  Many real-world complex systems are best modeled by multiplex networks. The
multiplexity has proved to have broad impact on the system's structure and
function. Most theoretical studies on multiplex networks to date, however, have
largely ignored the effect of link overlap across layers despite strong
empirical evidences for its significance. In this article, we investigate the
effect of link overlap in the viability of multiplex networks, both
analytically and numerically. Distinctive role of overlapping links in
viability and mutual connectivity is emphasized and exploited for setting up
proper analytic framework. A rich phase diagram for viability is obtained and
greatly diversified patterns of hysteretic behavior in viability are observed
in the presence of link overlap. Mutual percolation with link overlap is
revisited as a limit of multiplex viability problem, and controversy between
existing results is clarified. The distinctive role of overlapping links is
further demonstrated by the different responses of networks under random
removals of overlapping and non-overlapping links, respectively, as well as
under several removal strategies. Our results show that the link overlap
strongly facilitates viability and mutual percolation; at the same time, the
presence of link overlap poses challenge in analytical approach to the problem.
