Ranking and Drawing in Subexponential Time

Abstract

In this paper we obtain parameterized subexponential-time algorithms for p -Kemeny Aggregation (p-KAGG) — a problem in social choice theory — and for p -One-Sided Crossing Minimization (p-OSCM) – a problem in graph drawing (see the introduction for definitions). These algorithms run in time $\mathcal{O}^{*}(2^{\mathcal{O}(\sqrt{k}{\rm log} k)})$, where k is the parameter, and significantly improve the previous best algorithms with running times $\cal{O}^{*}$(1.403^k) and $\cal{O}^{*}$(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 $\mathcal{O}^{*}(2^{\mathcal{O}(\sqrt{k}{\rm log} k)})$, while if it is even then the problem cannot have a subexponential time algorithm unless the exponential time hypothesis fails (equivalently, unless FPT=M[1]).