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The Smoothed Complexity of Computing Kemeny and Slater Rankings PRELIMINARY VERSION: DO NOT CITE The AAAI Digital Library will contain the published version some time after the conference The Smoothed Complexity of Computing Kemeny and Slater Rankings Lirong Xia1 and Weiqiang Zheng2 1 RPI, 2 Peking University [email protected], [email protected] Abstract stances, which may itself be unrealistic. To tackle this prob- lem, Spielman and Teng (2004) introduced smoothed com- The computational complexity of winner determination un- plexity analysis to generalize and combine the worst-case der common voting rules is a classical and fundamental topic analysis and the average-case analysis. The idea is that the in the field of computational social choice. Previous work has input ~x of an algorithm Alg is often a noisy perception of the established the NP-hardness of winner determination under ∗ some commonly-studied voting rules, such as the Kemeny ground truth input ~x . Consequently, let TimeAlg(~x) denote rule and the Slater rule. In a recent position paper, Baumeis- the runtime of Alg when the input is ~x, the worst-case is an- ter, Hogrebe, and Rothe (2020) questioned the relevance of alyzed by assuming that an adversary chooses a ground truth the worst-case nature of NP-hardness in social choice and ~x∗ and then Nature adds a noise ~ (e.g. a Gaussian noise) to proposed to conduct smoothed complexity analysis (Spiel- it, so that the algorithm’s input becomes ~x = ~x∗ + ~. Then, man and Teng 2009) under Blaser¨ and Manthey’s (2015) the expected runtime is evaluated according to the noise in- framework. troduced by Nature. Formally, the smoothed runtime of Alg In this paper, we develop the first smoothed complexity re- is defined as: sults for winner determination in voting. We illustrate the ∗ inappropriateness of Blaser¨ and Manthey’s (2015) smoothed max~x∗ fE~ [TimeAlg(~x + ~)]g complexity framework in social choice contexts by pointing out that the framework categorizes an always-exponential- This generalizes the worst-case runtime ∗ time brute force search algorithm as being smoothed poly- max~x∗ [TimeAlg(~x )] and the average-case runtime ∗ time, under a natural noise model based on the well-studied E~x∗∼π[TimeAlg(~x )] under a distribution π over inputs. Mallows model in social choice and statistics. We then prove Smoothed complexity analysis has been applied to a wide the smoothed hardness of Kemeny and Slater using the clas- range of problems in mathematical programming, machine sical smoothed runtime analysis, and prove a parameterized learning, numerical analysis, discrete math, combinatorial typical-case smoothed easiness result for Kemeny. Overall, optimization, and equilibrium analysis and price of anar- our results show that smoothed complexity analysis in com- chy, see the survey by Spielman and Teng (2009). In a re- putational social choice is a challenging and fruitful topic. cent position paper, Baumeister, Hogrebe, and Rothe (2020) proposed to conduct smoothed complexity analysis in com- 1 Introduction putational social choice under Blaser¨ and Manthey’s (2015) framework and proposed a natural noise model that lever- The computational complexity of winner determination un- ages the celebrated Mallows (1957) model. However, we are der common voting rules is a classical and fundamental not aware of a technical result on the smoothed complexity topic in the field of computational social choice (Brandt of winner determination in voting. The following question et al. 2016, Section 1.2.3). A low computational complex- remains open. ity of winner determination is desirable and is indeed the case for many commonly-studied and widely-applied vot- What is the smoothed complexity of winner determination ing rules. On the other hand, winner determination has been under commonly-studied voting rules? proved to be NP-hard for some classical voting rules such As illustrated in the following example, the question is as the Kemeny rule, the Slater rule, the Dodgson rule, and highly relevant not only in the theory of computational so- the Young rule (Bartholdi, Tovey, and Trick 1989; Conitzer cial choice, but also in AI-aided group decision-making. 2006; Rothe, Spakowski, and Vogel 2003). To address the worst-case nature of NP-hardness, Example 1. Suppose the developer of an intelligent system average-case analysis was conducted to provide a more real- is planning to implement a voting rule for group decision istic analysis of algorithms. However, average-case analysis making. Kemeny is being considered, but the system needs is sensitive to the choice of the distribution over input in- to estimate the practical runtime of computing Kemeny to decide how much computational resource is sufficient for a Copyright c 2021, Association for the Advancement of Artificial timely decision. The system can learn and predict agents’ Intelligence (www.aaai.org). All rights reserved. preferences from their past