3,182 research outputs found
Improved Parameterized Algorithms for Constraint Satisfaction
For many constraint satisfaction problems, the algorithm which chooses a
random assignment achieves the best possible approximation ratio. For instance,
a simple random assignment for {\sc Max-E3-Sat} allows 7/8-approximation and
for every \eps >0 there is no polynomial-time (7/8+\eps)-approximation
unless P=NP. Another example is the {\sc Permutation CSP} of bounded arity.
Given the expected fraction of the constraints satisfied by a random
assignment (i.e. permutation), there is no (\rho+\eps)-approximation
algorithm for every \eps >0, assuming the Unique Games Conjecture (UGC).
In this work, we consider the following parameterization of constraint
satisfaction problems. Given a set of constraints of constant arity, can we
satisfy at least constraint, where is the expected fraction
of constraints satisfied by a random assignment? {\sc Constraint Satisfaction
Problems above Average} have been posed in different forms in the literature
\cite{Niedermeier2006,MahajanRamanSikdar09}. We present a faster parameterized
algorithm for deciding whether equations can be simultaneously
satisfied over . As a consequence, we obtain -variable
bikernels for {\sc boolean CSPs} of arity for every fixed , and for {\sc
permutation CSPs} of arity 3. This implies linear bikernels for many problems
under the "above average" parameterization, such as {\sc Max--Sat}, {\sc
Set-Splitting}, {\sc Betweenness} and {\sc Max Acyclic Subgraph}. As a result,
all the parameterized problems we consider in this paper admit -time
algorithms.
We also obtain non-trivial hybrid algorithms for every Max -CSP: for every
instance , we can either approximate beyond the random assignment
threshold in polynomial time, or we can find an optimal solution to in
subexponential time.Comment: A preliminary version of this paper has been accepted for IPEC 201
Computing Vertex Centrality Measures in Massive Real Networks with a Neural Learning Model
Vertex centrality measures are a multi-purpose analysis tool, commonly used
in many application environments to retrieve information and unveil knowledge
from the graphs and network structural properties. However, the algorithms of
such metrics are expensive in terms of computational resources when running
real-time applications or massive real world networks. Thus, approximation
techniques have been developed and used to compute the measures in such
scenarios. In this paper, we demonstrate and analyze the use of neural network
learning algorithms to tackle such task and compare their performance in terms
of solution quality and computation time with other techniques from the
literature. Our work offers several contributions. We highlight both the pros
and cons of approximating centralities though neural learning. By empirical
means and statistics, we then show that the regression model generated with a
feedforward neural networks trained by the Levenberg-Marquardt algorithm is not
only the best option considering computational resources, but also achieves the
best solution quality for relevant applications and large-scale networks.
Keywords: Vertex Centrality Measures, Neural Networks, Complex Network Models,
Machine Learning, Regression ModelComment: 8 pages, 5 tables, 2 figures, version accepted at IJCNN 2018. arXiv
admin note: text overlap with arXiv:1810.1176
On the Hardness and Inapproximability of Recognizing Wheeler Graphs
In recent years several compressed indexes based on variants of the Burrows-Wheeler transformation have been introduced. Some of these are used to index structures far more complex than a single string, as was originally done with the FM-index [Ferragina and Manzini, J. ACM 2005]. As such, there has been an increasing effort to better understand under which conditions such an indexing scheme is possible. This has led to the introduction of Wheeler graphs [Gagie et al., Theor. Comput. Sci., 2017]. Gagie et al. showed that de Bruijn graphs, generalized compressed suffix arrays, and several other BWT related structures can be represented as Wheeler graphs, and that Wheeler graphs can be indexed in a way which is space efficient. Hence, being able to recognize whether a given graph is a Wheeler graph, or being able to approximate a given graph by a Wheeler graph, could have numerous applications in indexing. Here we resolve the open question of whether there exists an efficient algorithm for recognizing if a given graph is a Wheeler graph. We present:
- The problem of recognizing whether a given graph G=(V,E) is a Wheeler graph is NP-complete for any edge label alphabet of size sigma >= 2, even when G is a DAG. This holds even on a restricted, subset of graphs called d-NFA\u27s for d >= 5. This is in contrast to recent results demonstrating the problem can be solved in polynomial time for d-NFA\u27s where d <= 2. We also show the recognition problem can be solved in linear time for sigma =1;
- There exists an 2^{e log sigma + O(n + e)} time exact algorithm where n = |V| and e = |E|. This algorithm relies on graph isomorphism being computable in strictly sub-exponential time;
- We define an optimization variant of the problem called Wheeler Graph Violation, abbreviated WGV, where the aim is to remove the minimum number of edges in order to obtain a Wheeler graph. We show WGV is APX-hard, even when G is a DAG, implying there exists a constant C >= 1 for which there is no C-approximation algorithm (unless P = NP). Also, conditioned on the Unique Games Conjecture, for all C >= 1, it is NP-hard to find a C-approximation;
- We define the Wheeler Subgraph problem, abbreviated WS, where the aim is to find the largest subgraph which is a Wheeler Graph (the dual of the WGV). In contrast to WGV, we prove that the WS problem is in APX for sigma=O(1);
The above findings suggest that most problems under this theme are computationally difficult. However, we identify a class of graphs for which the recognition problem is polynomial time solvable, raising the open question of which parameters determine this problem\u27s difficulty
Efficient Exact and Approximate Algorithms for Computing Betweenness Centrality in Directed Graphs
Graphs are an important tool to model data in different domains, including
social networks, bioinformatics and the world wide web. Most of the networks
formed in these domains are directed graphs, where all the edges have a
direction and they are not symmetric. Betweenness centrality is an important
index widely used to analyze networks. In this paper, first given a directed
network and a vertex , we propose a new exact algorithm to
compute betweenness score of . Our algorithm pre-computes a set
, which is used to prune a huge amount of computations that do
not contribute in the betweenness score of . Time complexity of our exact
algorithm depends on and it is respectively
and
for unweighted graphs and weighted graphs with positive weights.
is bounded from above by and in most cases, it
is a small constant. Then, for the cases where is large, we
present a simple randomized algorithm that samples from and
performs computations for only the sampled elements. We show that this
algorithm provides an -approximation of the betweenness
score of . Finally, we perform extensive experiments over several real-world
datasets from different domains for several randomly chosen vertices as well as
for the vertices with the highest betweenness scores. Our experiments reveal
that in most cases, our algorithm significantly outperforms the most efficient
existing randomized algorithms, in terms of both running time and accuracy. Our
experiments also show that our proposed algorithm computes betweenness scores
of all vertices in the sets of sizes 5, 10 and 15, much faster and more
accurate than the most efficient existing algorithms.Comment: arXiv admin note: text overlap with arXiv:1704.0735
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