The fractional independence number (Willis 2011), denoted (Shannon 1956, Acín et al. 2016) or
(Willis 2011), also called the fractional packing number (Shannon 1956, Acín
et al. 2016) or Rosenfeld number (Acín et al. 2016), is a graph
parameter defined by relaxing the weight condition in the computation of the independence
number from allowing only weights 0 and 1 to any real numbers in the interval
.

In other words, the fractional independence number of a graph with vertex set and edge set

(1)

where
is the weight on the th vertex. This is a linear program that can be solved efficiently.
Furthermore, a maximum weighting can always be obtained using the weights (Nemhauser 1975, Willis 2011), meaning that the fractional
independence number must be an integer or half-integer.

For a graph on nodes, the fractional independence number satisfies

Acín, A.; Duan, R.; Roberson, D. E.; Belén Sainz, A.; and Winter, A. "a New Property of the Lovász Number and Duality
Relations Between Graph Parameters." 5 Feb 2016. https://arxiv.org/abs/1505.01265.Nemhauser,
G. L. and Trotter, L. E. Jr. "Vertex Packings: Structural Properties
and Algorithms." Math. Programming8, 232-248, 1975.Shannon,
C. E. "The Zero-Error Capacity of a Noisy Channel." IRE Trans.
Inform. Th.2, 8-19, 1956.Willis, W. "Bounds for the
Independence Number of a Graph." Masters thesis. Richmond, VA: Virginia Commonwealth
University, 2011.