Efficient Algorithms for Lipschitz Selections of Set-Valued Mappings in ${\bf R}^2$: long version

Let $F$ be a set-valued mapping from an $N$-element metric space $({\mathcal M},\rho)$ into the family of all closed half-planes in ${\bf R}^2$. In this paper, we provide an efficient algorithm for a Lipschitz selection of $F$, i.e., a Lipschitz mapping $f:{\mathcal M}\to{\bf R}^2$ such that $f(x)\in F(x)$ for all $x\in{\mathcal M}$. Given a constant $\lambda>0$, this algorithm produces the following two outcomes: (1) The algorithm guarantees that there is no Lipschitz selection of $F$ with Lipschitz constant at most $\lambda$; (2) The algorithm returns a Lipschitz selection of $F$ with Lipschitz constant at most $3\lambda$. The total work and storage required by this selection algorithm are at most $CN^2$ where $C$ is an absolute constant.