The presence of certain elements within a star, and by extension its planet, strongly impacts the formation and evolution of the planetary system. The positive correlation between a host star's iron content and the presence of an orbiting giant exoplanet has been confirmed; however, the importance of other elements in predicting giant planet occurrence is less certain despite their central role in shaping internal planetary structure. We designed and applied a machine-learning algorithm to the Hypatia Catalog to analyze the stellar abundance patterns of known host stars to determine those elements important in identifying potential giant exoplanet host stars. We analyzed a variety of different elements ensembles-namely, volatiles, lithophiles, siderophiles, and Fe. We show that the relative abundances of oxygen, carbon, and sodium, in addition to iron, are influential indicators of the presence of a giant planet. We demonstrate the predictive power of our algorithm by analyzing stars with known giant planets and found that they had median 75% prediction score. We present a list of ~350 stars with no currently discovered planets that have a >=90% prediction probability likelihood of hosting a giant exoplanet. We investigated archival HARPS data and found significant trends that HIP 62345, HIP 71803, and HIP 10278 host long-period giant planet companions with estimated minimum M_p_sin(i) values of 3.7, 6.8, and 8.5M_J_, respectively. We anticipate that our findings will revolutionize future target selection, the role that elements play in giant planet formation, and the determination of giant planet interior structure models.