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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∗ {E~ [TimeAlg(~x + ~)]} 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 ) ∈ 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 ~π∈Πm 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 {Mm = (Θm, L(Am), Πm): m ≥ 3}, 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. Proof techniques developed for these theorems in our model in Proposition 1, which states that the always- may be useful in future work. exponential-time brute force search algorithm for Kemeny There is a large body of literature on the computa- and Slater is smoothed poly-time (in the sense of (Blaser¨ and tional complexity of Kemeny. The corresponding opti- Manthey 2015), see Definition 6) w.r.t. a large class of mod- mization problem, KEMENY RANKING, was proved to NP els including the model proposed by Baumeister, Hogrebe, be NP-hard (Bartholdi, Tovey, and Trick 1989) and Pk - and Rothe (2020). complete (Hemaspaandra, Spakowski, and Vogel 2005). Consequently, we conduct the smoothed analysis whose Approximation algorithms (Ailon, Charikar, and Newman expected runtime as defined in (1) is in accordance with the 2008; van Zuylen and Williamson 2007), PTAS (Kenyon- classical smoothed complexity analysis (Spielman and Teng Mathieu and Schudy 2007), and fixed-parameter efficient al- 2009), and prove smoothed hardness results for Kemeny gorithms (Betzler et al. 2009; Karpinski and Schudy 2010; (Theorem 1) and Slater (Theorem 2), respectively. Both the- Cornaz, Galand, and Spanjaard 2013) for KEMENY RANK- orems state that for a large class of models, if a smoothed ING have been developed. Practical algorithms have been proposed (Davenport and Kalagnanam 2004; Conitzer, Dav- Θm. Mm is neutral if for any θ ∈ Θm and any permu- 0 enport, and Kalagnanam 2006) and Ali and Meila (2012) tation σ over Am, there exists θ ∈ Θm such that for all compared the performance of 104 algorithms. Conitzer V ∈ L(Am), we have πθ(V ) = πθ0 (σ(V )). Mm is P- (2006) proved that the decision variant of Slater is NP-hard samplable if there exists a poly-time sampling algorithm for and proposed an efficient heuristic algorithm for computing each distribution in Πm. SLATER RANKING. Technically Mm is completely determined by Πm. Fol- There is a large body of literature on smoothed com- lowing the convention in statistics, we still keep the parame- plexity of algorithms (Spielman and Teng 2009). Blaser¨ ter space Θm and sample space L(Am) in the definition. For and Manthey (2015) established a complexity theory for example, let us recall the definition of single-agent Mallows smoothed complexity analysis by defining the counterparts model as follows. to P and NP, called SMOOTHED-P and DIST-NP re- para Definition 3. In a single-agent Mallows model M , spectively, together with a smoothed reduction and com- Ma,m Θ = L(A ) × (0, 1], where in each (R, ϕ) ∈ Θ , R plete problems. Their definitions are closely related to m m m is called the central ranking and ϕ is called the disper- the average-case complexity theory established by Levin sion parameter. For any W ∈ L(Am), we have π(R,ϕ) = (1986). However, as we will show in Section 3, it may not Qm i KT(R,W ) i=2(1−ϕ ) be suitable for computational social choice. Our Theorem 1 ϕ /Zϕ, where Zϕ = (1−ϕ)m−1 is the normaliza- and 2 illustrate the hardness of KEMENY RANKING and [ϕ,ϕ] tion constant. For any 0 < ϕ ≤ ϕ ≤ 1, we let M SLATER RANKING using the same pattern in (Huang and Ma,m