When Centerbridge Partners, a New York–based private equity and distressed-debt investor, gained the right this spring to sponsor the reorganization of bankrupt Extended Stay Hotels, it was more than just a corporate coup. For Centerbridge managing directors Vivek Melwani and William Rahm, it was a personal triumph. For close to two years, Melwani, 38, at one time a bankruptcy lawyer, and Rahm, 31, formerly a private equity specialist for Blackstone Group, had been painstakingly combing through the convoluted debt of the 684-hotel, Spartanburg, South Carolina, chain.