Ranking and Drawing in Subexponential Time

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)})\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}^{*}\cal{o}^{*}(1.403 k ) and \cal{o}^{*}\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)})\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]).