Teng 2007), which states that the existence of a smoothed denote the sub-model whose parameter space is L(Am) × poly-time algorithm would lead to a surprise in complexity [ϕ, ϕ]. theory. [ϕ,ϕ] It is not hard to verify that MMa,m is neutral and P- samplable (Doignon, Pekec,ˇ and Regenwetter 2004). See 2 Preliminaries Appendix ?? for another class of neutral and P-samplable 1 Basics of Voting. For any m ≥ 3, let Am = {a1, . . . , am} models based on the Plackett-Luce model. denote the set of m alternatives. A (preference) profile P ∈ When there are n ≥ 2 agents, the adversary chooses n ~π = (π , . . . , π ) ∈ Πn j L(Am) is a collection of n rankings (linear orders). Let 1 n m, and then agent ’s ranking will WMG(P ) denote the weighted majority graph of P , which be independently (but not necessarily identically) generated π is a directed weighted graph whose vertices are Am and for from j. each pair of alternatives a, b, the weight on edge a → b, de- Example 2. Suppose m = 3 and n = 2, and the model is [0.3,1] noted by wP (a, b), is the winning margin of a over b in their MMa,3 . Then, the adversary can set the first (respectively, pairwise competition. That is, wP (a, b) = −wP (b, a) = second) agent’s distribution to be the Mallows distribution #{R ∈ P : a b} − #{R ∈ P : b a}. Let UMG(P ) R R given ground truth (a1 a2 a3, 0.4) (respectively, (a3 denote the unweighted majority graph of P , which is the a2 a1, 0.8)). Then, the probability of generating profile unweighted directed graph obtained from WMG(P ) by re- (a2 a1 a3, a1 a3 a2) is Pr(a2 a1 a3|(a1 moving edges whose weights are ≤ 0. a2 a3, 0.4)) × Pr(a1 a3 a2|(a3 a2 a1, 0.8)) = 2 The Kendall’s Tau distance between two linear orders 0.4 × 0.8 ≈ 0.026. R,W ∈ L(A), denoted by KT(R,W ), is the number of Z0.4 Z0.8 As another example, the Mallows-based model proposed pairwise disagreements between R and W . Given a pro- by Baumeister, Hogrebe, and Rothe (2020) corresponds to file P and a linear order R, the Kemeny score of R in P P the single-agent Mallows model with fixed ϕ, i.e. M[ϕ,ϕ] . is W ∈P KT(R,W ) and the Slater score of R in P is Ma,m KT(R, UMG(P )), where KT is extended to measure the dis- Because KEMENY SCORE is in P when m is bounded tance between a linear order and an unweighted graph in the above by a constant, the smoothed complexity analy- natural way—any pair of alternatives {a, b} such that a b sis ought to be done for variable m. Therefore, fol- in the linear order but b → a in the graph contribute one to lowing (Blaser¨ and Manthey 2015), we are given a se- KT. The Kemeny rule (respectively, the Slater rule) aims at ries of single-agent preference models M~ = {Mm = selecting the linear order with the minimum Kemeny score (Θm, L(Am), Πm): m ≥ 3}. In particular, we will focus (respectively, Slater score) in P . The corresponding winner on the Mallows series, defined as follows. determination problems are defined as follows. ~ [ϕ,ϕ] Definition 4. For any 0 < ϕ ≤ ϕ ≤ 1, we let MMa = Definition 1 (KEMENY RANKING and SLATER RANKING). [ϕ,ϕ] n {M : m ≥ 3} denote the Mallows series. Given m ≥ 3, n ∈ N, and P ∈ L(Am) , in KEMENY Ma,m RANKING (respectively, SLATER RANKING), we are asked In most part of this paper (except Section 3), we focus to compute a ranking with minimum Kemeny score (respec- on the classical smoothed poly-time algorithms proposed tively, Slater score). by Spielman and Teng (2009) w.r.t. M~ .
