Model Counting Aditya A. Shrotri Dept. of Computer Science, Rice - - PowerPoint PPT Presentation

Theory and Practice of Efficient Approximate Model Counting

Aditya A. Shrotri

• Dept. of Computer Science,

Rice University

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• SAT: Does there exist a satisfying assignment?
• #SAT: How many satisfying assignments?
• Complexity: #P-Complete (contains entire polynomial hierarchy)
• In Practice: Harder than SAT

Beyond SAT: #SAT

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Probabilistic Inference Measuring Information Leakage Network Reliability Probabilistic Databases #SAT

• F = ¬𝑌1 ∧ 𝑌2 ∨ 𝑌2 ∧ 𝑌3 ∧ 𝑌4 ∨ (𝑌1 ∧ 𝑌3 ∧ ¬𝑌5)
• Disjunction of Cubes
• DNF-SAT is in P
• #DNF is #P-Complete [Valiant, ’79]
• Need to Approximate!

The Disjunctive Normal Form

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Input: DNF Formula F Tolerance ε 0 < ε < 1 Confidence δ 0 < δ < 1 Output: Approximate Count C s.t. Pr [#𝐆 ⋅(1-ε) < C < #𝐆 ⋅(1+ε) ] > 1-δ Challenge: Design a poly(m, n, 1

𝜗 , log( 1 𝜀 )) time algorithm

Fully Polynomial Randomized Approximation Scheme

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where m = #cubes n = #vars

• Monte Carlo Sampling [Karp et al., ‘89]
• Complexity: 𝑷(𝒏 ⋅ 𝒐 ⋅ 𝒎𝒑𝒉(

𝟐 𝜺) ⋅ 𝟐 𝜻𝟑)

• Hashing – based [Chakraborty et al., ‘16]
• Complexity: O(𝒏𝟒 ⋅ 𝒐 ⋅ 𝒎𝒑𝒉(

𝟐 𝜺) ⋅ 𝟐 𝜻𝟑)

Paradigms for #DNF FPRAS

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U F U F

• What is the power of hashing?
• Is hashing as powerful as Monte Carlo?
• How do algorithms compare empirically?
• No rigorous empirical comparison

Open Questions

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1.

New hashing-based algorithm SymbolicDNFApproxMC

1.

Complexity 𝑷(𝒏 ⋅ 𝒐 ⋅ 𝒎𝒑𝒉(

𝟐 𝜺) ⋅ 𝟐 𝜻𝟑)

• Significance: General technique of hashing is as powerful as specialized technique of Monte

Carlo for DNF-Counting!

2.

Introduced generic new concepts applicable beyond DNF formulas

• Symbolic Hashing
• A new highly space and time efficient 2-Universal Hash Family
• Stochastic Cell Counting

2.

Improved complexity of older hashing algorithm DNFApproxMC by 𝑃(𝑜)

3.

First large scale experimental comparison of 5 FPRASs

Our Contributions

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Experimental Results

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• Worst-case complexity not the last word
• DNFApproxMC is most robust and solves

largest number of benchmarks!

• Previous works used Vazirani Counter – not

the best choice

• Best complexity possible?
• Extend to Weighted DNF-Counting
• Hashing-based FPRAS for other problems
• Counting Linear Extensions, Perfect Matchings, Knapsacks …
• Practically efficient Exact Counting

Future Directions

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“Not All FPRASs are Equal: Demystifying FPRASs for DNF- Counting” Kuldeep S. Meel, Aditya A. Shrotri, and Moshe Y. Vardi Constraints 1-24 (2018) “On Hashing-Based Approaches to Approximate DNF-Counting” Kuldeep S. Meel, Aditya A. Shrotri, and Moshe Y. Vardi FST&TCS (2017)

Publications

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