Double Digestion Problem

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Double Digestion Problem
http//www.math.iastate.edu/wu/math597.html

• Math/BCB/ComS597
• Zhijun Wu
• Department of Mathematics

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Double Digestion Problem
a 1, 3, 3, 12
1 3 12
3
b 1, 2, 3, 3, 4, 6
2 4 6
3 3 1
c 1, 1, 1, 1, 2, 2, 2, 3, 6
1 1 2 2 6
3 1 2 1
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DDP is NP-complete!
An Example S 1,3,5,6,8,9 a
1,3,5,6,8,9 b 16,16 c 1,3,5,6,8,9 a
1 6 9 3 5 8 b 16
16 c 1 6 9 3 5 8 S1
1,6,9 S2 3,5,8
DDP is equivalent to SPP
Given a set of integers, S ai 1 i n, we
partition S by reducing the problem to a DDP
Construct a DDP problem with a ai 1 i
n, b s/2, s/2, c ai 1 i n,
where s Sa?S a.
Let a, b, and c be a solution to the DDP. Then
the elements in the first and second parts of a
form a solution to the partition problem for S.
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Problem Formulation
a1 a a2 . .
. an
b1 b b2 . .
. bm
c1 c c2 . .
. ck
x11 x12 x1k x21 x22 x2k X
xn1 xn2 xnk
y11 y12 y1k y21 y22 y2k Y
ym1 ym2 ymk
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Problem Formulation
a
1 3 12
3
b
2 4 6
3 3 1
c
1 1 2 2 6
3 1 2 1
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Integer Programming
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Example Representation
1 0 0 0 0 0 0 0 0 X 0 1 0 0 1 0 0 0 0
0 0 0 1 0 0 1 0 0 0 0 1 0 0 1 0 1 1
1 3 12
3
2 4 6
3 3 1
1 0 0 0 0 0 0 0 0 Y 0 1 1 0 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0
0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0
0 0 0 1
1 1 2 2 6
3 1 2 1
a (1, 3, 3, 12)T
b (1, 2, 3, 3, 4, 6)T
a Xc, b Yc
c (1, 1, 1, 1, 2, 2, 2, 3, 6)T
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Further Reading
Design and Analysis of Computer Algorithms by
Aho, Hopcroft, and Ullman Introduction to
Automata Theory, Languages, and Computation by
Hopcroft and Ullman Computers and Intractability
by Garey and Johnson Introduction to
Computational Biology by Waterman