Definition 2 (Single-agent preference model (Xia 2020)). 1We note that in this paper ϕ and ϕ are constants that do not de- A single-agent preference model for m alternatives is de- pend on m. We believe that this is a natural model in social choice, noted by Mm = (Θm, L(Am), Πm), where Πm is the set of and the case of variable ϕ and ϕ is an interesting direction for fu- distributions over L(Am) indexed by the parameter space ture work. Definition 5 (Smoothed poly-time). Given a series This is done in the following series of inequalities. ~ of single-agent preference models M = {Mm = 2 (Θ , L(A ), Π ): m ≥ 3}, an algorithm Alg is smoothed m > 2(3−ϕ)/(1−ϕ) ⇔ m log m > m( + 1) m m m 1 − ϕ poly-time, if for any m ≥ 3 and n ≥ 1, RT (Alg, m, n) as fΠm 1 2 defined in (1) is polynomial in m and n. ⇒(m + ) log m − m > (m − 1) 2 1 − ϕ Blaser¨ and Manthey’s (2015) framework. In this frame- 2 work, the smoothed runtime of an algorithm is analyzed ⇒ log m! > (m − 1) (Stirling’s formula) w.r.t. a perturbation model over the input, denoted by D = 1 − ϕ 1− 1 m−1 {D`,x,φ}, where each D`,x,φ is a distribution over data, x ⇒(m!) 2n (1 − ϕ) > 1 represents the “ground truth”, φ is the maximum probabil- ⇒nm log m(m!(1 − ϕ)m−1)n > (m!nm2)1/2 ity of any data point under D`,x,φ, and ` is the size of x. Let N`,x denote the size of support of D`,x,φ, i.e. the num- ber of data points that receive non-zero probability under D`,x,φ. Blaser¨ and Manthey (2015) defined the following Proposition 1 may appear paradoxical because it calls notion of smoothed poly-time algorithms (some technical the always-exponential-time BF (BM-smoothed) poly-time. assumptions are omitted for better presentation). Technically, this is allowed in Blaser¨ and Manthey’s (2015) Definition 6 (BM-Smoothed poly-time (Blaser¨ and Man- framework. As Blaser¨ and Manthey (2015) commented, the they 2015)). Given a perturbation model D = {D`,x,φ}, expected runtime in (2) is allowed to be exponentially an algorithm Alg is BM-smoothed poly-time if there exists large when N`,xφ is exponentially large, i.e. the perturba- > 0 such that for all D`,x,φ ∈ D, tion (which corresponds to 1/φ) is relatively small compared to the size of support of the distributions (i.e. N ). There- `,x Ey∼D`,x,φ (TimeAlg(y) ) = O(` · φ · N`,x) (2) fore, we believe that the real power of Blaser¨ and Manthey’s In words, Alg is BM-smoothed poly-time if for every dis- (2015) framework is in the analysis of scenarios where the perturbation is large compared to the size of the support of tribution D`,x,φ, the expected runtime for some constant the distributions, i.e. N`,xφ is not too large. Unfortunately, > 0 is linear in n and φ · N`,x. As commented by Blaser¨ ~ [ϕ,ϕ] and Manthey (2015), this does not mean that the expected this is not the case for the Mallows-series model MMa , runtime of the algorithm is polynomial in the input size. which we believe to be a reasonable model for social choice. While this formulation is adopted to build a sound theory It is possible that when ϕ and ϕ are allowed to change as m for smoothed complexity analysis, we feel that its relevance increases, BF is no longer a smoothed-poly-time algorithm in social choice is not clear, as discussed in the next section. in Blaser¨ and Manthey’s (2015) framework. 