Towards explaining the speed of kk-means

The $k$-means method is a popular algorithm for clustering, known for its speed in practice. This stands in contrast to its exponential worst-case running-time. To explain the speed of the $k$-means method, a smoothed analysis has been conducted. We sketch this smoothed analysis and a generalization to Bregman divergences

Smoothed Efficient Algorithms and Reductions for Network Coordination Games

Worst-case hardness results for most equilibrium computation problems have raised the need for beyond-worst-case analysis. To this end, we study the smoothed complexity of finding pure Nash equilibria in Network Coordination Games, a PLS-complete problem in the worst case. This is a potential game where the sequential-better-response algorithm is known to converge to a pure NE, albeit in exponential time. First, we prove polynomial (resp. quasi-polynomial) smoothed complexity when the underlying game graph is a complete (resp. arbitrary) graph, and every player has constantly many strategies. We note that the complete graph case is reminiscent of perturbing all parameters, a common assumption in most known smoothed analysis results. Second, we define a notion of smoothness-preserving reduction among search problems, and obtain reductions from $2$-strategy network coordination games to local-max-cut, and from $k$-strategy games (with arbitrary $k$) to local-max-cut up to two flips. The former together with the recent result of [BCC18] gives an alternate $O(n^8)$-time smoothed algorithm for the $2$-strategy case. This notion of reduction allows for the extension of smoothed efficient algorithms from one problem to another. For the first set of results, we develop techniques to bound the probability that an (adversarial) better-response sequence makes slow improvements on the potential. Our approach combines and generalizes the local-max-cut approaches of [ER14,ABPW17] to handle the multi-strategy case: it requires a careful definition of the matrix which captures the increase in potential, a tighter union bound on adversarial sequences, and balancing it with good enough rank bounds. We believe that the approach and notions developed herein could be of interest in addressing the smoothed complexity of other potential and/or congestion games

Smoothed Complexity Theory

Smoothed analysis is a new way of analyzing algorithms introduced by Spielman and Teng (J. ACM, 2004). Classical methods like worst-case or average-case analysis have accompanying complexity classes, like P and AvgP, respectively. While worst-case or average-case analysis give us a means to talk about the running time of a particular algorithm, complexity classes allows us to talk about the inherent difficulty of problems. Smoothed analysis is a hybrid of worst-case and average-case analysis and compensates some of their drawbacks. Despite its success for the analysis of single algorithms and problems, there is no embedding of smoothed analysis into computational complexity theory, which is necessary to classify problems according to their intrinsic difficulty. We propose a framework for smoothed complexity theory, define the relevant classes, and prove some first hardness results (of bounded halting and tiling) and tractability results (binary optimization problems, graph coloring, satisfiability). Furthermore, we discuss extensions and shortcomings of our model and relate it to semi-random models.Comment: to be presented at MFCS 201

On smoothed analysis of quicksort and Hoare's find

We provide a smoothed analysis of Hoare's find algorithm, and we revisit the smoothed analysis of quicksort. Hoare's find algorithm - often called quickselect or one-sided quicksort - is an easy-to-implement algorithm for finding the k-th smallest element of a sequence. While the worst-case number of comparisons that Hoare’s find needs is Theta(n^2), the average-case number is Theta(n). We analyze what happens between these two extremes by providing a smoothed analysis. In the first perturbation model, an adversary specifies a sequence of n numbers of [0,1], and then, to each number of the sequence, we add a random number drawn independently from the interval [0,d]. We prove that Hoare's find needs Theta(n/(d+1) sqrt(n/d) + n) comparisons in expectation if the adversary may also specify the target element (even after seeing the perturbed sequence) and slightly fewer comparisons for finding the median. In the second perturbation model, each element is marked with a probability of p, and then a random permutation is applied to the marked elements. We prove that the expected number of comparisons to find the median is Omega((1−p)n/p log n). Finally, we provide lower bounds for the smoothed number of comparisons of quicksort and Hoare’s find for the median-of-three pivot rule, which usually yields faster algorithms than always selecting the first element: The pivot is the median of the first, middle, and last element of the sequence. We show that median-of-three does not yield a significant improvement over the classic rule

