Social choice meets graph drawing: How to get subexponential time algorithms for ranking and drawing problems

Abstract

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 V social choice theory-and for p-OSCM-a problem in graph drawing. These algorithms run in time O*(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(√ k log k)</sup> ), where k is the parameter, and significantly improve the previous best algorithms with running times O*(1.403 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> ) and O*(1.4656 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</sup> ), 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 oddthe V above guarantee version can still be solved in time O*(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O(√ k log k)</sup> ), 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]).