307 research outputs found
Heuristics for Sparsest Cut Approximations in Network Flow Applications
The Maximum Concurrent Flow Problem (MCFP) is a polynomially bounded problem that has been used over the years in a variety of applications. Sometimes it is used to attempt to find the Sparsest Cut, an NP-hard problem, and other times to find communities in Social Network Analysis (SNA) in its hierarchical formulation, the HMCFP. Though it is polynomially bounded, the MCFP quickly grows in space utilization, rendering it useful on only small problems. When it was defined, only a few hundred nodes could be solved, where a few decades later, graphs of one to two thousand nodes can still be too much for modern commodity hardware to handle.
This dissertation covers three approaches to heuristics to the MCFP that run significantly faster in practice than the LP formulation with far less memory utilization. The first two approaches are based on the Maximum Adjacency Search (MAS) and apply to both the MCFP and the HMCFP used for community detection. We compare the three approaches to the LP performance in terms of accuracy, runtime, and memory utilization on several classes of synthetic graphs representing potential real-world applications. We find that the heuristics are often correct, and run using orders of magnitude less memory and time
On Brambles, Grid-Like Minors, and Parameterized Intractability of Monadic Second-Order Logic
Brambles were introduced as the dual notion to treewidth, one of the most
central concepts of the graph minor theory of Robertson and Seymour. Recently,
Grohe and Marx showed that there are graphs G, in which every bramble of order
larger than the square root of the treewidth is of exponential size in |G|. On
the positive side, they show the existence of polynomial-sized brambles of the
order of the square root of the treewidth, up to log factors. We provide the
first polynomial time algorithm to construct a bramble in general graphs and
achieve this bound, up to log-factors. We use this algorithm to construct
grid-like minors, a replacement structure for grid-minors recently introduced
by Reed and Wood, in polynomial time. Using the grid-like minors, we introduce
the notion of a perfect bramble and an algorithm to find one in polynomial
time. Perfect brambles are brambles with a particularly simple structure and
they also provide us with a subgraph that has bounded degree and still large
treewidth; we use them to obtain a meta-theorem on deciding certain
parameterized subgraph-closed problems on general graphs in time singly
exponential in the parameter.
The second part of our work deals with providing a lower bound to Courcelle's
famous theorem, stating that every graph property that can be expressed by a
sentence in monadic second-order logic (MSO), can be decided by a linear time
algorithm on classes of graphs of bounded treewidth. Using our results from the
first part of our work we establish a strong lower bound for tractability of
MSO on classes of colored graphs
Combinatorial Fiedler Theory and Graph Partition
Partition problems in graphs are extremely important in applications, as
shown in the Data science and Machine learning literature. One approach is
spectral partitioning based on a Fiedler vector, i.e., an eigenvector
corresponding to the second smallest eigenvalue of the Laplacian matrix
of the graph . This problem corresponds to the minimization of a
quadratic form associated with , under certain constraints involving the
-norm. We introduce and investigate a similar problem, but using the
-norm to measure distances. This leads to a new parameter as the
optimal value. We show that a well-known cut problem arises in this approach,
namely the sparsest cut problem. We prove connectivity results and different
bounds on this new parameter, relate to Fiedler theory and show explicit
expressions for for trees. We also comment on an -norm
version of the problem
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
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
On the Parameterized Complexity of Sparsest Cut and Small-set Expansion Problems
We study the NP-hard \textsc{-Sparsest Cut} problem (SC) in which,
given an undirected graph and a parameter , the objective is to
partition vertex set into subsets whose maximum edge expansion is
minimized. Herein, the edge expansion of a subset is defined as
the sum of the weights of edges exiting divided by the number of vertices
in . Another problem that has been investigated is \textsc{-Small-Set
Expansion} problem (SSE), which aims to find a subset with minimum edge
expansion with a restriction on the size of the subset. We extend previous
studies on SC and SSE by inspecting their parameterized complexity. On
the positive side, we present two FPT algorithms for both SSE and 2SC
problems where in the first algorithm we consider the parameter treewidth of
the input graph and uses exponential space, and in the second we consider the
parameter vertex cover number of the input graph and uses polynomial space.
Moreover, we consider the unweighted version of the SC problem where is fixed and proposed two FPT algorithms with parameters treewidth and
vertex cover number of the input graph. We also propose a randomized FPT
algorithm for SSE when parameterized by and the maximum degree of the
input graph combined. Its derandomization is done efficiently.
\noindent On the negative side, first we prove that for every fixed integer
, the problem SC is NP-hard for graphs with vertex cover
number at most . We also show that SC is W[1]-hard when parameterized
by the treewidth of the input graph and the number~ of components combined
using a reduction from \textsc{Unary Bin Packing}. Furthermore, we prove that
SC remains NP-hard for graphs with maximum degree three and also graphs with
degeneracy two. Finally, we prove that the unweighted SSE is W[1]-hard for
the parameter
Compression bounds for Lipschitz maps from the Heisenberg group to
We prove a quantitative bi-Lipschitz nonembedding theorem for the Heisenberg
group with its Carnot-Carath\'eodory metric and apply it to give a lower bound
on the integrality gap of the Goemans-Linial semidefinite relaxation of the
Sparsest Cut problem
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