directed hypergraphs algorithms and applications - Free

DIRECTED HYPERGRAPHS
ALGORITHMS AND APPLICATIONS
Giorgio Ausiello
Joint work with L. Laura, G.F. Italiano, U. Nanni
September 2011

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.

Reduction from SET-COVER
Let us consider an instance of the
SET-COVER decision problem:
a universe A = {e 1, e 2, …, e n},
a family of sets S1, S2, …, Sm an integer k.
A,
We can construct a hypergraph such
that there exists a hyperpath from
node p to node t with k+n+1
hyperarcs iff there exists a set cover
with k sets.

Measuring hyperpaths
On directed hypergraphs a variety of measures
can be defined , all collapsing into the usual
notion of cost of a path when the hypergraph is
a digraph. In some cases such measures lead to
tractable (polynomial time) minimum hyperpath
problems that are also easy to maintain in the
dynamic case.

Examples
• Rank of a hyperpath: maximum number of arcs
traversed.
• Gap of a hyperpath: minimum number of arcs
traversed.
(See next figure - A formal definition will be
provided later)
Interpretation in transition systems: no information
can arrive from source to target of a hyperpath
before gap ( ) steps; all information arrives after
rank ( ) steps.

Left Hyperpath
Gap=3
Rank=4
Gap and Rank
Right Hyperpath
Gap=2
Rank=5

In order to characterize tractable minimum hyperpath
problems and provide algorithms we need to introduce
inductively defined cost measures based on the structure
of the unfolded representation of hyperpath (hyperpath
tree).

An example of the folded an unfolded representation of
hyperpath (hyperpath tree).

Let H = be a weighted hypergraph
An inductively defined measure µ over a weighted directed
hypergraph is characterized by a triple µ = (µ 0, ψ, f) where:
µ 0 : measure of an empty hyperpath (normally equal to zero)
ψ : combination of costs on the source set
f : composition of ψ and w
and is defined as follows:
µ (∅) = µ 0
µ (h Z, t) = f (w X,t, ψ (µ(h Z, x1), µ(h Z, x2), …, µ(h Z, xn)) if h Z, t is
the hyperpath: (Z, x 1), (Z, x 2), …, (Z, x n), (X, t) and X = {x 1,
x 2, …, x n}

Example.
In the most intuitive case of the ‘traversal cost’
(c) of an unfolded hyperpath (where we sum up
over the cost of all traversed hyperarcs) we have
: c = (0, +, +), that is:
c(∅) = 0
c(h Z, t) = w X,t + ∑ (c(h Z, x1), c(h Z, x2), …, c(h Z, xn))

Other examples of inductively defined cost measures:
Gap: g = (0, min, +)
g (∅) = 0
g (h Z, t) = w X,t + min (g(h Z, x1), g(h Z, x2), …, g(h Z, xn))
Rank: r = (0, max, +)
r (∅) = 0
r (h Z, t) = w X,t + max (r(h Z, x1), r(h Z, x2), …, r(h Z, xn))
Bottleneck: b = (∞, min, min)
b (∅) = ∞
b (h Z, t) = min{w X,t, min (b(h Z, x1), b(h Z, x2), …, b(h Z, xn))

Threshold: t = (0, max, max)
t (∅) = 0
t (h Z, t) = max{w X,t, max (t(h Z, x1), t(h Z, x2), …, t(h Z, xn))
Average-depth: avg = (0, average, +)
avg (∅) = 0
avg (h Z, t) = w X,t + average (avg(h Z, x1), avg(h Z, x2), …,
avg(h Z, xn))
Path-product: pp = (1, ∏, ∗)
pp (∅) = 1
pp (h Z, t) = w X,t ∗ ∏ (pp(h Z, x1), pp(h Z, x2), …, pp(h Z, xn))

Algorithms for the computation of shortest
hyperpath heavily depend on the properties of
functions ψ and f
In order to point out such differences see the
following examples.

