Download presentation

Presentation is loading. Please wait.

Published byAbigail O'Donnell Modified over 3 years ago

1
A New SAT Encoding of the At- Most-One Constraint Jingchao Chen Donghua University, China

2
2 Definition At-Most-One (AMO) constraint: Given X = {x 1,x 2,…,x n } of n Boolean variables, at most one out of n variables in X is allowed to be true. AMO encoding: Convert AMO constraint to SAT problem in CNF

3
3 Known AMO encodings standard AMO encoding: AMO(X)={ x i ∨ x j | x i, x j ∈ X,i

4
4 A summary of AMO encodings Methodinventorclausesaux. vars standardfolkloren*(n-1)/20 bitwiseFrisch et al.n log nlog n sequentialSinz3n-4n-1 2-productThis paper

5
5 Basic Idea of a Product Encoding u 1 u 2 · · · · · · · · · · · · ·u i · · · · · · u p vqvjv2v1vqvjv2v1 x 1 x 2 · · · · · · · · · · · · · · · · · · · · ·x p x p+1 x p+2 · · · · · · · · · · · · · · · ·· ·x 2p x jp-p+1 x jp-p+2 · · · · · · · · x k · · · · x jp x qp-p+1 x qp-p+2 · · · · · · · · · · · · · x pq n≈pq x k →

6
6 Example n=5, p=3, q=2 v2v1v2v1 u 1 u 2 u 3 x 1 x 2 x 3 x 4 x 5

7
7 Basic formula of 2-product encoding where X={x 1,x 2,…x n }, U={u 1,u 2,…u p }, V={v 1,v 2,…v q }

8
8 Property (1) of 2-product encoding If using the sequential encoding to encode sub-constraints AMO (U) and AMO (V), the 2-product encoding requires 2n + 3p-4 +3q-4 ≈ clauses and auxiliary variables.

9
9 Property (2) of 2-product encoding If using the standard encoding to encode sub-constraints AMO (U) and AMO (V), the 2-product encoding requires 2n + p(p-1)/2 + q(q-1)/2 ≈ clauses and auxiliary variables.

10
10 Property (3) of 2-product encoding If encoding sub-constraints AMO (U) and AMO (V) in a recursive way, the 2-product encoding requires clauses and auxiliary variables.

11
11 k-product encoding map(X,W 1,W 2,…W k ) denotes each point in X is defined by a point in W 1 ×W 2 ×…×W k. It consists of the following clauses. |W 1 |=|W 2 |=…=|W k |=p

12
12 Property of k-product encoding When |W 1 |=|W 2 |=…=|W k |=p=2, k-product encoding become a bitwise encoding. If using the standard encoding to encode sub- constraints AMO(W i ), |W i |=p=, the k-product encoding of AMO requires clauses and auxiliary variables.

13
13 Empirical evaluation Table 1. The number of clauses and auxiliary variables required to encode AMO constraints of edge-matching problems.

14
14 Table 2. Runtime (in seconds) required by CircleSAT to solve edge-matching problems based on various AMO encodings.

15
15 Conclusions Present four versions of the product AMO encoding 2-product encoding requires the minimal clauses Unit propagation on product encoding achieves arc- consistency.

16
16 Thank you

Similar presentations

OK

1 Polynomial Time Reductions Polynomial Computable function : For any computes in polynomial time.

1 Polynomial Time Reductions Polynomial Computable function : For any computes in polynomial time.

© 2018 SlidePlayer.com Inc.

All rights reserved.

To ensure the functioning of the site, we use **cookies**. We share information about your activities on the site with our partners and Google partners: social networks and companies engaged in advertising and web analytics. For more information, see the Privacy Policy and Google Privacy & Terms.
Your consent to our cookies if you continue to use this website.

Ads by Google

Ppt on review writing services By appt only movie page Ppt on brand marketing resume Ppt on statistics for class 11th Ppt on health food in hindi Ppt on water activity chart Ppt on computer malwares bytes Ppt on organic farming in india Ppt on haunted places in india Ppt on design patterns in java