  Given a polygon and a visibility range, the Myopic Watchman Problem with
Discrete Vision (MWPDV) asks for a closed path P and a set of scan points S,
such that (i) every point of the polygon is within visibility range of a scan
point; and (ii) path length plus weighted sum of scan number along the tour is
minimized. Alternatively, the bicriteria problem (ii') aims at minimizing both
scan number and tour length. We consider both lawn mowing (in which tour and
scan points may leave P) and milling (in which tour, scan points and visibility
must stay within P) variants for the MWPDV; even for simple special cases,
these problems are NP-hard.
  We show that this problem is NP-hard, even for the special cases of
rectilinear polygons and L_\infty scan range 1, and negligible small travel
cost or negligible travel cost. For rectilinear MWPDV milling in grid polygons
we present a 2.5-approximation with unit scan range; this holds for the
bicriteria version, thus for any linear combination of travel cost and scan
cost. For grid polygons and circular unit scan range, we describe a bicriteria
4-approximation. These results serve as stepping stones for the general case of
circular scans with scan radius r and arbitrary polygons of feature size a, for
which we extend the underlying ideas to a pi(r/a}+(r+1)/2) bicriteria
approximation algorithm. Finally, we describe approximation schemes for MWPDV
lawn mowing and milling of grid polygons, for fixed ratio between scan cost and
travel cost.