3 Brute Force Search is BM-Smoothed 4 Smoothed Hardness of Kemeny and Slater Poly-Time for Kemeny and Slater In this section, we follow the classical smoothed runtime Let BF denote the brute force search algorithm that first analysis (Definition 5) to analyze KEMENY RANKING and computes the Kemeny scores (respectively, Slater scores) SLATER RANKING. We first recall the orthogonal decom- for all m! rankings, then chooses a ranking with the mini- position of weighted directed graphs (Young 1974; Zwicker mum score. Note that the runtime of BF is Ω(m!) for any 2018). input. The following simple observation states that BF is A WMG Gcyc is called a cycle, if the absolute weight BM-smoothed poly-time for a large class of models. of any edge is 0 or 1, and the edges with positive weights forms a cycle. Let a ∈ A , a WMG G is called a co- Proposition 1. For any fixed 0 < ϕ ≤ ϕ < 1, when m ≥ m a cycle centered at a, if for any b ∈ Am such that b 6= a, 2(3−ϕ)/(1−ϕ), BF is BM-smoothed poly-time for KEMENY wGa (a, b) = −wGa (b, a) = 1 and all other edges have ~ [ϕ,ϕ] 0 RANKING and SLATER RANKING w.r.t. MMa , where the weight . dispersion parameter is discretized. Note that any WMG G can be viewed as a vector in m(m−1) 2 whose components are indexed by (i , i ), where ~ [ϕ,ϕ] R 1 2 Proof. We first translate MMa to the D`,x,φ notation. 1 ≤ i1 < i2 ≤ m, with value wG(ai , ai ). Given any pair n 1 2 For any m, n, and any ~π ∈ Πm, let x = ~π represent of WMGs G1 and G2, we define their dot product as the central rankings and dispersion parameters for the n n X agents. Therefore, ` = Θ(nm log m), N`,x = (m!) , and G1 · G2 = wG1 (ai1 , ai2 ) × wG2 (ai1 , ai2 ) 1 n n(m−1) 1 φ ≥ ( ) ≥ (1 − ϕ) . Let = . For any 1≤i1
Definition 7. For any alternative a, let Oa denote the fol- Θ∗ Let n = |PG |. Because the number of different types of lowing set of 2(m − 1) permutations over Am. Θ∗ 5 K 5 parameters in PG is O(m ), we have n = m − O(m ), K i i Θ∗ Θ m 5 Oa = {σ , σ ◦ ηa : 1 ≤ i ≤ m − 1} kWMG(PG ) − WMG(PG · Θ )k∞ = O(m ), and a a |PG | Θ∗ mK 5 Like OG ,Oa can be naturally applied to linear orders, kWMG(PG ) − G · Θ )k∞ = O(m ). 3c |PG | fractional profiles, distributions over L(Am), parameters in We now prove that Alg can be leveraged to provide an RP Θ Θm, and weighted majority graphs. Let Pa = Oa(θ3c). We algorithm (Algorithm 1) for EFAS. have the following claim. 0 Θ∗ Notice that sampling P from PG takes polynomial time Θ Θ ~ Claim 2. |Pa | = 2(m − 1) and Pa consists of 2(m − 1) because M is P-samplable (Assumption 1 (i)). It follows Θ types of parameters. WMG(Pa ) is a co-cycle whose cen- that Algorithm 1 is a polynomial-time randomized algo- ter is a and the absolute weight on any non-zero edge is rithm. Clearly, if (G, t) is a NO instance, then Algorithm 1 P returns NO. Therefore, to prove that Algorithm 1 is an RP 2 (d ,d )∈{a}×A\{a} wπ3c (d1, d2). 1 2 algorithm it suffices to prove that if (G, t) is a YES instance, Let P Θ = O (QΘ ). We have the following claim.. then Algorithm 1 returns YES with > 1 probability. 1 a1 G3c 2 ALGORITHM 1: Algorithm for EFAS. EFAS is NP-hard (Perrot and Pham 2015) and RP⊆NP. It follows that RP=NP. Input: An EFAS instance (G, t), Alg for KEMENY RANKING whose smoothed runtime is T . Θ∗ Compute a parameter profile PG according to (3). 0 ~ Θ∗ 0 We now prove a similar theorem for SLATER RANKING Sample a profile P from Mm given PG and run Alg on P . with an additional condition. The theorem requires one more if Alg returns R∗ within 3T time and R∗ is a solution to (G, t) then richness assumption in addition to Assumption 1, which can return YES be viewed as the co-cycle counterpart to (iii)—the new as- else sumption requires that Πm contains a distribution whose return NO WMG has a non-negligible co-cycle component. While this end assumption is quite technical, we expect it to hold for many