Clearing Financial Networks with Derivatives: From Intractability to Algorithms

Financial networks raise a significant computational challenge in identifying insolvent firms and evaluating their exposure to systemic risk. This task, known as the clearing problem, is computationally tractable when dealing with simple debt contracts. However under the presence of certain derivatives called credit default swaps (CDSes) the clearing problem is $\textsf{FIXP}$-complete. Existing techniques only show $\textsf{PPAD}$-hardness for finding an $\epsilon$-solution for the clearing problem with CDSes within an unspecified small range for $\epsilon$. We present significant progress in both facets of the clearing problem: (i) intractability of approximate solutions; (ii) algorithms and heuristics for computable solutions. Leveraging $\textsf{Pure-Circuit}$ (FOCS'22), we provide the first explicit inapproximability bound for the clearing problem involving CDSes. Our primal contribution is a reduction from $\textsf{Pure-Circuit}$ which establishes that finding approximate solutions is $\textsf{PPAD}$-hard within a range of roughly 5%. To alleviate the complexity of the clearing problem, we identify two meaningful restrictions of the class of financial networks motivated by regulations: (i) the presence of a central clearing authority; and (ii) the restriction to covered CDSes. We provide the following results: (i.) The $\textsf{PPAD}$-hardness of approximation persists when central clearing authorities are introduced; (ii.) An optimisation-based method for solving the clearing problem with central clearing authorities; (iii.) A polynomial-time algorithm when the two restrictions hold simultaneously

On the Smoothed Complexity of Combinatorial Local Search

We propose a unifying framework for smoothed analysis of combinatorial local optimization problems, and show how a diverse selection of problems within the complexity class PLS can be cast within this model. This abstraction allows us to identify key structural properties, and corresponding parameters, that determine the smoothed running time of local search dynamics. We formalize this via a black-box tool that provides concrete bounds on the expected maximum number of steps needed until local search reaches an exact local optimum. This bound is particularly strong, in the sense that it holds for any starting feasible solution, any choice of pivoting rule, and does not rely on the choice of specific noise distributions that are applied on the input, but it is parameterized by just a global upper bound ϕ on the probability density. The power of this tool can be demonstrated by instantiating it for various PLS-hard problems of interest to derive efficient smoothed running times (as a function of ϕ and the input size). Most notably, we focus on the important local optimization problem of finding pure Nash equilibria in Congestion Games, that has not been studied before from a smoothed analysis perspective. Specifically, we propose novel smoothed analysis models for general and Network Congestion Games, under various representations, including explicit, step-function, and polynomial resource latencies. We study PLS-hard instances of these problems and show that their standard local search algorithms run in polynomial smoothed time. Further applications of our framework to a wide range of additional combinatorial problems can be found in the full version of our paper

Smoothed analysis of deterministic discounted and mean-payoff games

We devise a policy-iteration algorithm for deterministic two-player discounted and mean-payoff games, that runs in polynomial time with high probability, on any input where each payoff is chosen independently from a sufficiently random distribution. This includes the case where an arbitrary set of payoffs has been perturbed by a Gaussian, showing for the first time that deterministic two-player games can be solved efficiently, in the sense of smoothed analysis. More generally, we devise a condition number for deterministic discounted and mean-payoff games, and show that our algorithm runs in time polynomial in this condition number. Our result confirms a previous conjecture of Boros et al., which was claimed as a theorem and later retracted. It stands in contrast with a recent counter-example by Christ and Yannakakis, showing that Howard's policy-iteration algorithm does not run in smoothed polynomial time on stochastic single-player mean-payoff games. Our approach is inspired by the analysis of random optimal assignment instances by Frieze and Sorkin, and the analysis of bias-induced policies for mean-payoff games by Akian, Gaubert and Hochart