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Disparate impact doctrine offers an important legal apparatus for targeting unfair data-driven algorithmic decisions. A recent body of work has focused on conceptualizing and operationalizing one particular construct from this doctrine -- the less discriminatory alternative, an alternative policy that reduces disparities while meeting the same business needs of a status quo or baseline policy. This paper puts forward four fundamental results, which each represent limits to searching for and using less discriminatory algorithms (LDAs). (1) Statistically, although LDAs are almost always identifiable in retrospect on fixed populations, making conclusions about how alternative classifiers perform on an unobserved distribution is more difficult. (2) Mathematically, a classifier can only exhibit certain combinations of accuracy and selection rate disparity between groups, given the size of each group and the base rate of the property or outcome of interest in each group. (3) Computationally, a search for a lower-disparity classifier at some baseline level of utility is NP-hard. (4) From a modeling and consumer welfare perspective, defining an LDA only in terms of business needs can lead to LDAs that leave consumers strictly worse off, including members of the disadvantaged group. These findings, which may seem on their face to give firms strong defenses against discrimination claims, only tell part of the story. For each of our negative results limiting what is attainable in this setting, we offer positive results demonstrating that there exist effective and low-cost strategies that are remarkably effective at identifying viable lower-disparity policies.