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6. MINIMUM HYPERPATHS IN DIRECTED HYPERGRAPHS In the case of directed graphs the computation of shortest path is a well known ‘easy’ problem. This is not the case for directed hypergraphs. Again a long history: • Martelli, Montanari 1973 • Knuth 1977 • Nguyen, Pallottino 1988 • Italiano, Nanni 1992 • Ramalingam, Reps 1996 • A., Laura, Italiano, Nanni 1998 - 2011
Shortest Hyperpath Problems The following shortest hyperpath problems are NP-hard: • minimum number of hyperarcs • minimum size • minimum source area • minimum number of source sets (!) Reduction from SET-COVER. By means of approximation preserving reductions it is easy to show that all problems are log-APX-hard to approximate.
Page 1: DIRECTED HYPERGRAPHS ALGORITHMS AND
Page 5 and 6: Measuring hyperpaths On directed hy
Page 7 and 8: Left Hyperpath Gap=3 Rank=4 Gap and
Page 9 and 10: An example of the folded an unfolde
Page 11 and 12: Example. In the most intuitive case
Page 13 and 14: Threshold: t = (0, max, max) t (∅
Page 15 and 16: The most general formulation of (si
Page 17 and 18: Generalization: Bellman-Ford equati
Page 19 and 20: Functional directed hypergraph H F
Page 21 and 22: Minimization of superior context fr
Page 23 and 24: Weakly superior functions (WSUP): g
Page 25 and 26: In the following we will see how (a
Page 27 and 28: Algorithm 1 - Let H = be a directe
Page 29 and 30: Algorithm 1 can be applied to minim
Page 31 and 32: In the previous example we have the
Page 33 and 34: Note that in this case convergence
Page 35 and 36: Algorithm 2 computes all shortest d
Page 37: Essential references • Ausiello,

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