The Robust Set Problem: Parameterized Complexity and Approximation

In this paper, we introduce the

Robust Set

problem: given a graph

G

 = (

V

,

E

), a threshold function

t

:

V

 → 

N

and an integer

k

, find a subset of vertices

V

′ ⊆ 

V

of size at least

k

such that every vertex

v

in

G

has less than

t

(

v

) neighbors in

V

′. This problem occurs in the context of the spread of undesirable agents through a network (virus, ideas, fire, …). Informally speaking, the problem asks to find the largest subset of vertices with the property that if anything bad happens in it then this will have no consequences on the remaining graph. The threshold

t

(

v

) of a vertex

v

represents its reliability regarding its neighborhood; that is, how many neighbors can be infected before

v

gets himself infected.

We study in this paper the parameterized complexity of

Robust Set

and the approximation of the associated maximization problem. When the parameter is

k

, we show that this problem is W[2]-complete in general and W[1]-complete if all thresholds are constant bounded. Moreover, we prove that, if

P

 ≠ 

NP

, the maximization version is not

n

1 − 

ε

- approximable for any

ε

 > 0 even when all thresholds are at most two. When each threshold is equal to the degree of the vertex, we show that

k

-Robust Set

is fixed-parameter tractable for parameter

k

and the maximization version is APX-complete. We give a polynomial-time algorithm for graphs of bounded treewidth and a PTAS for planar graphs. Finally, we show that the parametric dual problem (

n

 − 

k

)-

Robust Set

is fixed-parameter tractable for a large family of threshold functions.