72 research outputs found
Computational Complexity of Certifying Restricted Isometry Property
Given a matrix with rows, a number , and , is
-RIP (Restricted Isometry Property) if, for any vector , with at most non-zero co-ordinates, In many applications, such as
compressed sensing and sparse recovery, it is desirable to construct RIP
matrices with a large and a small . Given the efficacy of random
constructions in generating useful RIP matrices, the problem of certifying the
RIP parameters of a matrix has become important.
In this paper, we prove that it is hard to approximate the RIP parameters of
a matrix assuming the Small-Set-Expansion-Hypothesis. Specifically, we prove
that for any arbitrarily large constant and any arbitrarily small
constant , there exists some such that given a matrix , it
is SSE-Hard to distinguish the following two cases:
- (Highly RIP) is -RIP.
- (Far away from RIP) is not -RIP.
Most of the previous results on the topic of hardness of RIP certification
only hold for certification when . In practice, it is of interest
to understand the complexity of certifying a matrix with being close
to , as it suffices for many real applications to have matrices
with . Our hardness result holds for any constant
. Specifically, our result proves that even if is indeed very
small, i.e. the matrix is in fact \emph{strongly RIP}, certifying that the
matrix exhibits \emph{weak RIP} itself is SSE-Hard.
In order to prove the hardness result, we prove a variant of the Cheeger's
Inequality for sparse vectors
Many Sparse Cuts via Higher Eigenvalues
Cheeger's fundamental inequality states that any edge-weighted graph has a
vertex subset such that its expansion (a.k.a. conductance) is bounded as
follows: \phi(S) \defeq \frac{w(S,\bar{S})}{\min \set{w(S), w(\bar{S})}}
\leq 2\sqrt{\lambda_2} where is the total edge weight of a subset or a
cut and is the second smallest eigenvalue of the normalized
Laplacian of the graph. Here we prove the following natural generalization: for
any integer , there exist disjoint subsets ,
such that where
is the smallest eigenvalue of the normalized Laplacian and
are suitable absolute constants. Our proof is via a polynomial-time
algorithm to find such subsets, consisting of a spectral projection and a
randomized rounding. As a consequence, we get the same upper bound for the
small set expansion problem, namely for any , there is a subset whose
weight is at most a \bigO(1/k) fraction of the total weight and . Both results are the best possible up to constant
factors.
The underlying algorithmic problem, namely finding subsets such that the
maximum expansion is minimized, besides extending sparse cuts to more than one
subset, appears to be a natural clustering problem in its own right
Finding Connected Dense -Subgraphs
Given a connected graph on vertices and a positive integer ,
a subgraph of on vertices is called a -subgraph in . We design
combinatorial approximation algorithms for finding a connected -subgraph in
such that its density is at least a factor
of the density of the densest -subgraph
in (which is not necessarily connected). These particularly provide the
first non-trivial approximations for the densest connected -subgraph problem
on general graphs
Detecting and Characterizing Small Dense Bipartite-like Subgraphs by the Bipartiteness Ratio Measure
We study the problem of finding and characterizing subgraphs with small
\textit{bipartiteness ratio}. We give a bicriteria approximation algorithm
\verb|SwpDB| such that if there exists a subset of volume at most and
bipartiteness ratio , then for any , it finds a set
of volume at most and bipartiteness ratio at most
. By combining a truncation operation, we give a local
algorithm \verb|LocDB|, which has asymptotically the same approximation
guarantee as the algorithm \verb|SwpDB| on both the volume and bipartiteness
ratio of the output set, and runs in time
, independent of the size of the
graph. Finally, we give a spectral characterization of the small dense
bipartite-like subgraphs by using the th \textit{largest} eigenvalue of the
Laplacian of the graph.Comment: 17 pages; ISAAC 201
Coverage Centrality Maximization in Undirected Networks
Centrality metrics are among the main tools in social network analysis. Being
central for a user of a network leads to several benefits to the user: central
users are highly influential and play key roles within the network. Therefore,
the optimization problem of increasing the centrality of a network user
recently received considerable attention. Given a network and a target user
, the centrality maximization problem consists in creating new links
incident to in such a way that the centrality of is maximized,
according to some centrality metric. Most of the algorithms proposed in the
literature are based on showing that a given centrality metric is monotone and
submodular with respect to link addition. However, this property does not hold
for several shortest-path based centrality metrics if the links are undirected.
In this paper we study the centrality maximization problem in undirected
networks for one of the most important shortest-path based centrality measures,
the coverage centrality. We provide several hardness and approximation results.
We first show that the problem cannot be approximated within a factor greater
than , unless , and, under the stronger gap-ETH hypothesis, the
problem cannot be approximated within a factor better than , where
is the number of users. We then propose two greedy approximation
algorithms, and show that, by suitably combining them, we can guarantee an
approximation factor of . We experimentally compare the
solutions provided by our approximation algorithm with optimal solutions
computed by means of an exact IP formulation. We show that our algorithm
produces solutions that are very close to the optimum.Comment: Accepted to AAAI 201
A Cheeger Inequality for Small Set Expansion
The discrete Cheeger inequality, due to Alon and Milman (J. Comb. Theory
Series B 1985), is an indispensable tool for converting the combinatorial
condition of graph expansion to an algebraic condition on the eigenvalues of
the graph adjacency matrix. We prove a generalization of Cheeger's inequality,
giving an algebraic condition equivalent to small set expansion. This algebraic
condition is the p-to-q hypercontractivity of the top eigenspace for the graph
adjacency matrix. Our result generalizes a theorem of Barak et al (STOC 2012)
to the low small set expansion regime, and has a dramatically simpler proof;
this answers a question of Barak (2014)
- …