models in social choice, as Example 4 shows. (iv) There exist constants k∗ ≥ 0 and B > 0 such that for K m any m ≥ 3, there exist πco ∈ Πm such that WMG(πco) has Let Gn = G · Θ . We first prove in the following claim |PG | B a co-cycle component G with WMG(π ) · G > ∗ . that with exponentially small probability WMG(P 0) is dif- co co co mk K+1 ferent from Gn by more than Ω(m 2 ). Theorem 2 (Smoothed Hardness of SLATER RANKING). For any series of single-agent preference models M~ that 0 K+1 Claim 5. Pr(kWMG(P ) − Gnk∞ > Ω(m 2 )) < satisfies Assumption 1 and (iv), if there exists a smoothed −Ω(m) exp . poly-time algorithm for SLATER RANKING w.r.t. M~ , then Proof. We first show that for each pairwise comparison b NP=RP. K K+1 m 2 vs. c, Pr(|wP 0 (b, c) − wG(b, c) · Θ | = Ω(m )) < Proof sketch. The high-level idea of the proof is similar to |PG | exp−Ω(m), then apply union bound to all pairwise compar- the proof of Theorem 1. The difference is that in this proof we use the TOURNAMENT FEEDBACK ARC SET (TFAS) isons. Notice that wP 0 (b, c) can be viewed as the sum of n independent (not necessarily identical) bounded random problem, which is NP-hard (Alon 2006; Conitzer 2006). variables, each of which corresponds to the pairwise com- Since it is unknown whether EULERIAN TOURNAMENT parison between b and c in a ranking—if b c in the rank- FEEDBACK ARC SET is NP-hard, in a TFAS instance (G, t) ing, then the random variable takes 1, otherwise the random it is possible that G is not Eulerian. Therefore, we need a variable takes −1. By Hoeffding’s inequality for bounded co-cycle component to construct a parameter profile whose random variables, we have: WMG is G. Moreover, the weight on the co-cycle compo- nent cannot be too small, otherwise the construction will not K+1 Pr(|wP 0 (b, c) − E(wP 0 (b, c))| > Ω(m 2 )) be polynomial. Condition (iv) is used to guarantee the exis- tence of a desirable co-cycle component as described. K+1 2 2 Ω(m 2 /n) n < exp{− } = exp{−Ω(m)} Slightly more formally, the proof proceeds in four steps. 4n Step 1 is the same as the Step 1 in the proof of Theorem 1. In Step 2, we use permutations of π , which is guaranteed by Also notice that (wP 0 (b, c)) = w Θ∗ (b, c) and co E PG Θ∗ 5 K+1 (iv), to construct a parameter profile whose WMG is a co- kWMG(PG ) − Gnk = O(m ) = O(m 2 ). cycle. The reduction from TFAS will be presented in Step Suppose (G, t) is a YES instance. That is, there exists a 3. In Step 4 we show that a smoothed poly-time algorithm ranking R0 whose KT distance to G is no more than t. Due for SLATER RANKING can be used to prove that TFAS is in to Markov’s inequality, Alg returns a Kemeny ranking R∗ RP. 2 2 ∗ The following example shows that Assumption 1 and (iv) with probability ≥ 3 . We now prove that R is a solution to (G, t) with probability 1 − exp−Ω(m). hold for a large class of Mallows-based models. 0 Note that for any ranking R, |KT(R,P )−KT(R,Gn)| = ~ [ϕ,ϕ] K+5 −Ω(m) Example 4. We show that for any ϕ 6= 1, MMa satisfies O(m 2 ) holds with probability 1 − exp , which Assumption 1 and (iv). (i) and (ii) have been discussed in follows after applying Claim 5 and the union bound Example 3. For any ϕ ∈ [ϕ, ϕ] ∩ (0, 1), we show that (iii) to all Θ(m2) pairwise comparisons. In this case R∗ and (iv) hold for π = (a · · · a , ϕ). is a solution to (G, t), because if it is not, then 1 m ∗ 0 mK K−k−2 For (iii), let G3c = a1 → a2 → a3 → a1. Mallows KT(R ,Gn) − KT(R ,Gn) > Θ = Ω(m ). We |PG | (1957) proved that under Mallows’ model, the probability 