14,376 research outputs found
Computing Exact Minimum Cuts Without Knowing the Graph
We give query-efficient algorithms for the global min-cut and the s-t cut problem in unweighted, undirected graphs. Our oracle model is inspired by the submodular function minimization problem:
on query S subset V, the oracle returns the size of the cut between S and V S.
We provide algorithms computing an exact minimum - cut in with ~{O}(n^{5/3}) queries, and computing an exact global minimum cut of G with only ~{O}(n) queries (while learning the graph requires ~{Theta}(n^2) queries)
Weighted Min-Cut: Sequential, Cut-Query and Streaming Algorithms
Consider the following 2-respecting min-cut problem. Given a weighted graph
and its spanning tree , find the minimum cut among the cuts that contain
at most two edges in . This problem is an important subroutine in Karger's
celebrated randomized near-linear-time min-cut algorithm [STOC'96]. We present
a new approach for this problem which can be easily implemented in many
settings, leading to the following randomized min-cut algorithms for weighted
graphs.
* An -time sequential algorithm:
This improves Karger's and bounds when the input graph is not extremely
sparse or dense. Improvements over Karger's bounds were previously known only
under a rather strong assumption that the input graph is simple [Henzinger et
al. SODA'17; Ghaffari et al. SODA'20]. For unweighted graphs with parallel
edges, our bound can be improved to .
* An algorithm requiring cut queries to compute the min-cut of
a weighted graph: This answers an open problem by Rubinstein et al. ITCS'18,
who obtained a similar bound for simple graphs.
* A streaming algorithm that requires space and
passes to compute the min-cut: The only previous non-trivial exact min-cut
algorithm in this setting is the 2-pass -space algorithm on simple
graphs [Rubinstein et al., ITCS'18] (observed by Assadi et al. STOC'19).
In contrast to Karger's 2-respecting min-cut algorithm which deploys
sophisticated dynamic programming techniques, our approach exploits some cute
structural properties so that it only needs to compute the values of cuts corresponding to removing pairs of tree edges, an
operation that can be done quickly in many settings.Comment: Updates on this version: (1) Minor corrections in Section 5.1, 5.2;
(2) Reference to newer results by GMW SOSA21 (arXiv:2008.02060v2), DEMN
STOC21 (arXiv:2004.09129v2) and LMN 21 (arXiv:2102.06565v1
Integrality gaps of semidefinite programs for Vertex Cover and relations to embeddability of Negative Type metrics
We study various SDP formulations for {\sc Vertex Cover} by adding different
constraints to the standard formulation. We show that {\sc Vertex Cover} cannot
be approximated better than even when we add the so called pentagonal
inequality constraints to the standard SDP formulation, en route answering an
open question of Karakostas~\cite{Karakostas}. We further show the surprising
fact that by strengthening the SDP with the (intractable) requirement that the
metric interpretation of the solution is an metric, we get an exact
relaxation (integrality gap is 1), and on the other hand if the solution is
arbitrarily close to being embeddable, the integrality gap may be as
big as . Finally, inspired by the above findings, we use ideas from the
integrality gap construction of Charikar \cite{Char02} to provide a family of
simple examples for negative type metrics that cannot be embedded into
with distortion better than 8/7-\eps. To this end we prove a new
isoperimetric inequality for the hypercube.Comment: A more complete version. Changed order of results. A complete proof
of (current) Theorem
Minimum Sparsity of Unobservable Power Network Attacks
Physical security of power networks under power injection attacks that alter
generation and loads is studied. The system operator employs Phasor Measurement
Units (PMUs) for detecting such attacks, while attackers devise attacks that
are unobservable by such PMU networks. It is shown that, given the PMU
locations, the solution to finding the sparsest unobservable attacks has a
simple form with probability one, namely, , where
is defined as the vulnerable vertex connectivity of an augmented
graph. The constructive proof allows one to find the entire set of the sparsest
unobservable attacks in polynomial time. Furthermore, a notion of the potential
impact of unobservable attacks is introduced. With optimized PMU deployment,
the sparsest unobservable attacks and their potential impact as functions of
the number of PMUs are evaluated numerically for the IEEE 30, 57, 118 and
300-bus systems and the Polish 2383, 2737 and 3012-bus systems. It is observed
that, as more PMUs are added, the maximum potential impact among all the
sparsest unobservable attacks drops quickly until it reaches the minimum
sparsity.Comment: submitted to IEEE Transactions on Automatic Contro
Learning and Testing Variable Partitions
Let be a multivariate function from a product set to an
Abelian group . A -partition of with cost is a partition of
the set of variables into non-empty subsets such that is -close to
for some with
respect to a given error metric. We study algorithms for agnostically learning
partitions and testing -partitionability over various groups and error
metrics given query access to . In particular we show that
Given a function that has a -partition of cost , a partition
of cost can be learned in time
for any .
In contrast, for and learning a partition of cost is NP-hard.
When is real-valued and the error metric is the 2-norm, a
2-partition of cost can be learned in time
.
When is -valued and the error metric is Hamming
weight, -partitionability is testable with one-sided error and
non-adaptive queries. We also show that even
two-sided testers require queries when .
This work was motivated by reinforcement learning control tasks in which the
set of control variables can be partitioned. The partitioning reduces the task
into multiple lower-dimensional ones that are relatively easier to learn. Our
second algorithm empirically increases the scores attained over previous
heuristic partitioning methods applied in this context.Comment: Innovations in Theoretical Computer Science (ITCS) 202
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