  We develop a method to identify cosmic voids from the matter density field by
adopting a physically-motivated concept that voids are the counterpart of
massive clusters. To prove the concept we use a pair of $\Lambda$CDM
simulations, a reference and its initial density-inverted mirror simulation,
and study the relation between the effective size of voids and the mass of
corresponding clusters. Galaxy cluster-scale dark matter halos are identified
in the Mirror simulation at $z=0$ by linking dark matter particles. The void
corresponding to each cluster is defined in the Reference simulation as the
region occupied by the member particles of the cluster. We study the voids
corresponding to the halos more massive than $10^{13}h^{-1}M_{\odot}$. We find
a power-law scaling relation between the void size and the corresponding
cluster mass. Voids with corresponding cluster mass above
$10^{15}h^{-1}M_{\odot}$ occupy $\sim1\%$ of the total simulated volume,
whereas this fraction increases to $\sim54\%$ for voids with corresponding
cluster mass above $10^{13}h^{-1}M_{\odot}$. It is also found that the density
profile of the identified voids follows a universal functional form. Based on
these findings, we propose a method to identify cluster-counterpart voids
directly from the matter density field without their mirror information by
utilizing three parameters such as the smoothing scale, density threshold, and
minimum core fraction. We recover voids corresponding to clusters more massive
than $3\times10^{14}h^{-1}M_{\odot}$ at 70--74 \% level of completeness and
reliability. Our results suggest that we are able to identify voids in a way to
associate them with clusters of a particular mass-scale.