0 0 0 K+1 note that |KT(R ,P ) − KT(R ,Gn)| = O(m 2 ) × for a b only depends on m, ϕ, and the difference in K+5 the ranks of a and b in the central ranking. Therefore, for 2 2 ∗ 0 ∗ O(m ) = O(m ) and |KT(R ,P ) − KT(R ,Gn)| = 1 1+2ϕ K+5 ∗ 0 0 0 any m, WMG(π) · G3c = 2 · − 2 = O(m 2 ), which means that KT(R ,P ) − KT(R ,P ) = 1+ϕ (1+ϕ)(1+ϕ+ϕ ) 1+2ϕ2 K−k−2 K+5 Θ(m ) − 2O(m 2 ) > 0 (because K = 11 + 2k), (1+ϕ)(1+ϕ+ϕ2) = Θ(1), see Figure 1. This means that which contradicts the optimality of R∗. k = 0 and A = Θ(1). Therefore, Algorithm 1 returns YES with probability at For (iv), it is not hard to verify that k∗ = 0 and B = Θ(1) 2 −Ω(m) least 3 −exp , which proves that EFAS is in RP. Since for the co-cycle centered at a1. 5 Parameterized Typical-Case Smoothed The proof is done by directly calculating Complexity of KEMENY RANKING EW ∼θKT(R,W ) using Mallows (1957)’s closed-form formulas for probability of pairwise comparisons. Following the idea in probably polynomial smoothed com- plexity of perceptron (Blum and Dunagan 2002), in this sec- Claim 7. We have: tion we prove a similar result on the parameterized typical- 2nt2 Pr(KT(P 0) > KT(P ) + 2ϕ∗ + t) ≤ exp − case smoothed complexity of Betzler et al. (2009)’s dynamic m2(m − 1)2 programming algorithm for KEMENY RANKING, denoted by Alg , under the Mallows series M~ (Definition 4). Un- Proof. For any pair of agents 1 ≤ j1 < j2 ≤ n, let KS Ma R0 and R0 denote their rankings in P 0, respectively. We ~ j1 j2 der MMa, for convenience, sometimes we rewrite a param- 0 0 0 have KT(R ,R ) ≤ KT(Rj ,Rj ) + KT(R ,Rj ) + ~ n n j1 j2 1 2 j1 1 eter θ ∈ Θm as (P, ~ϕ), where P ∈ L(Am) is called the 0 0 n KT(R ,Rj ). Therefore, (KT(P )) = KT(P ) + central profile and ~ϕ ∈ (0, 1] is called the dispersion vec- j2 2 E (n−1) Pn KT(R0 ,R ) 2 Pn KT(R0 ,R ) tor. For any n-profile P , the average KT distance is defined j=1 j j j=1 j j n(n−1)/2 = KT(P ) + n . No- as j ≤ n (R0 ,R ) 1 X tice that for each , KT j j is a ran- KT(P ) = KT(R1,R2) dom variable in [0, m(m − 1)/2] whose mean is n(n − 1) 2 mϕj R1,R2∈P no more than min(m ϕj, 2 2 ). Let Sn = (1−ϕj ) (1−ϕj ) The high-level idea behind Theorem 3 below is as fol- Pn 0 ∗ 2 j=1 KT(Rj,Rj). We have that E(Sn) ≤ 2ϕ n. There- lows. Betzler et al. (2009) proposed AlgKS and proved that fore, for any t > 0, by Hoeffding’s inequality, we have: for any profile P , its runtime is O(exp(KT(P ))poly(mn)). Sn ∗ Sn Sn Therefore, to show that with high probability AlgKS runs in Pr( > 2ϕ + t) ≤ Pr( > E( ) + t) polynomial time, it suffices to show that with high probabil- n n n ity, KT(P 0) is upper bounded by a constant, where P 0 is a 2(nt)2 ≤ Pr(S > (S ) + nt) ≤ exp − ~ n E n 2 randomly generated profile according to MMa given some n(m(m − 1)) central profile P and dispersion vector ~ϕ. Intuitively, when 2nt2 P is sufficiently centralized, i.e. KT(P ) is upper bounded by = exp − m2(m − 1)2 a constant, and ~ϕ is not too large on average, P 0 should also be sufficiently centralized. The formal relationship between centralization of P , ~ϕ, probability for AlgKS to be in P , and The theorem follows after Claim 7. the runtime is stated in the following theorem. ∗ m2 Pn ∗ Theorem 3 (Parameterized Typical-Case Smoothed We note that ϕ ≤ n j=1 ϕj and ϕ ≤ m Pn ϕ Complexity of KEMENY RANKING). For any m ≥ 3, n j=1 (1−ϕ)2(1−ϕ2) , and the former upper bound is the n n ≥ 1, any central profile P ∈ L(Am) , any disper- average dispersion of the agents multiplied by m2. Nei- n ∗ sion vector ~ϕ ∈ (0, 1] , and any t > 0, let ϕ = ther upper bound implies the other. The former is stronger 1 P 2 mϕj j≤n min(m ϕj, 2 2 ) and d = dKT(P ) + when some ϕj is close to 1. The latter is stronger when all n (1−ϕj ) (1−ϕj ) ∗ 1 2ϕ + te. We have ϕj = O( m ). We immediately obtain the following corollary by combining the former upper bound with Theorem 3. 