Complexity of the General Chromatic Art Gallery Problem

In the original Art Gallery Problem (AGP), one seeks the minimum number of guards required to cover a polygon $P$. We consider the Chromatic AGP (CAGP), where the guards are colored. As long as $P$ is completely covered, the number of guards does not matter, but guards with overlapping visibility regions must have different colors. This problem has applications in landmark-based mobile robot navigation: Guards are landmarks, which have to be distinguishable (hence the colors), and are used to encode motion primitives, \eg, "move towards the red landmark". Let $\chi_G(P)$, the chromatic number of $P$, denote the minimum number of colors required to color any guard cover of $P$. We show that determining, whether $\chi_G(P) \leq k$ is \NP-hard for all $k \geq 2$. Keeping the number of colors minimal is of great interest for robot navigation, because less types of landmarks lead to cheaper and more reliable recognition

Reallocation Problems in Scheduling

In traditional on-line problems, such as scheduling, requests arrive over time, demanding available resources. As each request arrives, some resources may have to be irrevocably committed to servicing that request. In many situations, however, it may be possible or even necessary to reallocate previously allocated resources in order to satisfy a new request. This reallocation has a cost. This paper shows how to service the requests while minimizing the reallocation cost. We focus on the classic problem of scheduling jobs on a multiprocessor system. Each unit-size job has a time window in which it can be executed. Jobs are dynamically added and removed from the system. We provide an algorithm that maintains a valid schedule, as long as a sufficiently feasible schedule exists. The algorithm reschedules only a total number of O(min{log^* n, log^* Delta}) jobs for each job that is inserted or deleted from the system, where n is the number of active jobs and Delta is the size of the largest window.Comment: 9 oages, 1 table; extended abstract version to appear in SPAA 201