An Improved Approximation Algorithm for the Maximum Weight Independent Set Problem in d-Claw Free Graphs · arXivDesk
2106.03545Jun 7, 2021full version of the paper "An Improved Approximation Algorithm for the Maximum Weight Independent Set Problem in d-Claw Free Graphs" published in the proceedings of STACS 2021, 30 pages, 4 figures
An Improved Approximation Algorithm for the Maximum Weight Independent Set Problem in d-Claw Free Graphs
In this paper, we consider the task of computing an independent set of maximum weight in a given d-claw free graph G=(V,E) equipped with a positive weight function w:V→R+
Nearby in the stack
. In doing so,
d≥2
is considered a constant. The previously best known approximation algorithm for this problem is the local improvement algorithm SquareImp proposed by Berman. It achieves a performance ratio of
2d+ε
in time
O(∣V(G)∣d+1⋅(∣V(G)∣+∣E(G)∣)⋅(d−1)2⋅(2εd+1)2)
for any
ε>0
, which has remained unimproved for the last twenty years. By considering a broader class of local improvements, we obtain an approximation ratio of
2d−63,700,9921+ε
for any
ε>0
at the cost of an additional factor of
O(∣V(G)∣(d−1)2)
in the running time. In particular, our result implies a polynomial time
2d
-approximation algorithm. Furthermore, the well-known reduction from the weighted
k
-Set Packing Problem to the Maximum Weight Independent Set Problem in
k+1
-claw free graphs provides a
2k+1−63,700,9921+ε
-approximation algorithm for the weighted
k
-Set Packing Problem for any
ε>0
. This improves on the previously best known approximation guarantee of