Social Choice Meets Graph Drawing: How to Get Subexponential Time Algorithms for Ranking and Drawing Problems

We analyze a common feature of p-Kemeny AGGregation (p-KAGG) and p-One-Sided Crossing Minimization (p-OSCM) to provide new insights and findings of interest to both the graph drawing community and the social choice community. We obtain parameterized subexponential-time algorithms for p-KAGG — a problem in social choice theory — and for p-OSCM — a problem in graph drawing. These algorithms run intime O*(2^O(√k log k)), where k is the parameter, and significantly improve the previous best algorithms with running times O*(1.403^k) and O*(1.4656^k), respectively. We also study natural “above-guarantee” versions of these problems and show them to be fixed parameter tractable. In fact, we show that the above-guarantee versions of these problems are equivalent to a weighted variant of p-directed feedback arc set. Our results for the above-guarantee version of p-KAGG reveal an interesting contrast. We show that when the number of “votes” in the input to p-KAGG is odd the above guarantee version can still be solved in time O*(2^O(√k log k)), while if it is eventhen the problem cannot have a subexponential time algorithm unless the exponential time hypothesis fails (equivalently, unless FPT = M[1]).