2nd/10_Wiki/Topics/AI_and_ML/Constraint Satisfaction Problems (CSP).md

---
id: wiki-2026-0508-csp
title: Constraint Satisfaction Problems (CSP)
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [CSP, 제약 충족 문제, AC-3, backtracking, MIP, OR-Tools, scheduling]
duplicate_of: none
source_trust_level: A
confidence_score: 0.93
verification_status: applied
tags: [csp, constraint-programming, optimization, scheduling, sat, smt, ortools, backtracking]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python / C++
  framework: OR-Tools / Z3 / MiniZinc / Choco
---

# Constraint Satisfaction Problems (CSP)

## 매 한 줄
> **"매 rule 의 break X + 매 fill"**. 매 (Variables, Domains, Constraints) 의 triple. 매 backtracking + 매 propagation. 매 modern: OR-Tools, Z3 SMT, MiniZinc. 매 scheduling, 매 routing, 매 puzzle 의 NP-hard 의 practical 접근.

## 매 핵심

### 매 components
- **Variables** (X): 매 assignable.
- **Domains** (D): 매 possible values.
- **Constraints** (C): 매 relations.

### 매 type
- **Boolean**: SAT.
- **Integer / discrete**: pure CSP.
- **Continuous**: linear / convex / nonlinear programming.
- **Mixed Integer**: MIP.
- **SMT**: 매 first-order theory.

### 매 algorithm

#### Backtracking
- 매 DFS + 매 backtrack on constraint violation.
- 매 baseline.

#### Constraint Propagation
- **AC-3** (Arc Consistency): 매 inconsistent value 의 prune.
- **Forward Checking**: 매 variable assignment 시 의 future variable 의 prune.

#### Heuristic
- **MRV** (Minimum Remaining Values): 매 가장 constrained variable first.
- **Degree heuristic**: 매 매 connected variable.
- **LCV** (Least Constraining Value): 매 future flexibility maximize.

#### Local search
- **Min-conflicts**: 매 random init + 매 conflict reduce.
- **Simulated annealing**.
- **Tabu search**.

### 매 SAT (special case)
- 매 boolean only.
- 매 CNF form.
- 매 modern solver: Glucose, MiniSat, Kissat.
- 매 reduction: 매 다른 NP-complete 의 SAT.

### SMT (extended)
- 매 first-order theory + decision procedure.
- 매 theory: arithmetic, arrays, bitvectors, strings.
- 매 Z3, CVC5, Yices.

### 매 MIP (Mixed Integer Programming)
- 매 LP relaxation + branch & bound.
- 매 Gurobi, CPLEX, OR-Tools.

### CP-SAT (modern)
- 매 OR-Tools 의 hybrid.
- 매 CP + SAT.
- 매 매 fastest 의 industrial scheduling.

### 매 응용
1. **Scheduling**: 매 work, 매 sport, 매 exam.
2. **Routing** (VRP).
3. **Resource allocation**.
4. **Configuration** (car options).
5. **Puzzle** (Sudoku, N-Queens, Zebra).
6. **Verification** (SMT).
7. **Compiler** (register allocation).

## 💻 패턴

### N-Queens (backtracking)
```python
def solve_n_queens(n):
    queens = [-1] * n
    
    def backtrack(row):
        if row == n: return [queens[:]]
        solutions = []
        for col in range(n):
            if all(queens[r] != col and abs(queens[r] - col) != row - r 
                   for r in range(row)):
                queens[row] = col
                solutions.extend(backtrack(row + 1))
        return solutions
    
    return backtrack(0)

print(len(solve_n_queens(8)))  # 92
```

### AC-3 (constraint propagation)
```python
def ac3(domains, constraints):
    """매 arc consistency."""
    queue = [(x, y) for (x, y) in constraints]
    while queue:
        (x, y) = queue.pop(0)
        if revise(domains, x, y, constraints):
            if not domains[x]: return False
            for z in neighbors(x):
                if z != y: queue.append((z, x))
    return True

def revise(domains, x, y, constraints):
    revised = False
    for vx in list(domains[x]):
        if not any(constraint_holds(x, vx, y, vy, constraints) for vy in domains[y]):
            domains[x].remove(vx)
            revised = True
    return revised
```

### Sudoku (OR-Tools CP-SAT)
```python
from ortools.sat.python import cp_model

def solve_sudoku(grid):
    model = cp_model.CpModel()
    
    # 매 9×9 variable
    cells = [[model.NewIntVar(1, 9, f'c{r}{c}') for c in range(9)] for r in range(9)]
    
    # 매 given clues
    for r in range(9):
        for c in range(9):
            if grid[r][c] != 0:
                model.Add(cells[r][c] == grid[r][c])
    
    # 매 row / col / box uniqueness
    for r in range(9):
        model.AddAllDifferent(cells[r])
    for c in range(9):
        model.AddAllDifferent([cells[r][c] for r in range(9)])
    for br in range(3):
        for bc in range(3):
            model.AddAllDifferent([cells[3*br + i][3*bc + j] for i in range(3) for j in range(3)])
    
    solver = cp_model.CpSolver()
    if solver.Solve(model) == cp_model.OPTIMAL:
        return [[solver.Value(cells[r][c]) for c in range(9)] for r in range(9)]
    return None
```

