<h2>🧠 What is OR-Tools?</h2>
<p><a href="https://developers.google.com/optimization" target="_blank" rel="noopener nofollow">OR-Tools</a> is an open-source mathematical optimization suite developed by Google. It supports a variety of optimization problems, including:</p>
<ul>
<li>Constraint Programming (CP)</li>
<li>Linear Programming (LP)</li>
<li>Mixed Integer Programming (MIP)</li>
<li>Routing problems (e.g., TSP, VRP)</li>
</ul>
<h2>🚀 What is the CP-SAT Solver?</h2>
<p>CP-SAT is OR-Tools’ <strong>state-of-the-art Constraint Programming solver</strong>. It is:</p>
<ul>
<li>Designed for solving combinatorial optimization problems.</li>
<li>Based on SAT (Boolean satisfiability) solving techniques.</li>
<li>Capable of modeling complex constraints like scheduling, assignment, resource allocation, and more.</li>
</ul>
<p>This cheat sheet focuses on <strong>modeling with the CP-SAT solver using the Python API</strong>.</p>
<hr>
<h2>📘 About This Cheat Sheet</h2>
<blockquote>
<p><strong>Why this exists:</strong><br>
I had been looking for a consolidated API reference or blog for OR-Tools, but couldn’t find one that covered everything in one place.<br>
So, I created this cheat sheet to serve as a quick, unified reference for myself and others working with the CP-SAT solver.</p>
</blockquote>
<blockquote>
<p><strong>Language Note:</strong><br>
This cheat sheet is based on the <strong>Python API</strong>, but the modeling patterns and constraints are quite similar across other supported languages (C++, Java, C#).<br>
It can still serve as a helpful guide for applying constraints effectively in any language.</p>
</blockquote>
<hr>
<h2>🧩 Interfaces to Extend</h2>
<ul>
<li><strong><code>CpSolverSolutionCallback</code></strong><br>
Abstract base class to customize solution output and logic.</li>
</ul>
<h3>✅ Default Implementations</h3>
<table>
<thead>
<tr>
<th>Class</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>ObjectiveSolutionPrinter</code></td>
<td>Prints solution time &amp; objective value.</td>
</tr>
<tr>
<td><code>VarArraySolutionPrinter</code></td>
<td>Prints only variable values.</td>
</tr>
<tr>
<td><code>VarArrayAndObjectiveSolutionPrinter</code></td>
<td>Prints variables, time, and objective value.</td>
</tr>
</tbody>
</table>
<hr>
<h2>📌 Variable Creation</h2>
<table>
<thead>
<tr>
<th>Method</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>NewBoolVar(name)</code></td>
<td>Boolean variable (0 or 1).</td>
</tr>
<tr>
<td><code>NewIntVar(lb, ub, name)</code></td>
<td>Integer variable with lower and upper bounds.</td>
</tr>
<tr>
<td><code>NewConstant(value)</code></td>
<td>Constant value wrapped as a variable.</td>
</tr>
<tr>
<td><code>NewIntVarFromDomain(domain, name)</code></td>
<td>Integer variable from a custom domain.</td>
</tr>
<tr>
<td><code>NewIntervalVar(start, size, end, name)</code></td>
<td>Fixed interval variable.</td>
</tr>
<tr>
<td><code>NewOptionalIntervalVar(start, size, end, is_present, name)</code></td>
<td>Optional interval (active via <code>BoolVar</code>).</td>
</tr>
</tbody>
</table>
<hr>
<h2>🧮 Constraints</h2>
<table>
<thead>
<tr>
<th>Method</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>Add(expr)</code></td>
<td>Adds a constraint to the model.</td>
</tr>
<tr>
<td><code>OnlyEnforceIf(bool_var)</code></td>
<td>Makes a constraint conditional.</td>
</tr>
</tbody>
</table>
<h3>🔁 Boolean Logic</h3>
<table>
<thead>
<tr>
<th>Method</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>AddBoolAnd(vars)</code></td>
<td>All vars must be true.</td>
</tr>
<tr>
<td><code>AddBoolOr(vars)</code></td>
<td>At least one var must be true.</td>
</tr>
<tr>
<td><code>AddBoolXOr(vars)</code></td>
<td>Odd number of true vars.</td>
</tr>
<tr>
<td><code>AddImplication(a, b)</code></td>
<td>Adds <code>a ⇒ b</code>.</td>
</tr>
<tr>
<td><code>Negated(var)</code></td>
<td>Returns logical NOT of <code>var</code>.</td>
</tr>
</tbody>
</table>
<h3>➗ Arithmetic</h3>
<table>
<thead>
<tr>
<th>Method</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>AddAbsEquality(target, var)</code></td>
<td>`target =</td>
</tr>
<tr>
<td><code>AddDivisionEquality(target, num, denom)</code></td>
<td><code>target = num / denom</code>.</td>
</tr>
<tr>
<td><code>AddModuloEquality(target, var, mod)</code></td>
<td><code>target = var % mod</code>.</td>
</tr>
<tr>
<td><code>AddMultiplicationEquality(target, vars)</code></td>
<td>Product of vars equals target.</td>
</tr>
<tr>
<td><code>AddLinearConstraint(expr, lb, ub)</code></td>
<td>Linear constraint <code>lb ≤ expr ≤ ub</code>.</td>
