The race for the Liberal Democratic Party presidency began Monday, with five candidates vying to lead a ruling party reeling from election defeats over the past year and a minority government that will be forced to compromise with the opposition in order to ensure a parliamentary majority. The five are: Yoshimasa Hayashi, 64, the chief Cabinet secretary; Takayuki Kobayashi, 50, a former economic security minister; Shinjiro Koizumi, 44, the agriculture minister; Toshimitsu Motegi, 69, former LDP secretary-general; and Sanae Takaichi, 64, a former economic security minister.