200 research outputs found
Approximating the Regular Graphic TSP in near linear time
We present a randomized approximation algorithm for computing traveling
salesperson tours in undirected regular graphs. Given an -vertex,
-regular graph, the algorithm computes a tour of length at most
, with high probability, in time. This improves upon a recent result by Vishnoi (\cite{Vishnoi12}, FOCS
2012) for the same problem, in terms of both approximation factor, and running
time. The key ingredient of our algorithm is a technique that uses
edge-coloring algorithms to sample a cycle cover with cycles with
high probability, in near linear time.
Additionally, we also give a deterministic
factor approximation algorithm
running in time .Comment: 12 page
List covering of regular multigraphs
A graph covering projection, also known as a locally bijective homomorphism,
is a mapping between vertices and edges of two graphs which preserves
incidencies and is a local bijection. This notion stems from topological graph
theory, but has also found applications in combinatorics and theoretical
computer science.
It has been known that for every fixed simple regular graph of valency
greater than 2, deciding if an input graph covers is NP-complete. In recent
years, topological graph theory has developed into heavily relying on multiple
edges, loops, and semi-edges, but only partial results on the complexity of
covering multigraphs with semi-edges are known so far. In this paper we
consider the list version of the problem, called \textsc{List--Cover}, where
the vertices and edges of the input graph come with lists of admissible
targets. Our main result reads that the \textsc{List--Cover} problem is
NP-complete for every regular multigraph of valency greater than 2 which
contains at least one semi-simple vertex (i.e., a vertex which is incident with
no loops, with no multiple edges and with at most one semi-edge). Using this
result we almost show the NP-co/polytime dichotomy for the computational
complexity of \textsc{ List--Cover} of cubic multigraphs, leaving just five
open cases.Comment: Accepted to IWOCA 202
Spotting Trees with Few Leaves
We show two results related to the Hamiltonicity and -Path algorithms in
undirected graphs by Bj\"orklund [FOCS'10], and Bj\"orklund et al., [arXiv'10].
First, we demonstrate that the technique used can be generalized to finding
some -vertex tree with leaves in an -vertex undirected graph in
time. It can be applied as a subroutine to solve the
-Internal Spanning Tree (-IST) problem in
time using polynomial space, improving upon previous algorithms for this
problem. In particular, for the first time we break the natural barrier of
. Second, we show that the iterated random bipartition employed by
the algorithm can be improved whenever the host graph admits a vertex coloring
with few colors; it can be an ordinary proper vertex coloring, a fractional
vertex coloring, or a vector coloring. In effect, we show improved bounds for
-Path and Hamiltonicity in any graph of maximum degree
or with vector chromatic number at most 8
Approximation algorithms for variants of the traveling salesman problem
The traveling salesman problem, hereafter abbreviated and referred to as TSP, is a very well known NP-optimization problem and is one of the most widely researched problems in computer science. Classical TSP is one of the original NP - hard problems [1]. It is also known to be NP - hard to approximate within any factor and thus there is no approximation algorithm for TSP for general graphs, unless P = NP. However, given the added constraint that edges of the graph observe triangle inequality, it has been shown that it is possible achieve a good approximation to the optimal solution [2]. TSP has a number of variants that have been deeply researched over the years. Approximations of varying degrees have been achieved depending on the complexity presented by the problem setup. An obvious variant is that of finding a maximum weight hamiltonian tour, also informally known as the taxicab ripoff problem . The problem is not equivalent to the minimization problem when the edge weights are non-negative and does allow good approximations. Also important is the problem when the graph is not symmetric. The problem in this case, as should be expected, is slightly tougher to approximate. Another very well researched problem is when weights of edges are drawn from the set { 1, 2}. This study was focused on gaining an understanding of these algorithms keeping in mind the primary endeavor of improving them. This thesis presents approximation algorithms for the aforementioned and other variants of the TSP, and is focused on the techniques and methods used for developing these algorithms
Assessing the Computational Complexity of Multi-Layer Subgraph Detection
Multi-layer graphs consist of several graphs (layers) over the same vertex
set. They are motivated by real-world problems where entities (vertices) are
associated via multiple types of relationships (edges in different layers). We
chart the border of computational (in)tractability for the class of subgraph
detection problems on multi-layer graphs, including fundamental problems such
as maximum matching, finding certain clique relaxations (motivated by community
detection), or path problems. Mostly encountering hardness results, sometimes
even for two or three layers, we can also spot some islands of tractability
Generating constrained random graphs using multiple edge switches
The generation of random graphs using edge swaps provides a reliable method
to draw uniformly random samples of sets of graphs respecting some simple
constraints, e.g. degree distributions. However, in general, it is not
necessarily possible to access all graphs obeying some given con- straints
through a classical switching procedure calling on pairs of edges. We therefore
propose to get round this issue by generalizing this classical approach through
the use of higher-order edge switches. This method, which we denote by "k-edge
switching", makes it possible to progres- sively improve the covered portion of
a set of constrained graphs, thereby providing an increasing, asymptotically
certain confidence on the statistical representativeness of the obtained
sample.Comment: 15 page
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