Questions on optimization constrained to integer variables.
An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers.
Integer problems may be defined as the problem of maximizing or minimizing a linear function subject to both linear and integer constraints. The constraints may be equalities or inequalities.
Integer programs are problems that can be expressed in canonical form as
$$\max\quad c^\top x$$ $$\text{s.t.}\quad Ax\le b$$ $$x\ge0$$ $$x\in\Bbb Z^n$$
where $x$ represents the vector of variables (to be determined), $c$ and $b$ are vectors of (known) coefficients, $A$ is a (known) matrix of coefficients, $(⋅)^⊤$ is the matrix transpose, and $\Bbb Z^n$ is the set of whole numbers of dimension $n$.
The expression to be maximized or minimized is called the objective function ($c^⊤x$ in this case).
The inequalities $Ax \le b$ and $x \ge 0$ are the constraints which specify a convex polytope over which the objective function is to be optimized. The inequality $x \ge 0$ is called non-negativity constraints and are often found in linear programming problems. The $x\in\Bbb Z^n$ constraint limits the to be determined vector variables $x$ to be whole integers. The other inequality $Ax \le b$ is called the main constraints.
Integer programming is NP-hard. A special case, $0-1$ integer linear programming, in which unknowns are binary, and only the restrictions must be satisfied, is one of Karp's $21$ NP-complete problems.
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