The most general formulation of (single source) shortest hyperpath problems can
be provided by Bellman-Ford equations.
Example Of Bellman-Ford equations for graphs
A=0
B=A+2
C=min(A+4, B+1)
D=B+1
E=min(C+2, D+1)
Solution:
A=0
B=2
C=3
D=3
E=4

Generalization: Bellman-Ford equations for shortest hyperpaths in hypergraphs.
The case of the rank measure. ψ=‘max’ and f= ‘+’
A=0
B=A+1
C=min(A+6, D+1)
D=max(B,C) +1
E=D+1
Solution:
A=0
B=1
C=6
D=7
E=8

Generalization: Bellman-Ford equations for shortest hyperpaths in hypergraphs.
The case of the average measure. ψ=‘average’ and f= ‘+’
A=0
B=A+1
C=min(A+6, D+1)
D=average(B,C) +1
E=D+1
Solution:
A=0
B=1
C=5
D=4
E=5

Note that more generally we may have that:
- the functions ψ and f may be different from
hyperarc to hyperarc
- the source set of a hyperarc should be defined
as an ordered tuple because the composition
function ψ might not be associative and
commutative.

Functional directed hypergraph
H F = where F establishes a
correspondence between any hyperarc (X, t) in H
and three entities:
- a weight w X,t
- a function ψ such that ψ X,t : D |X| → D
- a function f such that f X,t : D 2 → D

Related formulations
• Minimization of superior context-free
grammars (Knuth 1977)
• Minimization of additive weighting functions
(Nguyen, Pallottino 1988)
• Monotone systems of polynomial equations
(Esparza et al. 2008)

Minimization of superior
context free grammars
Productions:
θ : Y → g θ (X 1, …, X n) n≥ 0
Y, X 1, …, X n ∈ V N ; g θ , (, ), , ∈ V T
where ∀θ g θ is monotone non decreasing in each variable and
∀θ g θ (x 1, …, x n) ≥ max{ x i | i = 1, …, n}
(superior functions: e.g. max (x, y), (x + y) etc. )
Language:
L(Y) = { α | α ∈ V T* and Y ⇒ α}
(e.g. = max ( (3 +1), (4 + max(2, 3))))
Val (α ) ∈ ℜ +
Problem: Given a superior context free grammar and given
the axiom Y find α in L(Y) such that Val (α) is minimum.

Classification of the functions ψ and f
Superior functions (SUP): g monotone non decreasing in each
variable and g (x 1, …, x n) ≥ max{ x i | i = 1, …, n}
Examples: max (x, y), (x + y), etc.
Measures: traversal cost, rank, threshold, path-product [1, + ∞)
Strictly superior functions (SSUP): g monotone non
decreasing in each variable and g (x 1, …, x n) > max{ x i | i =
1, …, n},
Examples: x + y with positive variables
Measures: traversal cost, rank with positive weights, pathproduct
(1, + ∞)

Weakly superior functions (WSUP): g monotone non
decreasing in each variable and
g (x 1, …, x n) < x i → g (x 1, …, x i, …, x n) =
= g (x 1, …, ∞, …, x n)
Clearly SSUP ⊂ SUP ⊂ WSUP
Examples of WSUP\SUP functions are min {x, y} and
constant functions.
Measures: gap, bottleneck

Similar approach can be used in a suitable way
to define:
Inferior functions
Weakly inferior functions
Clearly SINF ⊂ INF ⊂ WINF
Measures based on functions in INF: bottleneck,
path-product [0, 1]
Measures based on functions in WINF\INF:
threshold

In the following we will see how (and under
what costs) Dijkstra algorithm can be extended
to compute optimum hyperpaths for measures
based on superior functions and on weakly
superior functions.
Similar approach can used for extending Dijkstra
algorithm to compute optimum hyperpaths for
measures based on inferior functions and on
weakly inferior functions.

Dijkstra algorithm scheme. Constructs single source shortest
path tree (spt) starting from node x. Let G = be a
weighted digraph.
For any n ∈ N set d(n)=∞ and set visited (n) = FALSE.
Enqueue x with priority d(x) = 0 in priority queue Q.
While Q not empty do
u = Extract-min (Q); visited (u) = TRUE;
insert edge (father (u), u) in spt;
for each (u,v) in A do
if v not visited and d(v) > d(u) + w(u,v)
then father (v) = u
d(v) = d(u) + w(u,v)
Enqueue (v) or Update (v) in Q with priority d(v)

