  We consider a multistage framework introduced recently where, given a time
horizon t=1,2,...,T, the input is a sequence of instances of a (static)
combinatorial optimization problem I_1,I_2,...,I_T, (one for each time step),
and the goal is to find a sequence of solutions S_1,S_2,...,S_T (one for each
time step) reaching a tradeoff between the quality of the solutions in each
time step and the stability/similarity of the solutions in consecutive time
steps. For several polynomial-time solvable problems, such as Minimum Cost
Perfect Matching, the multistage variant becomes hard to approximate (even for
two time steps for Minimum Cost Perfect Matching). In this paper, we study the
multistage variants of some important discrete minimization problems (including
Minimum Cut, Vertex Cover, Set Cover, Prize-Collecting Steiner Tree,
Prize-Collecting Traveling Salesman). We focus on the natural question of
whether linear-programming-based methods may help in developing good
approximation algorithms in this framework. We first show that Min Cut remains
polytime solvable in its multistage variant, and Vertex Cover remains
2-approximable, as particular case of a more general statement which easily
follows from the work of (Hochbaum, EJOR 2002) on monotone and IP2 problems.
Then, we tackle other problems and for this we introduce a new two-threshold
rounding scheme, tailored for multistage problems. As a first application, we
show that this rounding scheme gives a 2$f$-approximation algorithm for the
multistage variant of the f-Set Cover problem, where each element belongs to at
most f sets. More interestingly, we are able to use our rounding scheme in
order to propose a 3.53-approximation algorithm for the multistage variant of
the Prize-Collecting Steiner Tree problem, and a 3.034-approximation algorithm
for the multistage variant of the Prize-Collecting Traveling Salesman problem.
