Algorithms for stable and perturbation-resilient problems

Abstract

We study the notion of stability and perturbation resilience introduced by Bilu and Linial (2010) and Awasthi, Blum, and Sheffet (2012). A combinatorial optimization problem is α-stable or α-perturbation-resilient if the optimal solution does not change when we perturb all parameters of the problem by a factor of at most α. In this paper, we give improved algorithms for stable instances of various clustering and combinatorial optimization problems. We also prove several hardness results. We rst give an exact algorithm for 2-perturbation resilient instances of clustering problems with natural center-based objectives. The class of clustering problems with natural center-based objectives includes such problems as k-means, k-median, and k-center. Our result improves upon the result of Balcan and Liang (2016), who gave an algorithm for clustering 1 + √ 2 ≈ 2.41 perturbationresilient instances. Our result is tight in the sense that no polynomialtime algorithm can solve (2 − ε)-perturbation resilient instances of k-center unless N P = RP, as was shown by Balcan, Haghtalab, and White (2016). We then give an exact algorithm for (2 − 2/k)-stable instances of Minimum Multiway Cut with k terminals, improving the previous result of Makarychev, Makarychev, and Vijayaraghavan (2014), who gave an algorithm for 4-stable instances. We also give an algorithm for (2 − 2/k + δ)-weakly stable instances of Minimum Multiway Cut. Finally, we show that there are no robust polynomial-time algorithms for n 1−ε-stable instances of Set Cover, Minimum Vertex Cover, and Min 2-Horn Deletion (unless P = N P).

Beyond worst-case analysis

“…The empirical superiority of LRU is due to the special structure in real-world page request sequenceslocality of reference-and traditional worst-case analysis provides no vocabulary to speak about this structure. 5 This is what work on "beyond worst-case analysis" is all about: articulating properties of "real-world" inputs, and proving rigorous and meaningful algorithmic guarantees for inputs with these properties.…”

“…(b) For every f and k and every sequence that conforms to f , the page fault rate of the LRU policy is at most α f (k). 5 If worst-case analysis has an implicit model of data, then it's the "Murphy's Law" data model, where the instance to be solved is an adversarially selected function of the chosen algorithm. Outside of cryptographic applications, this is a rather paranoid and incoherent way to think about a computational problem.…”

“…Many such problems, including the k-means, k-median, and k-center problems, are polynomial-time solvable already in 2-perturbation-stable instances [5,10]. The algorithm in Angelidakis et al [5], like its precursor in Awasthi et al [8], is inspired by the widely-used single-linkage clustering algorithm. It computes a minimum spanning tree (where edge weights are the interpoint distances) and uses dynamic programming to optimally remove k − 1 edges to define k clusters.…”

Untitled

“…They show that under this condition, one can find the optimum solution in polynomial time so long as α ≥ min{n/d min , √ nd max } where d min and d max are the minimum and maximum degrees in the graph, respectively. Following that work, a series of results [7,11,10,3] have analyzed center-based clustering under this stability assumption, including k-median, k-means, and k-center objective functions, with the current best bounds finding the optimum solution when α ≥ 2 [3]. A survey of center-based clustering, including under various stability conditions, appears in [4].…”

“…Our results show that under the (widely believed) assumption that PPAD is not contained in quasi-polynomial time [17], such uniformly stable games are inherently easier for computation of approximate equilibria than general bimatrix games. 3 Moreover, variants of many games appearing commonly in experimental economics including the public goods game, matching pennies, and identical interest game [22] satisfy this condition. See Section 3, Section 5, and Appendix B for detailed examples.…”