The Federal Communications Commission's (FCC's) ongoing Incentive Auction will, if successful, transfer billions of dollars of radio spectrum from television broadcasters to mobile-network operators. Hundreds of broadcasters may go off the air. Most of those remaining on the air, including hundreds of Canadian broadcasters not bidding, will have to move to new channels to continue broadcasting. The auction can only end if all these broadcasters will fit into the spectrum remaining for television. Whether a given set of broadcasters fits is the broadcaster-repacking problem. The FCC must calculate its solutions thousands of times per round of bidding. Speed is essential. By reducing the broadcaster-repacking problem to the maximum independent set problem, we show that the former is $\mathcal{NP}$-complete. This reduction also allows us to expand on sparsity-exploiting heuristics in the literature, which have made the FCC's repacking-problem instances tractable. We conclude by relating the heuristics to satisfiability and integer programming reductions. These provide a basis for implementing algorithms in off-the-shelf software to solve the broadcaster-repacking problem.