31 research outputs found
Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut
We investigate the notion of stability proposed by Bilu and Linial. We obtain
an exact polynomial-time algorithm for -stable Max Cut instances with
for some absolute constant . Our
algorithm is robust: it never returns an incorrect answer; if the instance is
-stable, it finds the maximum cut, otherwise, it either finds the
maximum cut or certifies that the instance is not -stable. We prove
that there is no robust polynomial-time algorithm for -stable instances
of Max Cut when , where is the best
approximation factor for Sparsest Cut with non-uniform demands.
Our algorithm is based on semidefinite programming. We show that the standard
SDP relaxation for Max Cut (with triangle inequalities) is integral
if , where
is the least distortion with which every point metric space of negative
type embeds into . On the negative side, we show that the SDP
relaxation is not integral when .
Moreover, there is no tractable convex relaxation for -stable instances
of Max Cut when . That suggests that solving
-stable instances with might be difficult or
impossible.
Our results significantly improve previously known results. The best
previously known algorithm for -stable instances of Max Cut required
that (for some ) [Bilu, Daniely, Linial, and
Saks]. No hardness results were known for the problem. Additionally, we present
an algorithm for 4-stable instances of Minimum Multiway Cut. We also study a
relaxed notion of weak stability.Comment: 24 page
Certified Algorithms: Worst-Case Analysis and Beyond
In this paper, we introduce the notion of a certified algorithm. Certified algorithms provide worst-case and beyond-worst-case performance guarantees. First, a ?-certified algorithm is also a ?-approximation algorithm - it finds a ?-approximation no matter what the input is. Second, it exactly solves ?-perturbation-resilient instances (?-perturbation-resilient instances model real-life instances). Additionally, certified algorithms have a number of other desirable properties: they solve both maximization and minimization versions of a problem (e.g. Max Cut and Min Uncut), solve weakly perturbation-resilient instances, and solve optimization problems with hard constraints.
In the paper, we define certified algorithms, describe their properties, present a framework for designing certified algorithms, provide examples of certified algorithms for Max Cut/Min Uncut, Minimum Multiway Cut, k-medians and k-means. We also present some negative results
Improved Cheeger's Inequality: Analysis of Spectral Partitioning Algorithms through Higher Order Spectral Gap
Let \phi(G) be the minimum conductance of an undirected graph G, and let
0=\lambda_1 <= \lambda_2 <=... <= \lambda_n <= 2 be the eigenvalues of the
normalized Laplacian matrix of G. We prove that for any graph G and any k >= 2,
\phi(G) = O(k) \lambda_2 / \sqrt{\lambda_k}, and this performance guarantee
is achieved by the spectral partitioning algorithm. This improves Cheeger's
inequality, and the bound is optimal up to a constant factor for any k. Our
result shows that the spectral partitioning algorithm is a constant factor
approximation algorithm for finding a sparse cut if \lambda_k$ is a constant
for some constant k. This provides some theoretical justification to its
empirical performance in image segmentation and clustering problems. We extend
the analysis to other graph partitioning problems, including multi-way
partition, balanced separator, and maximum cut
Clustering Under Perturbation Stability in Near-Linear Time
We consider the problem of center-based clustering in low-dimensional Euclidean spaces under the perturbation stability assumption. An instance is ?-stable if the underlying optimal clustering continues to remain optimal even when all pairwise distances are arbitrarily perturbed by a factor of at most ?. Our main contribution is in presenting efficient exact algorithms for ?-stable clustering instances whose running times depend near-linearly on the size of the data set when ? ? 2 + ?3. For k-center and k-means problems, our algorithms also achieve polynomial dependence on the number of clusters, k, when ? ? 2 + ?3 + ? for any constant ? > 0 in any fixed dimension. For k-median, our algorithms have polynomial dependence on k for ? > 5 in any fixed dimension; and for ? ? 2 + ?3 in two dimensions. Our algorithms are simple, and only require applying techniques such as local search or dynamic programming to a suitably modified metric space, combined with careful choice of data structures
Stability and Recovery for Independence Systems
Two genres of heuristics that are frequently reported to perform much better on "real-world" instances than in the worst case are greedy algorithms and local search algorithms. In this paper, we systematically study these two types of algorithms for the problem of maximizing a monotone submodular set function subject to downward-closed feasibility constraints. We consider perturbation-stable instances, in the sense of Bilu and Linial [11], and precisely identify the stability threshold beyond which these algorithms are guaranteed to recover the optimal solution. Byproducts of our work include the first definition of perturbation-stability for non-additive objective functions, and a resolution of the worst-case approximation guarantee of local search in p-extendible systems
Exact Algorithms and Lower Bounds for Stable Instances of Euclidean k-Means
We investigate the complexity of solving stable or perturbation-resilient
instances of k-Means and k-Median clustering in fixed dimension Euclidean
metrics (or more generally doubling metrics). The notion of stable or
perturbation resilient instances was introduced by Bilu and Linial [2010] and
Awasthi et al. [2012]. In our context we say a k-Means instance is
\alpha-stable if there is a unique OPT solution which remains unchanged if
distances are (non-uniformly) stretched by a factor of at most \alpha. Stable
clustering instances have been studied to explain why heuristics such as
Lloyd's algorithm perform well in practice. In this work we show that for any
fixed \epsilon>0, (1+\epsilon)-stable instances of k-Means in doubling metrics
can be solved in polynomial time. More precisely we show a natural multiswap
local search algorithm in fact finds the OPT solution for (1+\epsilon)-stable
instances of k-Means and k-Median in a polynomial number of iterations. We
complement this result by showing that under a plausible PCP hypothesis this is
essentially tight: that when the dimension d is part of the input, there is a
fixed \epsilon_0>0 s.t. there is not even a PTAS for (1+\epsilon_0)-stable
k-Means in R^d unless NP=RP. To do this, we consider a robust property of CSPs;
call an instance stable if there is a unique optimum solution x^* and for any
other solution x', the number of unsatisfied clauses is proportional to the
Hamming distance between x^* and x'. Dinur et al. have already shown stable
QSAT is hard to approximate for some constant Q, our hypothesis is simply that
stable QSAT with bounded variable occurrence is also hard. Given this
hypothesis, we consider "stability-preserving" reductions to prove our hardness
for stable k-Means. Such reductions seem to be more fragile than standard
L-reductions and may be of further use to demonstrate other stable optimization
problems are hard.Comment: 29 page
Block Stability for MAP Inference
To understand the empirical success of approximate MAP inference, recent work
(Lang et al., 2018) has shown that some popular approximation algorithms
perform very well when the input instance is stable. The simplest stability
condition assumes that the MAP solution does not change at all when some of the
pairwise potentials are (adversarially) perturbed. Unfortunately, this strong
condition does not seem to be satisfied in practice. In this paper, we
introduce a significantly more relaxed condition that only requires blocks
(portions) of an input instance to be stable. Under this block stability
condition, we prove that the pairwise LP relaxation is persistent on the stable
blocks. We complement our theoretical results with an empirical evaluation of
real-world MAP inference instances from computer vision. We design an algorithm
to find stable blocks, and find that these real instances have large stable
regions. Our work gives a theoretical explanation for the widespread empirical
phenomenon of persistency for this LP relaxation