0 d 2 2 2 0 PrP ∼(P,~ϕ)(TimeAlgKS (P ) > O(16 (d n m log m)) 4 Corrollary 1. Given M~ Ma,m, any n = Ω(m ), and any 2nt2 < exp − parameter (P, ~ϕ) such that (1) KT(P ) = O(log m + log n), m2(m − 1)2 ~ϕ·~1 log m+log n and (2) n = O( m2 ), we have: Theorem 3 contains a parameter t to control the tradeoff 0 t 2 Pr (TimeAlg (P ) = ω(poly(mn) )) = exp(−Ω(t )) between the probability and the runtime guarantee. A larger P 0∼(P,~ϕ) KR t corresponds to a larger d, which corresponds to a higher Corollary 1 states that when n is sufficiently large, and upper bound on the runtime. Meanwhile, the probability for the average KT distance in P and the average dispersion are the runtime of AlgKS to exceed the upper bound in the theo- not too large, with high probability AlgKR solves KEMENY rem decreases exponentially in t. RANKING in polynomial time.
Proof. Betzler et al. (2009) proved that the runtime of AlgKS is O(16d¯(d¯2n2m2 log m)), where d¯ = dKT(P 0)e. There- 6 Summary and Future Work fore, it suffices to prove that with high probability, the aver- We show that the smoothed complexity framework by Blaser¨ age KT distance in P 0 is at most d. and Manthey (2015) may not be appropriate for computa- We first prove a claim on the expected KT distance be- tional social choice. We prove the smoothed hardness of Ke- tween the central ranking and randomly generated ranking. meny and Slater, and a parameterized typical-case smoothed easiness result for Kemeny. An immediate open question Claim 6. For any single-agent Mallows model and any θ = is the smoothed complexity of the Dodgson rule and the (R, ϕ), we have Young rule. Smoothed complexity of other problems in com- mϕ putational social choice is also an obvious open question KT(R,W ) ≤ min(m2ϕ, ) EW ∼θ (1 − ϕ)2(1 − ϕ2) as Baumeister, Hogrebe, and Rothe (2020) pointed out. Acknowledgments Cornaz, D.; Galand, L.; and Spanjaard, O. 2013. Kemeny elections with bounded single-peaked or single-crossing LX acknowledges support from NSF #1453542 and width. In Proceedings of IJCAI, 76–82. #1716333, ONR #N00014-171-2621, and a gift fund from Google. We thank the anonymous reviewers for their very Davenport, A.; and Kalagnanam, J. 2004. A Computational helpful comments and suggestions. Study of the Kemeny Rule for Preference Aggregation. In Proceedings of AAAI, 697–702. San Jose, CA, USA. Broader Impact Doignon, J.-P.; Pekec,ˇ A.; and Regenwetter, M. 2004. The repeated insertion model for rankings: Missing link between This paper aims to understand the smoothed complexity of two subset choice models. Psychometrika 69(1): 33–54. computing commonly-studied voting rules, which are im- portant tools for collective decision making. The results will Hemaspaandra, E.; Spakowski, H.; and Vogel, J. 2005. The be important to multi-agent systems, where voting is used complexity of Kemeny elections. Theoretical Computer Sci- to achieve consensus. Success of the research will benefit ence 349(3): 382–391. general public beyond the CS research community because Huang, L.-S.; and Teng, S.