 
Summary: Complexity of the Network Median Problem
on Grids
A. A. Ageev
Abstract
The ngrid is dened to be the adjacency graph of the n n
integer lattice. A graph is called a planar grid if it is isomorphic
to a subgraph of the ngrid. The paper shows that the network
median problem (an analog of the wellknown kmedian problem)
remains NPhard when restricted to the set of ngrids.
1 Introduction
Let P() be an optimization or decision problem whose instance includes
a graph taken from a prescribed set of undirected graphs . Choosing a
subset 0 and considering the problem P() on this subset we obtain
a subproblem P( 0 ). Even though P() is NPhard (NPcomplete), the
subproblem P( 0 ) may turn out to be polynomially solvable for some
0 . The question of complete characterization of such sets of graphs is
of greatest importance both for theoretical developments and practical
applications. In its general setting this question appears to be very dif
cult and remains unresolved for any concrete problem. However, the
situation may become much easier if we impose some restrictions on the
