Beyond worst-case analysis

Roughgarden, Tim

doi:10.1145/3232535

Cited by 48 publications

(45 citation statements)

References 43 publications

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“…Just like in our result, a γ-perturbation stable instance of the maximum cut problem is known [19] to be NP-hard for small values of γ and polynomial for larger values of γ. Since the problem of identifying the value of γ where the complexity switches has largely been resolved [14], we hope that the approaches reviewed by Roughgarden [19] will be helpful for our proposed study as well.…”

supporting

confidence: 78%

“…We hence propose to characterise the complexity classes of the problem RNNI(ρ) for values of ρ ≥ 0. This problem is similar in spirit to the beyond worst-case analysis (including parametrised complexity [7]) framework [19]. Just like in our result, a γ-perturbation stable instance of the maximum cut problem is known [19] to be NP-hard for small values of γ and polynomial for larger values of γ.…”

supporting

confidence: 72%

“…In light of this biological argument, the NP-hardness result can be interpreted as RNNI(ρ)-SP being hard only when the graph RNNI(ρ) is biologically irrelevant. Although in sharp contrast with the common belief in the field of computational phylogenetics [25], this interpretation resonates with the idea suggested in the beyond worst case analysis framework [19] that some problems are only computationally hard when their instances are practically irrelevant. The following question is hence natural.…”

Section: Additional Open Problemsmentioning

confidence: 59%

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Geometry of Ranked Nearest Neighbour Interchange Space of Phylogenetic Trees

Collienne¹,

Elmes²,

Fischer

et al. 2019

Preprint

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In this paper we study the graph of ranked phylogenetic trees where the adjacency relation is given by a local rearrangement of the tree structure. Our work is motivated by tree inference algorithms, such as maximum likelihood and Markov Chain Monte Carlo methods, where the geometry of the search space plays a central role for efficiency and practicality of optimisation and sampling. We hence focus on understanding the geometry of the space (graph) of ranked trees, the so-called ranked nearest neighbour interchange (RNNI) graph. We find the radius and diameter of the space exactly, improving the best previously known estimates. Since the RNNI graph is a generalisation of the classical nearest neighbour interchange (NNI) graph to ranked phylogenetic trees, we compare geometric and algorithmic properties of the two graphs. Surprisingly, we discover that both geometric and algorithmic properties of RNNI and NNI are quite different. For example, we establish convexity of certain natural subspaces in RNNI which are not convex is NNI. Our results suggest that the complexity of computing distances in the two graphs is different.

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supporting

confidence: 78%

supporting

confidence: 72%

Section: Additional Open Problemsmentioning

confidence: 59%

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Geometry of Ranked Nearest Neighbour Interchange Space of Phylogenetic Trees

Collienne¹,

Elmes²,

Fischer

et al. 2019

Preprint

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“…Is it possible to get a similar result for k-means and k-median, and possibly a unifying algorithm that solves all 2-perturbation-resilient instances of these 3 problems? P 2 (P R [17] [16,18]). Get a better upper bound and any lower bound on the stability parameter for Minimum Multiway Cut.…”

Section: Introductionmentioning

confidence: 99%

Algorithms for stable and perturbation-resilient problems

Angelidakis

Makarychev

2017

Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

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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).

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“…This result was later improved along multiple axes by . There are many other works that show positive results for different natural notions of stability in various settings [Arora et al, 2012, Awasthi et al, 2010, Gupta et al, 2014, Hardt and Roth, 2013, Kumar and Kannan, 2010, Kumar et al, 2004, Roughgarden, 2014].…”

Section: Problemmentioning

confidence: 99%

k -center Clustering under Perturbation Resilience

Balcan

Haghtalab

White

2020

ACM Trans. Algorithms

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The k-center problem is a canonical and long-studied facility location and clustering problem with many applications in both its symmetric and asymmetric forms. Both versions of the problem have tight approximation factors on worst case instances: a 2-approximation for symmetric k-center and an O(log * (k))-approximation for the asymmetric version. Therefore to improve on these ratios, one must go beyond the worst case.In this work, we take this approach and provide strong positive results both for the asymmetric and symmetric k-center problems under a natural input stability (promise) condition called α-perturbation resilience [Bilu and Linial, 2012], which states that the optimal solution does not change under any α-factor perturbation to the input distances. We provide algorithms that give strong guarantees simultaneously for stable and non-stable instances: our algorithms always inherit the worst-case guarantees of clustering approximation algorithms, and output the optimal solution if the input is 2-perturbation resilient. In particular, we show that if the input is only perturbation resilient on part of the data, our algorithm will return the optimal clusters from the region of the data that is perturbation resilient, while achieving the best worst-case approximation guarantee on the remainder of the data. Furthermore, we prove our result is tight by showing symmetric k-center under (2 − )-perturbation resilience is hard unless N P = RP .The impact of our results are multifaceted. First, to our knowledge, asymmetric k-center is the first problem that is hard to approximate to any constant factor in the worst case, yet can be optimally solved in polynomial time under perturbation resilience for a constant value of α. This is also the first tight result for any problem under perturbation resilience, i.e., this is the first time the exact value of α for which the problem switches from being NP-hard to efficiently computable has been found. Furthermore, our results illustrate a surprising relationship between symmetric and asymmetric k-center instances under perturbation resilience. Unlike approximation ratio, for which symmetric k-center is easily solved to a factor of 2 but asymmetric k-center cannot be approximated to any constant factor, both symmetric and asymmetric k-center can be solved optimally under resilience to 2-perturbations. Finally, our guarantees in the setting where only part of the data satisfies perturbation resilience makes these algorithms more applicable to real-life instances.

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Beyond worst-case analysis

Cited by 48 publications

References 43 publications

Geometry of Ranked Nearest Neighbour Interchange Space of Phylogenetic Trees

Geometry of Ranked Nearest Neighbour Interchange Space of Phylogenetic Trees

Algorithms for stable and perturbation-resilient problems

k -center Clustering under Perturbation Resilience

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