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**11 - 12**of**12**### 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.

### On the economic placement of monitors . . .

"... Network monitoring systems are very important components for protecting networks. Due to economical and technical constraints, it is necessary to provide a proper monitor placement strategy. In this paper, we discuss how to place monitors for a given router level topology to maximize the observati ..."

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Network monitoring systems are very important components for protecting networks. Due to economical and technical constraints, it is necessary to provide a proper monitor placement strategy. In this paper, we discuss how to place monitors for a given router level topology to maximize the observation of attack events. We set up a network model including routing strategies and a threat model for general network topology and then define the monitor placement problem. We give a simple proof to show that our problem is NP-complete and provide heuristic solutions with experimental results. Due to routing asymmetry, upstream traffic and downstream traffic often traverse different paths. We also extend the monitor placement problem to the asymmetric routing scenario and discuss how to optimize monitor placement to observe bi-directional data.