Presentation notes

Finding small OBDDs for incompletely specifed truth tables is hard Jesper Torp Kristensen Peter Bro Miltersen University of Aarhus, Denmark.

COCOON'06, Taipei, August 18, 2006.

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Outline 

Main result



Motivation and previous work



Some ingredients of the proof



An open problem

COCOON'06, Taipei, Augus

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Main result 

The optimization problem Minimum Consistent OBDD:  Input: The truth table of a partial Boolean function.  Output: A minimum sized OBDD consistent with the truth table.

is NP-hard.

COCOON'06, Taipei, Augus

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Representing Boolean functions by OBDDs x1 © x2 © x3

x1 0

1

x2

x2 1

0

x3

1

1

1

0

x3

0

0

0

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Representing Boolean functions by OBDDs X1

X2

X3

0

0

0

0

0

0

1

1

0

1

0

1

0

1

1

0

1

0

0

1

1

0

1

0

1

1

0

0

1

1

1

1

x1 0

1

x2

x2 1

0

x3

1

1

1

0

x3

0

0

0

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Representing Boolean strings by OBDDs 01101001

x1 0

1

x2

x2 1

0

x3

1

1

1

0

x3

0

0

0

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Representing Boolean strings by OBDDs 01101001

x1 0

1

x2 1

0

x3

“generic”

101000100100111

x2

1

1

1

0

x3

0

0

0

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OBDDs as a compression scheme ď Ž

Kiefer, Flajolet, Yang, 2000: OBDD representation is a universal compression scheme.

ď Ž

Meaning: For any constant k, and sufficiently long input, the compression rate asymptotically matches the kth order empirical block entropy of the input.

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OBDDs as a compression scheme 

OBBD representation is a universal locally decodable source code:  



It achieves the block entropy rate.

Individual bits of the compressed file can be reconstructed in polylogarithmic time.

Bryant (1985): Optimal-sized OBDD representation is efficiently computable.

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Chess endgame tables  

Used in chess playing software Example:  

 

Tradeoff between size of table and access time.

State-of-the-art representation: Nalimov 



KBBK endgame (King-Bishop-Bishop-King) Straightforward representation as array: 32 MB

KBBK representation ¼ 450 kB

MSc thesis project of J. T. Kristensen: Approach the state-of-the-art using OBDDs. COCOON'06, Taipei, Augus

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Endgame tables are partially defined functions efficient

?

0

127481

?

?

Typically 5%-50% of the table is ?

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Minimum consistent OBDD problem 

Given a partially defined table, find the minimum sized OBDD consistent with the table.



No polynomial time algorithm known….

In the MSc thesis of Kristensen, various heuristics are suggested for getting good solutions.

Is the use of heuristics necessary? Topic of this talk….

Interesting challenge: Further investigate and improve heuristics for this problem.







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NP-hardness of finding minimum OBDDs for partially defined functions    

Sauerhoff and Wegener, 1996.

Hirata, Shimozono, Shinohara, 1996.

Simon, 1990.

Pitt and Warmuth, 1993.

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Input model for all previous result 

The partially defined functions are given in compact form, such as an OBDD or as a list of positive and negative instances.



Also, the hardness proofs use inputs with compact (polylogarithmic sized) representation.



We want a hardness result for the case where the partially defined function is given as an uncompressed table.



Note that such a hardness result immediately implies the previous ones.

COCOON'06, Taipei, Augus

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Main result 

The optimization problem Minimum Consistent OBDD:  Input: The truth table of a partial Boolean function.  Output: A minimum sized OBDD consistent with the truth table.

is NP-hard.

COCOON'06, Taipei, Augus

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NP-hardness proof   

Graph colouring · Minimum String Cover · Minimum Consistent OBDD

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Minimum String Cover problem ď Ž

Given a set of S partially defined Boolean strings (strings over 0,1,?), a cover is a set of fully defined strings so that every string in S is consistent with at least one of them.

ď Ž

Example: 00,11 is a cover for 0?, ?0, 1?

ď Ž

MSC: Given a set of partially defined strings, find their minimum sized cover.

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Graph Coluring reduces to Minimum Set Cover 

Given a graph G=(V,E), we make n strings: s1, s2, …, sn, one for each vertex v in V = {1,…,n}.



sv has as its j’th letter:   



1 if v = j 0 if v is different from j and (v,j) is in E ? otherwise.

G has a coloring with k colors if and only if the strings have a cover of size k.

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MSC reduces to “Minimum middle-layer consistent OBDD”. 

Given s1, s2, …, sn define f(i,j) = “what is the j’th bit of s_i ?”

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MSC reduces to Minimum Consistent OBDD 

f(i,j,t,z) = si(z) if t=0



f(i,j,t,z) = bj(z) if t=1

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Hardness of approximation ď Ž

Minimum consistent OBDD is hard to approximate within a factor of n0.249.

ď Ž

More precise statement: Given a table of size n that is promised to be compressible to n0.5001, it is hard to compress it to size n0.7499.

ď Ž

Open Problem: Given a table of size n that is promised to be compressible to size n/10, is it possible to compress it efficiently to size n/2?

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