behavior, but is only able to do it probabilistically—the agents’ preferences can be modeled poly-time algorithm exists, then RP=NP, which is consid- as their ground truth preferences plus some random noise. Is ered very unlikely to hold. there an algorithm for Kemeny whose expected runtime is Finally, we consider parameterized typical-case smoothed low, no matter what the “ground truth” is? 2 complexity and prove a mildly positive result in Theorem 3, Our Model. In this paper we answer the question for the which implies that for a large class of Mallows-based mod- Kemeny rule and the Slater rule, for which winner deter- els, if the average Kendall Tau’s distance in the central rank- mination means computing an optimal consensus ranking. ings and the average dispersion parameters are not too large, Successfully addressing the question requires appropriate then the dynamic programming algorithm for Kemeny pro- choices of (1) a noise model for social choice scenarios, and posed by Betzler et al. (2009) runs in poly-time with high (2) a notion of expected runtime. probability. (1) Noise model. We adopt the smoothed social choice Related Work and Discussions. We are not aware of a pre- framework (Xia 2020), which covers a wide range of vious technical result on the smoothed complexity of social models, including the Mallows-based model proposed choice problems. As discussed above, Baumeister, Hogrebe, by Baumeister, Hogrebe, and Rothe (2020). In the frame- and Rothe (2020) proposed to conduct smoothed complex- work, for any number of alternatives and any number of ity analysis in computational social choice and proposed a agents, denoted by m ≥ 3 and n ≥ 1 respectively, the ad- natural Mallows-based model for such analysis. The authors versary chooses a distribution π for each agent j from a set j also suggested that smoothed analysis can be done for an- of distributions Π over all rankings. m alyzing ties and paradoxes in social choice. As a position (2) Expected runtime. Let ~π = (π ; : : : ; π ) 2 Πn de- 1 n m paper, (Baumeister, Hogrebe, and Rothe 2020) does not con- note the vector of distributions chosen by the adversary, one tain technical results. per agent. Then, given an algorithm Alg, the adversary aims at choosing ~π to maximize the expected runtime of Alg on Xia (2020) independently proposed to conduct smoothed the profile P where each ranking is generated independently analysis in social choice, provided a general framework from ~π, formally defined as follows. for doing so, and proved dichotomous characterizations for Condorcet’s paradox and the ANR impossibility theorem to RTΠ (Alg; m; n) = sup n Time (P ) (1) vanish in the smoothed sense. Our model adopts the noise f m ~π2Πm EP ∼~π Alg model by Xia (2020) combined with formulation of the In (1), the expectation is taken over randomly-generated pro- worst average-case runtime (1) as done by Spielman and file P according to the distributions ~π set by the adver- Teng (2009). We emphasize that (Spielman and Teng 2009) sary. Because the size of the input of Alg is the size of P , used a different noise model from the one used in this paper. i.e. Θ(nm log m), we desire RT (Alg; m; n) to be poly- f Πm Our paper aims at making the first technical attempt nomial in m and n. Following the convention in statistics of smoothed complexity analysis in computational social and the notation in (Xia 2020), for any m ≥ 3, we use a choice, and overall our results show that the topic is single-agent preference model Mm = (Θm; L(Am); Πm) highly challenging and fruitful. The seemingly paradoxi- to model the adversary’s capability, where Θm is the param- cal smoothed efficiency of brute force search under Blaser¨ eter space and L(Am) is the set of all rankings over m alter- and Manthey’s (2015) framework (Proposition 1) is indeed natives. not technically surprising and is deliberately allowed, as Note that when m is a constant, many commonly-studied Blaser¨ and Manthey (2015) commented. See more techni- voting rules, including Kemeny and Slater, are easy to com- cal discussions after Proposition 1. Therefore, this result pute. Therefore, to meaningfully analyze the smoothed com- does not mean that Blaser¨ and Manthey’s (2015) theory ~ plexity, we will consider an infinite series of models M = is wrong or inconsistent, but instead, it can be viewed as fMm = (Θm; L(Am); Πm): m ≥ 3g, following the con- a call for future research in the theory of smoothed com- vention in average-case complexity theory (Bogdanov and plexity analysis and statistical models in social choice con- Trevisan 2006) and the smoothed complexity theory pro- texts. The smoothed hardness of Kemeny (Theorem 1) and posed by Blaser¨ and Manthey (2015). Slater (Theorem 2) are negative news and the parameter- Our Contributions. We first illustrate the inappropriate- ized typical-case smoothed efficiency (Theorem 3) is pos- ness of applying Blaser¨ and Manthey’s (2015) framework itive news.
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