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**1 - 3**of**3**### Hitting Set, Spanning Trees, and the Minimum Length Corridor Problem

, 2008

"... 1 Introduction We consider a problem that is a combination of hitting set and minimum spanning trees that generalizes the minimum length corridor problem. In the minimum length corridor problem, we are given a rectangle aligned with the axes in the plane that is subdivided into smaller rectangles al ..."

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1 Introduction We consider a problem that is a combination of hitting set and minimum spanning trees that generalizes the minimum length corridor problem. In the minimum length corridor problem, we are given a rectangle aligned with the axes in the plane that is subdivided into smaller rectangles also aligned with the axes in the plane. This defines a planar graph whose vertices are the corners of the rectangles and whose edges are segments of the sides joining them, where the edges have edge lengths. The aim is to find a subtree of this graph that touches all the faces, including the outer face, in at least one vertex, and of minimal total edge length.

### Approximating Corridors and Tours via Restriction and Relaxation Techniques

, 2010

"... Abstract. Given a rectangular boundary partitioned into rectangles, the Minimum-Length Corridor (MLC-R) problem consists of finding a corridor of least total length. A corridor is a set of connected line segments, each of which must lie along the line segments that form the rectangular boundary and/ ..."

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Abstract. Given a rectangular boundary partitioned into rectangles, the Minimum-Length Corridor (MLC-R) problem consists of finding a corridor of least total length. A corridor is a set of connected line segments, each of which must lie along the line segments that form the rectangular boundary and/or the boundary of the rectangles, and must include at least one point from the boundary of every rectangle and from the rectangular boundary. The MLC-R problem is known to be NP-hard. We present the first polynomial-time constant ratio approximation algorithm for the MLC-R and MLCk problems. The MLCk problem is a generalization of the MLC-R problem where the rectangles are rectilinear c-gons, for c ≤ k and k is a constant. We also present the first polynomial-time constant ratio approximation algorithm for the Group Traveling Salesperson Problem (GTSP) for a rectangular boundary partitioned into rectilinear c-gons as in the MLCk problem. Our algorithms are based on the restriction and relaxation approximation techniques.