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https://github.com/janecea/exact-cover-sat
Encodes, solves, and decodes the exact cover problem via reduction to SAT
https://github.com/janecea/exact-cover-sat
dimacs-cnf exact-cover sat-solver
Last synced: 11 days ago
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Encodes, solves, and decodes the exact cover problem via reduction to SAT
- Host: GitHub
- URL: https://github.com/janecea/exact-cover-sat
- Owner: JANECEA
- Created: 2024-10-27T13:10:53.000Z (about 2 months ago)
- Default Branch: main
- Last Pushed: 2024-11-15T00:32:40.000Z (about 1 month ago)
- Last Synced: 2024-11-15T01:24:37.943Z (about 1 month ago)
- Topics: dimacs-cnf, exact-cover, sat-solver
- Language: Python
- Homepage:
- Size: 43.9 KB
- Stars: 0
- Watchers: 1
- Forks: 0
- Open Issues: 0
-
Metadata Files:
- Readme: README.md
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README
# Exact-Cover-SAT
Encodes, solves, and decodes the exact cover problem via reduction to SAT.
# Documentation
## Problem description
*"Given a collection* $S$ *of subsets of a set* $X$ *, an exact cover is a subcollection* $S^{\*}$ *of* $S$ *such that each element in* $X$ *is contained in exactly one subset in* $S^*$ *.
It is a non-deterministic polynomial time* $(NP)$ *complete problem and has a variety of applications, ranging from the optimization of airline flight schedules, cloud computing, and electronic circuit design."*
[Source](https://en.wikipedia.org/wiki/Exact_cover).
Example of a valid input file format:
```
{1 2 3 4}
{
{ }
{1 3}
{1 2 3}
{2 4}
}
```
**Input format**:
* First line only contains the main set: $X$.
* Following lines only contain the set of subsets of the main set: $S$.
* Both subsets of literals and the set of subsets $S$ need to be contained in `{ }`.
* Lines after the main set can contain multiple subsets.
* Literals need to be delimited by (one or more) spaces.
* Literals can be strings of various length but without whitespace.
This input file format is also valid:
```
{alpha bravo charlie delta}
{ { } {alpha charlie} {alpha bravo charlie} {bravo delta} }
```
*Any deviation from the problem's specifications (e.g., subset contains an element that is not present in the main set) should trigger an exception during parsing.*
## Encoding
The problem is encoded using one set of variables. Variable $s_i$ represents that subset $S_i$ is included in the exact cover $S^{\*}$.
Analogically, variable $\neg s_i$ means the subset is not included in $S^*$.
The conditions for a collection of subsets to be an exact cover of the main set $X$ can be written as the following constraints:
- Every element of the main set $X$ has to be included in at least one subset in $S^*$.
```math
\bigwedge_{e \in X} \left( \bigvee_{\substack{ S_i \in S \\ e \in S_i}} (s_i) \right)
```
- Every element of the main set $X$ can be included in at most one subset in $S^*$.
```math
\bigwedge_{e \in X}
\left(
\bigwedge_{\substack{S_i, S_j \in S\\ S_i \neq S_j}}
\left(\neg s_i \lor \neg s_j \right)
\right)
```
## User documentation
Basic usage:
```
main.py [-h] [-i INPUT] [-o OUTPUT] [-s SOLVER] [-v {0,1}]
```
Command-line options:
* `-h`, `--help` : Show a help message and exit.
* `-i INPUT`, `--input INPUT` : The instance file. Default: "input.in".
* `-o OUTPUT`, `--output OUTPUT` : Output file for the DIMACS cnf formula. Default: "output.cnf".
* `-s SOLVER`, `--solver SOLVER` : The SAT solver to be used. Default "glucose-syrup" *
* `-v {0,1}`, `--verb {0,1}` : Verbosity of the SAT solver used.
\* *If the provided path cannot be found, the script assumes this to be a global command.*
## Example instances
* `input-easy-sat.in`: Simple satisfiable instance provided in the [Exact cover wikipedia page](https://en.wikipedia.org/wiki/Exact_cover) where $|X| = 4$, $|S| = 4$.
* `input-easy-unsat.in`: Simple unsatisfiable instance where $|X| = 5$, $|S| = 5$.
* `input-hard-sat.in`: Complex satisfiable instance generated using [instance_generator.py](instance_generator.py) where $|X| = 100$, $|S| = 560$.
* `input-hard-unsat.in`: Complex unsatisfiable instance generated using [instance_generator.py](instance_generator.py) where $|X| = 100$, $|S| = 500$.
## Experiments
Experiments were run on Intel Core i7-7500U CPU (2.70GHz) and 12 GB RAM on Fedora Linux 40.
The following values are averages over 5 runs of `glucose-syrup` on various instances generated by the [instance_generator.py](instance_generator.py) with the given parameters and `GUARANTEED_SAT` set to `True`.
#### For `SUBSET_COUNT_MULT` set to `5`:
| **Main set size** | **Variable count** | **Clause count** | **Runtime (s)** |
|-------------------|--------------------|------------------|-----------------|
| **20** | 113 | 25360 | 0.123 |
| **40** | 226 | 196185 | 1.358 |
| **60** | 338 | 707458 | 6.892 |
| **80** | 451 | 1586590 | 20.177 |
| **100** | 561 | 3081377 | 46.554 |
---
#### For `SUBSET_COUNT_MULT` set to `10`:
| **Main set size** | **Variable count** | **Clause count** | **Runtime (s)** |
|-------------------|--------------------|------------------|-----------------|
| **20** | 214 | 105004 | 0.763 |
| **40** | 425 | 831246 | 10.331 |
| **60** | 641 | 2813321 | 49.826 |
| **80** | 849 | 6402951 | 12.339 |
| **100** | 1064 | 12434209 | 24.045 |
---
*`SUBSET_COUNT_MULT` specifies how many random subsets of* $X$ *will be created.
For example if `MAIN_SET_SIZE = 20` and `SUBSET_COUNT_MULT = 5`, then* $100$ *random subsets will be created.*