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 social choice theory-and for p-OSCM-a problem in graph drawing. These algorithms run inwhere 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. p k 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 D MOE1).

Diversity in Kemeny Rank Aggregation: A Parameterized Approach

“…, π r } of r linear orders on C such that the Kemeny score for each order π i is at distance at most δ of the optimum, and we find that KT-Div(R) ≥ d and that scatteredness is at least s. k) ). In contrast, for the more general problem of PCO, only algorithms with running time O * (k √ k ) were known before, where k is the cost parameter [Fernau et al, 2014]. Here, we prove that PCO also admits algorithms of the form O * (2 O( √ k) ), by making use of several structural insights for cocomparability graphs.…”

“…Given: A partial order ρ ⊆ C × C over a set C, a cost function c : C × C → N, and some k ∈ N. Output: Is there a linear order τ ⊇ ρ with c(τ \ ρ) = (x,y)∈τ \ρ c(x, y) ≤ k? Intuitively, given a partial order ρ and a cost function c, the goal is to find a linear extension of ρ incurring a cost of at most k. The only difference between CO and the original PCO problem introduced in [Dujmovic et al, 2003;Fernau, 2005] is that, in the latter, for every pair (x, y) ∈ C × C such that x and y are incomparable in ρ, the cost of (x, y) is strictly positive (c(x, y) > 0) whereas in CO, the cost can be zero (c(x, y) = 0).…”

“…For certain problems of practical relevance, this framework may not be satisfactory because it precludes the user from the possibility of choosing among optimal solutions in case more than one exists, or from choosing a slightly less optimal solution that may be a better fit for the intended application, due to subjective factors. A recent upcoming trend of research in artificial intelligence, called diversity of solutions [Petit and Trapp, 2019;Baste et al, 2020;Ingmar et al, 2020;Baste et al, 2019;Fomin et al, 2020], has focused on the development of notions of optimality that may be more appropriate in settings where subjectivity is essential. The idea is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another.…”

Parameterized Complexity of Multi-winner Determination: More Effort Towards Fixed-Parameter Tractability

“…A remarkable advantage of using the reduction scheme is that we can immediately obtain and continually update our results from the state-of-the-art of these graph/set problems. In fact, this scheme has been advocated by several researchers (see, e.g., [25,31]).…”

Problems on Finite Automata and the Exponential Time Hypothesis

“…However, as some NP-hard problems involving cycles in directed graphs admit subexponential-time algorithms, see [24,25], our lower bound could be even matched. Nonetheless, this stays an open question.…”