</tr>
<tr>
<td><code>AddLinearExpressionInDomain(expr, domain)</code></td>
<td>Constrain expression to domain.</td>
</tr>
</tbody>
</table>
<h3>🧠 Other Constraints</h3>
<table>
<thead>
<tr>
<th>Method</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>AddAllDifferent(vars)</code></td>
<td>All values must be different.</td>
</tr>
<tr>
<td><code>AddAllowedAssignments(vars, tuples)</code></td>
<td>Allow only specific combinations.</td>
</tr>
<tr>
<td><code>AddForbiddenAssignments(vars, tuples)</code></td>
<td>Forbid specific combinations.</td>
</tr>
<tr>
<td><code>AddCircuit(vars)</code></td>
<td>Hamiltonian circuit constraint.</td>
</tr>
<tr>
<td><code>AddAutomaton(vars, initial, states, transitions, final)</code></td>
<td>Finite automaton constraint.</td>
</tr>
<tr>
<td><code>AddElement(index, array, target)</code></td>
<td>Lookup: <code>target = array[index]</code>.</td>
</tr>
<tr>
<td><code>AddHint(var, value)</code></td>
<td>Provide a starting value to guide the solver.</td>
</tr>
<tr>
<td><code>AddDecisionStrategy(vars, var_strategy, value_strategy)</code></td>
<td>Add heuristic for branching.</td>
</tr>
<tr>
<td><code>AddInverse(fwd, rev)</code></td>
<td>Bi-directional mapping: <code>fwd[i] = j ⇔ rev[j] = i</code>.</td>
</tr>
<tr>
<td><code>AddMapDomain(var, bools, offset)</code></td>
<td><code>var == i + offset ⇔ bools[i] == true</code>.</td>
</tr>
<tr>
<td><code>AddMaxEquality(target, vars)</code></td>
<td><code>target = max(vars)</code>.</td>
</tr>
<tr>
<td><code>AddMinEquality(target, vars)</code></td>
<td><code>target = min(vars)</code>.</td>
</tr>
</tbody>
</table>
<hr>
<h2>📦 Scheduling Constraints</h2>
<table>
<thead>
<tr>
<th>Method</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>AddNoOverlap(intervals)</code></td>
<td>Intervals must not overlap.</td>
</tr>
<tr>
<td><code>AddNoOverlap2D(x_intervals, y_intervals)</code></td>
<td>2D non-overlapping constraint.</td>
</tr>
<tr>
<td><code>AddCumulative(intervals, demands, capacity)</code></td>
<td>Resource usage constraint.</td>
</tr>
<tr>
<td><code>AddReservoirConstraint(times, deltas, min_level, max_level)</code></td>
<td>Reservoir level tracking.</td>
</tr>
<tr>
<td><code>AddReservoirConstraintWithActive(times, deltas, actives, min_level, max_level)</code></td>
<td>Conditional reservoir.</td>
</tr>
</tbody>
</table>
<hr>
<h2>🎯 Objective Functions</h2>
<table>
<thead>
<tr>
<th>Method</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>Minimize(expr)</code></td>
<td>Minimize a linear expression.</td>
</tr>
<tr>
<td><code>Maximize(expr)</code></td>
<td>Maximize a linear expression.</td>
</tr>
<tr>
<td><code>HasObjective()</code></td>
<td>Check if model has an objective.</td>
</tr>
</tbody>
</table>
<hr>
<h2>📊 Model Tools &amp; Validation</h2>
<table>
<thead>
<tr>
<th>Method</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>ModelStats()</code></td>
<td>Get stats: variables, constraints, search space.</td>
</tr>
<tr>
<td><code>Validate()</code></td>
<td>Validates the model.</td>
</tr>
<tr>
<td><code>AssertIsBooleanVariable(var)</code></td>
<td>Raises error if <code>var</code> is not boolean.</td>
</tr>
</tbody>
</table>
<hr>
<h2>⚙️ Internal Utilities</h2>
<table>
<thead>
<tr>
<th>Method</th>
<th>Description</th>
</tr>
</thead>
<tbody>
<tr>
<td><code>GetOrMakeBooleanIndex(var)</code></td>
<td>Get/create internal boolean index.</td>
</tr>
<tr>
<td><code>GetOrMakeIndex(var)</code></td>
<td>Get/create index for a variable.</td>
</tr>
<tr>
<td><code>GetOrMakeIndexFromConstant(val)</code></td>
<td>Get index for a constant.</td>
</tr>
<tr>
<td><code>GetIntervalIndex(interval)</code></td>
<td>Get proto index for interval.</td>
</tr>
<tr>
<td><code>VarIndexToVarProto(index)</code></td>
<td>Return variable proto for index.</td>
</tr>
<tr>
<td><code>Proto()</code></td>
<td>Get full model proto (low-level).</td>
</tr>
<tr>
<td><code>Index()</code></td>
<td>Get index of a constraint.</td>
</tr>
</tbody>
</table>
<hr>
<h2>🧠 Constraint Reification Pattern</h2>
<pre><code class="language-python">b1 = model.NewBoolVar(<span class="hljs-string">&#x27;x_gt_0&#x27;</span>)  <span class="hljs-comment"># Boolean flag for x &gt; 0</span>

<span class="hljs-comment"># Enforce: b1 ⇔ (x &gt; 0)</span>
model.Add(x &gt; <span class="hljs-number">0</span>).OnlyEnforceIf(b1)
model.Add(x &lt;= <span class="hljs-number">0</span>).OnlyEnforceIf(b1.Not())
</code></pre>

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Optimization-Tools CP-SAT(Constraint Programming - Satisfiability) Cheat Sheet