  In this paper, we reveal a relation between joint winner property (JWP) in
the field of valued constraint satisfaction problems (VCSPs) and
M${}^\natural$-convexity in the field of discrete convex analysis (DCA). We
introduce the M${}^\natural$-convex completion problem, and show that a
function $f$ satisfying the JWP is Z-free if and only if a certain function
$\overline{f}$ associated with $f$ is M${}^\natural$-convex completable. This
means that if a function is Z-free, then the function can be minimized in
polynomial time via M${}^\natural$-convex intersection algorithms. Furthermore
we propose a new algorithm for Z-free function minimization, which is faster
than previous algorithms for some parameter values.