-H. 2007. On the Approximation voting is a key component of democracy. and Smoothed Complexity of Leontief Market Equilibria. In Proceedings of FAW, 96–107. References Karpinski, M.; and Schudy, W. 2010. Faster Algorithms for Ailon, N.; Charikar, M.; and Newman, A. 2008. Aggregat- Feedback Arc Set Tournament, Kemeny Rank Aggregation ing inconsistent information: Ranking and clustering. Jour- and Betweenness Tournament. In Proceedings of ISAAC, nal of the ACM 55(5): Article No. 23. 3–14. Kenyon-Mathieu, C.; and Schudy, W. 2007. How to Rank Ali, A.; and Meila, M. 2012. Experiments with Kemeny with Few Errors: A PTAS for Weighted Feedback Arc Set on ranking: What works when? Mathematical Social Sciences Tournaments. In Proceedings of STOC, 95–103. San Diego, 64(1): 28–40. California, USA. Alon, N. 2006. Ranking tournaments. SIAM Journal of Dis- Levin, L. A. 1986. Average case complete problems. SIAM crete Mathematics 20: 137–142. Journal on Computing 15(1): 285—286. Bartholdi, III, J.; Tovey, C.; and Trick, M. 1989. Voting Mallows, C. L. 1957. Non-null ranking model. Biometrika schemes for which it can be difficult to tell who won the 44(1/2): 114–130. election. Social Choice and Welfare 6: 157–165. Perrot, K.; and Pham, T. V. 2015. Feedback Arc Set Prob- Baumeister, D.; Hogrebe, T.; and Rothe, J. 2020. To- lem and NP-Hardness of Minimum Recurrent Configuration wards Reality: Smoothed Analysis in Computational Social Problem of Chip-Firing Game on Directed Graphs. In An- Choice. In Proceedings of AAMAS, 1691–1695. nals of Combinatorics, 19, 373–396. Betzler, N.; Fellows, M. R.; Guo, J.; Niedermeier, R.; and Rothe, J.; Spakowski, H.; and Vogel, J. 2003. Exact Com- Rosamond, F. A. 2009. Fixed-parameter algorithms for Ke- plexity of the Winner Problem for Young Elections. In meny rankings. Theoretical Computer Science 410: 4554– Theory of Computing Systems, volume 36(4), 375–386. 4570. Springer-Verlag. Blaser,¨ M.; and Manthey, B. 2015. Smoothed Complexity Spielman, D. A.; and Teng, S.-H. 2004. Smoothed analy- Theory. ACM Transactions on Computation Theory Article sis of algorithms: Why the simplex algorithm usually takes 6. polynomial time. Journal of the ACM 51(3). Spielman, D. A.; and Teng, S.-H. 2009. Smoothed Analy- Blum, A.; and Dunagan, J. D. 2002. Smoothed Analysis sis: An Attempt to Explain the Behavior of Algorithms in of the Perceptron Algorithm for Linear Programming. In Practice. Communications of the ACM 52(10): 76–84. Proceedings of SODA, 905–914. van Zuylen, A.; and Williamson, D. P. 2007. Deterministic Bogdanov, A.; and Trevisan, L. 2006. Average-Case Com- Algorithms for Rank Aggregation and Other Ranking and plexity. Foundations and Trends in Theoretical Computer Clustering Problems. In Proceedings of WOWA, 260–273. Science 2(1): 1–106. Xia, L. 2019. Learning and Decision-Making from Rank Brandt, F.; Conitzer, V.; Endriss, U.; Lang, J.; and Procac- Data. Synthesis Lectures on Artificial Intelligence and Ma- cia, A. D., eds. 2016. Handbook of Computational Social chine Learning. Morgan & Claypool Publishers. Choice. Cambridge University Press. Xia, L. 2020. The Smoothed Possibility of Social Choice. Conitzer, V. 2006. Computing Slater Rankings Using Sim- In Proceedings of NeurIPS. ilarities among Candidates. In Proceedings of AAAI, 613– Young, H. P. 1974. An axiomatization of Borda’s rule. Jour- 619. Boston, MA, USA. nal of Economic Theory 9(1): 43–52. Conitzer, V.; Davenport, A.; and Kalagnanam, J. 2006. Im- Zwicker, W. S. 2018. Cycles and Intractability in a Large proved Bounds for Computing Kemeny Rankings. In Pro- Class of Aggregation Rules. Journal of Artificial Intelli- ceedings of AAAI, 620–626. gence Research 61(1).