Algorithm 1 - Let H = be a directed hypergraph. Let µ
be an inductively defined measure based on superior functions.
Algorithm 1 computes all single source shortest distances starting
from source set S.
For any n ∈ N set d(n)=∞ and set visited (n) = FALSE.
∀x ∈ S Enqueue x with priority d(x) = µ 0 in priority queue Q.
While Q not empty do
u = Extract-min (Q); visited (u) = TRUE;
for each hyperarc (X, v) in H where X = {x 1 … x k}
such that u ∈ {x 1,.., x k} and ∀x i x i visited
if v not visited and
d(v) > µ(S, v) = f (w X,v, ψ (µ(h S, x1), µ(h S, x2), …, µ(h S, xn))
then d(v) = µ(S, v)
Enqueue (v) or Update (v) in Q with priority d(v)

Algorithm 1 computes all shortest distances from
source set S in time O(|H| log |N| + |H |).
Cost can be reduced to O(|N| log |N| + |H |) by
making use of Fibonacci heaps for implementing
the priority queue.

Algorithm 1 can be applied to minimization of
measures based on superior functions.
It can be easily extended to deal with
maximization of measures based on inferior
functions.
See also Martelli, Montanari 1973: computation
of traversal cost in and-or graphs.

A hard instance with a weakly superior function: gap
Remember: g (h Z, t) = w X,t + min (g(h Z, x1), g(h Z, x2), …, g(h Z, xn))

In the previous example we have the following sequence of
steps:
Extract-min (Q) = a 3 g(a 3) = 6
Extract-min (Q) = a 2 g(a 2) = 7
Extract-min (Q) = a 1 g(a 1) = 8
Extract-min (Q) = a 0 g(a 0) = 9
Enqueue (c 0) g(c 0) = 10 Extract-min (Q) = c 0
Enqueue (c 1) g(c 1) = 11 Extract-min (Q) = c 1
Enqueue (b 1) g(b 1) = 12 Extract-min (Q) = b 1
Now a new link (a 1 b 1, c 0) is created and c 0 has to be
enqueued again with g(c 0) =9 and so on: g(c 0) =8, g(c 0) =7,
etc.

In the example a major role is played by two facts:
1) a node can become reachable from a new source set during
the computation and therefore it has to be repeatedly visited
during shortest path computation
2) the measure is based on weakly superior functions and
therefore the value of the measure can decrease after a cycle
has been visited; this cannot happen with superior functions
As a consequence the approach of Algorithm 1 would incur
into cost
O(|N| log |N| + W |H |)
where W = # of values of domain (possible values of measure)

Note that in this case convergence is achieved after one scan of
cycles (this is true for all measures based on WSUP functions)
The next algorithm exploits this fact in order to overcome the
difficulty and obtain better performance (same as Algorithm 1).
Measures for which convergence is achieved after repeated visits
of cycles can also be found and we also saw an example
(average) where convergence requires an unbounded number of
scans of cycles.

Algorithm 2 - Let H = be a directed
hypergraph. Let µ be an inductively defined
measure based on superior functions. Algorithm
2 computes all single source shortest distances
starting from source set S.
1) Propagate reachability
2) Perform Algorithm 1

Algorithm 2 computes all shortest distances from
source set S in time O(|H| log |N| + |H |).
Again cost can be reduced to O(|N| log |N| + |H |)
by making use of Fibonacci heaps for
implementing the priority queue.
The algorithm can be extended to deal with
maximization problems for measures based on
weakly inferior functions.

Exercises
Prove that the following problems are NP-hard:
• hyperpath with minimum size
• hyperpath with minimum source area
• hyperpath with minimum number of source
sets

Essential references
• Ausiello, D’Atri, Sacca’, Minimal representation of directed
hypergraphs, SIAM on Comp., 1986
• Ausiello, Italiano, Online algorithms for polynomially solvable
satisfiability problems, J. of Logic Prog., 1991
• Gallo, Longo, Nguyen, Pallottino, Directed hypergraphs and
applications, Discrete Applied Math., 1993
• Ramalingam, Reps, An incremental algorithm for a generalization of
the shortest path problem, J. of Algorithms, 1996
• Esparza, Kiefer, Luttenberger, Solving monotone polynomial
equations, 5° IFIP Conf. on Theor. Comp. Sc., 2008
• Ausiello, Italiano, Laura, Nanni, Sarracco, Classification and traversal
algorithmic techniques for optimization problems on directed
hyperpaths, Tech. Rep. DIS, Sapienza University of Rome 2011