Lexicographically optimal integer points: structural properties and complexity.

We study lexicographically maximal and minimal, under different permutations, integer points in a compact convex set $X$. First, we give a necessary and sufficient condition for a knapsack polytope to be equivalent to a lex-ordered set. Then we show that if $X\subseteq[0,1]^{n}$ and $X\cap\{0,1\}^{n}$ is a independence system, then $X\cap\{0,1\}^{n}$ is equal to the intersection of all the lex-ordered sets corresponding to $X$. This implies that facet-defining inequalities can be obtained for the convex hull of $X\cap\{0,1\}^{n}$ by studying the combinatorial interplay between different lex sets. We further show that for any $X\subseteq[0,1]^{n}$ and $c\ge\mathbf{0}$, there is some lex optimal point that maximizes $c^{\top}x$ over $X\cap\mathbb{Z}^{n}$. For general discrete sets, we argue that lex optimal points yield a arbitrarily tight approximation factor of $1/n$ to the integer optimum. In the second half, we consider the question of how hard it is to compute lex optimal points. A finitely convergent cutting plane algorithm is given. Then we show that lex optimization is $\operatorname{NP-hard}$ even if integer feasibility of $X$ with respect to the trivial box is in $\mathrm{P}$, and on the contrary, if the integer feasibility of $X$ is in $\mathrm{P}$ for any box, then lex optimization is in $\mathrm{P}$. If $X$ is defined using arbitrarily many lex orders, then the complexity remains $\operatorname{NP-hard}$. Finally, we propose the question of finding the largest power of 2 such that the binary encoding of some integer larger than this power satisfies arbitrarily many lex orders. This problem is shown to be $\operatorname{NP-hard}$ to approximate, assuming $\mathrm{P}\neq\mathrm{NP}$, within a factor of 2.402, or more generally, within $(\sqrt{n+1}-1)^{1-\epsilon}$ for any $\epsilon>0$.