### Job Shop Scheduling (OR-Tools)
```python
from ortools.sat.python import cp_model

def schedule_jobs(jobs):
    model = cp_model.CpModel()
    horizon = sum(t for job in jobs for _, t in job)
    
    all_tasks = {}
    machine_to_intervals = collections.defaultdict(list)
    
    for j_id, job in enumerate(jobs):
        for t_id, (machine, duration) in enumerate(job):
            start = model.NewIntVar(0, horizon, f'start_{j_id}_{t_id}')
            end = model.NewIntVar(0, horizon, f'end_{j_id}_{t_id}')
            interval = model.NewIntervalVar(start, duration, end, f'interval_{j_id}_{t_id}')
            all_tasks[(j_id, t_id)] = (start, end, interval)
            machine_to_intervals[machine].append(interval)
    
    # 매 매 machine 의 1 task 만 의 동시.
    for intervals in machine_to_intervals.values():
        model.AddNoOverlap(intervals)
    
    # 매 매 job 의 sequence (precedence).
    for j_id, job in enumerate(jobs):
        for t_id in range(len(job) - 1):
            model.Add(all_tasks[(j_id, t_id+1)][0] >= all_tasks[(j_id, t_id)][1])
    
    # 매 minimize makespan
    makespan = model.NewIntVar(0, horizon, 'makespan')
    model.AddMaxEquality(makespan, [all_tasks[(j_id, len(job)-1)][1] for j_id, job in enumerate(jobs)])
    model.Minimize(makespan)
    
    solver = cp_model.CpSolver()
    solver.Solve(model)
    return solver.ObjectiveValue()
```

### Z3 SMT
```python
from z3 import *

# 매 example: 매 8-puzzle solvability check
s = Solver()
x = [Int(f'x_{i}') for i in range(9)]
s.add([0 <= xi for xi in x])
s.add([xi <= 8 for xi in x])
s.add(Distinct(x))

s.add(x[0] == 1)
s.add(x[1] == 2)
# ...

if s.check() == sat:
    print(s.model())
```

### Vehicle Routing (VRP)
```python
from ortools.constraint_solver import routing_enums_pb2, pywrapcp

manager = pywrapcp.RoutingIndexManager(num_nodes, num_vehicles, depot)
routing = pywrapcp.RoutingModel(manager)

def distance_callback(from_index, to_index):
    return distance_matrix[manager.IndexToNode(from_index)][manager.IndexToNode(to_index)]

transit_idx = routing.RegisterTransitCallback(distance_callback)
routing.SetArcCostEvaluatorOfAllVehicles(transit_idx)

params = pywrapcp.DefaultRoutingSearchParameters()
params.first_solution_strategy = routing_enums_pb2.FirstSolutionStrategy.PATH_CHEAPEST_ARC

solution = routing.SolveWithParameters(params)
```

### MiniZinc (declarative)
```minizinc
% n-queens.mzn
int: n = 8;
array[1..n] of var 1..n: q;

constraint forall(i, j in 1..n where i < j) (
  q[i] != q[j] /\
  q[i] - q[j] != i - j /\
  q[i] - q[j] != j - i
);

solve satisfy;
```

### ML-aided heuristic (RL for branching)
```python
# 매 modern: RL 의 branching variable selection
class LearnedHeuristic:
    def __init__(self, model):
        self.model = model
    
    def select_variable(self, state):
        """매 state 의 features → 매 best variable to branch."""
        features = encode_state(state)
        return self.model.predict(features)
```

## 🤔 결정 기준
| 문제 | Tool |
|---|---|
| Boolean SAT | Glucose, Kissat |
| SMT (math) | Z3, CVC5 |
| Discrete CSP | OR-Tools CP-SAT |
| MIP (large) | Gurobi, CPLEX |
| Scheduling | OR-Tools CP-SAT |
| Routing | OR-Tools |
| Continuous | scipy.optimize |
| Declarative | MiniZinc |

**기본값**: OR-Tools CP-SAT 의 baseline (free + fast).

## 🔗 Graph
- 부모: [[Combinatorial-Optimization]] · [[Graph_Theory|Graph-Theory]]
- 변형: [[MIP]]
- 응용: [[Scheduling]] · [[Routing]]
- Tool: [[OR-Tools]]
- Adjacent: [[Black-Box-Optimization]] · [[Automated-Theorem-Proving]] · [[Bayesian Statistics]]

## 🤖 LLM 활용
**언제**: 매 scheduling, 매 routing, 매 configuration. 매 verification. 매 puzzle.
**언제 X**: 매 differentiable problem (gradient descent). 매 black-box (BO).

## ❌ 안티패턴
- **No propagation**: 매 backtracking 만.
- **MRV / LCV 무시**: 매 inefficient.
- **Wrong solver for problem class**: 매 SAT for continuous.
- **No problem decomposition**: 매 huge instance 의 fail.
- **Constraint 의 redundant 의 add**: 매 solver 의 hint.
- **Single solution mode**: 매 enumerate 의 expensive.

## 🧪 검증 / 중복
- Verified (Russell-Norvig AI book, OR-Tools docs, Handbook of CP).
- 신뢰도 A.
- Related: [[Black-Box-Optimization]] · [[Automated-Theorem-Proving]] · [[Bayesian Statistics]] · [[Causal-Inference]].

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — type + algorithm + 매 N-Queens / AC-3 / Sudoku / JSP / VRP code |