Approximation algorithms for solving the 1-line Euclidean minimum Steiner tree problem

In this paper, we consider the 1-line Euclidean minimum Steiner tree problem, which is a variation of the Euclidean minimum Steiner tree problem and defined as follows. Given a set $$P=\{r_1,r_2,\ldots , r_n\}$$ P = { r 1 , r 2 , … , r n } of n points in the Euclidean plane $$\mathbb {R}^2$$ R 2 , we are asked to find the location of a line l and an Euclidean Steiner tree T ( l ) in $$\mathbb {R}^2$$ R 2 such that at least one Steiner point is located at such a line l , the objective is to minimize total weight of such an Euclidean Steiner tree T ( l ), i.e. , $$\min \{\sum _{e\in T(l)} w(e)~|~T(l)$$ min { ∑ e ∈ T ( l ) w ( e ) | T ( l ) is an Euclidean Steiner tree as mentioned-above $$\}$$ } , where we define weight $$w(e)=0$$ w ( e ) = 0 if the end-points u , v of each edge $$e=uv \in T(l)$$ e = u v ∈ T ( l ) are both located at such a line l and otherwise we denote weight w ( e ) to be the Euclidean distance between  u  and  v . Given a fixed line l as an input in $$\mathbb {R}^2$$ R 2 , we refer this problem as the 1-line-fixed Euclidean minimum Steiner tree problem; In addition, when Steiner points added are all located at such a fixed line l , we refer this problem as the constrained Euclidean minimum Steiner tree problem. We obtain the following two main results. (1) Using a polynomial-time exact algorithm to find a constrained Euclidean minimum Steiner tree, we can design a 1.214-approximation algorithm to solve the 1-line-fixed Euclidean minimum Steiner tree problem, and this algorithm runs in time $$O(n\log n)$$ O ( n log n ) ; (2) Using a combination of the algorithm designed in (1) for many times, a technique of finding linear facility location and an important lemma proved by some techniques of computational geometry, we can provide a 1.214-approximation algorithm to solve the 1-line Euclidean minimum Steiner tree problem, and this new algorithm runs in time $$O(n^3\log n)$$ O ( n 3 log n ) .