

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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).

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An Example of a Minimal Cut-set
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Independence Systems and our Problem How do They
Relate?

• Given a set E1,2, ,n, a family F of subsets
• Ii E is independent when any subset of Ii
  is also in the family of sets.
• An independent I set is maximal when there exists
  no I I in the family of subsets.

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Application of Independence Systems

• Define an independent set I in our case as a
  subset of a graphs 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.

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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.

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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.

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Another Definition

• Given E1,2, ,n, we say a family of
  independent sets Ik is maximal through some kltn
  when all independent sets are maximal in the
  given subset K1,2, ,k.

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The I j Problem

• The problem stated formally is Given the family
  of maximally independent sets within j-1,
  consider, for each Ij-1 the addition of the next
  element j. Test for maximality for each
  independent set.
• Recall we defined E1,2, ,n. This implies
  that labeling is an essential part of our
  procedure.

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Labeling Problems?
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The Algorithm
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• Black edges are labeled arbitrarily.
• Red edges are labeled according to a hierarchy
  based on the edges distance from the center
  vertex.

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The Algorithm
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• Black edges are labeled arbitrarily.
• Red edges are labeled according to a hierarchy
  based on the edges distance from the center
  vertex.

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The Algorithm Getting Started
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• Pick threshold L 4.
• Begin with the empty set of red edges.
• Add red edges in lexicographic order.

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• Add next lexicographic red edge, j1.
• Do shortest path algorithm.
• If there are no short paths, then the whole set
  of edges is an independent set I1.

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• Add next lexicographic red edge, j2.
• Do shortest path algorithm.
• If there are no short paths, then the whole set
  of edges is an independent set I2.

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• List short paths within
• 12, l, n, 14
• 13, m, n, 14
• Find all maximal independent sets

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• 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.

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But

• How can we generate all maximal independent sets
  for each in a systematic way?

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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.

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The Boolean Function
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• Form a tree of the short paths, with each edge
  of a short path represented by a node of the tree

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The Boolean Function
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• Form a tree of the short paths, with each edge
  of a short path represented by a node of the tree

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The Boolean Function

• 4 short paths

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• To find minimal cuts
• compute the conjunctive normal form
• of all the short paths

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k
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Reducing the Complexity
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• 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.

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j
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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.

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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.