CP vs. MIP: When to Choose Which

How to decide between a constraint-programming solver (CP-SAT) and a mixed-integer-programming solver (MIP) for a given combinatorial optimization problem, based on objective linearity, LP-relaxation tightness, presence of continuous variables, and combinatorial structure.

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Tutorial

Introduction: Two Solvers, Two Internal Engines

We compare two solver families that both handle integer decision variables but rely on very different internal machinery.

Mixed Integer Programming (MIP) solvers use branch-and-bound driven by linear-programming relaxations. They prune the search tree by comparing the LP bound against the best known incumbent. MIP excels when the constraints and objective are linear and the LP relaxation is tight — close to the integer optimum.

Constraint Programming (CP) solvers — including CP-SAT — propagate finite-domain constraints and search by backtracking. They excel at problems with rich combinatorial structure: $\texttt{alldifferent}$ , $\texttt{no\_overlap}$ , $\texttt{cumulative}$ , $\texttt{element}$ , table constraints, and reified logical implications.

A practical decision heuristic:

Linear objective + linear constraints + meaningful LP relaxation $\Rightarrow$ MIP.
Any continuous decision variable present $\Rightarrow$ MIP (CP-SAT is integer-only).
Scheduling / sequencing / assignment dominated by $\texttt{no\_overlap}$ , precedence, or $\texttt{alldifferent}$ $\Rightarrow$ CP.
Pure feasibility puzzle with no useful relaxation (Sudoku, graph coloring) $\Rightarrow$ CP.
Many disjunctive or 'if-then' logical rules forcing big-M formulations $\Rightarrow$ CP.

Quick litmus test: drop the integrality constraints and solve the resulting LP. If its optimum is close to the integer optimum, MIP will be fast. If the LP optimum is uninformative — equal to a trivial bound — CP is usually the better bet.