14,170 research outputs found
The List Coloring Reconfiguration Problem for Bounded Pathwidth Graphs
We study the problem of transforming one list (vertex) coloring of a graph
into another list coloring by changing only one vertex color assignment at a
time, while at all times maintaining a list coloring, given a list of allowed
colors for each vertex. This problem is known to be PSPACE-complete for
bipartite planar graphs. In this paper, we first show that the problem remains
PSPACE-complete even for bipartite series-parallel graphs, which form a proper
subclass of bipartite planar graphs. We note that our reduction indeed shows
the PSPACE-completeness for graphs with pathwidth two, and it can be extended
for threshold graphs. In contrast, we give a polynomial-time algorithm to solve
the problem for graphs with pathwidth one. Thus, this paper gives precise
analyses of the problem with respect to pathwidth
Bumblebees: a multiagent combinatorial optimization algorithm inspired by social insect behaviour
This paper introduces a multiagent optimization algorithm inspired by the collective behavior of social insects. In our method, each agent encodes a possible solution of the problem to solve, and evolves in a way similar to real life insects. We test the algorithm on a classical difficult problem, the -coloring of a graph, and we compare its performance in relation to a standard genetic algorithm and another multiagent
system. The results show that this algorithm
is faster and outperforms the other methods for a range of random graphs with different orders and densities. Moreover, the method is easy to adapt to solve different NP-complete problems
On the Complexity of Local Distributed Graph Problems
This paper is centered on the complexity of graph problems in the
well-studied LOCAL model of distributed computing, introduced by Linial [FOCS
'87]. It is widely known that for many of the classic distributed graph
problems (including maximal independent set (MIS) and -vertex
coloring), the randomized complexity is at most polylogarithmic in the size
of the network, while the best deterministic complexity is typically
. Understanding and narrowing down this exponential gap
is considered to be one of the central long-standing open questions in the area
of distributed graph algorithms. We investigate the problem by introducing a
complexity-theoretic framework that allows us to shed some light on the role of
randomness in the LOCAL model. We define the SLOCAL model as a sequential
version of the LOCAL model. Our framework allows us to prove completeness
results with respect to the class of problems which can be solved efficiently
in the SLOCAL model, implying that if any of the complete problems can be
solved deterministically in rounds in the LOCAL model, we can
deterministically solve all efficient SLOCAL-problems (including MIS and
-coloring) in rounds in the LOCAL model. We show
that a rather rudimentary looking graph coloring problem is complete in the
above sense: Color the nodes of a graph with colors red and blue such that each
node of sufficiently large polylogarithmic degree has at least one neighbor of
each color. The problem admits a trivial zero-round randomized solution. The
result can be viewed as showing that the only obstacle to getting efficient
determinstic algorithms in the LOCAL model is an efficient algorithm to
approximately round fractional values into integer values
Notes on complexity of packing coloring
A packing -coloring for some integer of a graph is a mapping
such that any two vertices of color
are in distance at least . This concept
is motivated by frequency assignment problems. The \emph{packing chromatic
number} of is the smallest such that there exists a packing
-coloring of .
Fiala and Golovach showed that determining the packing chromatic number for
chordal graphs is \NP-complete for diameter exactly 5. While the problem is
easy to solve for diameter 2, we show \NP-completeness for any diameter at
least 3. Our reduction also shows that the packing chromatic number is hard to
approximate within for any .
In addition, we design an \FPT algorithm for interval graphs of bounded
diameter. This leads us to exploring the problem of finding a partial coloring
that maximizes the number of colored vertices.Comment: 9 pages, 2 figure
Global Optimization Using Local Search Approach for Course Scheduling Problem
Course scheduling problem is a combinatorial optimization problem which is defined over a finite discrete problem whose candidate solution structure is expressed as a finite sequence of course events scheduled in available time and space resources. This problem is considered as non-deterministic polynomial complete problem which is hard to solve. Many solution methods have been studied in the past for solving the course scheduling problem, namely from the most traditional approach such as graph coloring technique; the local search family such as hill-climbing search, taboo search, and simulated annealing technique; and various population-based metaheuristic methods such as evolutionary algorithm, genetic algorithm, and swarm optimization. This article will discuss these various probabilistic optimization methods in order to gain the global optimal solution. Furthermore, inclusion of a local search in the population-based algorithm to improve the global solution will be explained rigorously
Solving Hard Computational Problems Efficiently: Asymptotic Parametric Complexity 3-Coloring Algorithm
Many practical problems in almost all scientific and technological
disciplines have been classified as computationally hard (NP-hard or even
NP-complete). In life sciences, combinatorial optimization problems frequently
arise in molecular biology, e.g., genome sequencing; global alignment of
multiple genomes; identifying siblings or discovery of dysregulated pathways.In
almost all of these problems, there is the need for proving a hypothesis about
certain property of an object that can be present only when it adopts some
particular admissible structure (an NP-certificate) or be absent (no admissible
structure), however, none of the standard approaches can discard the hypothesis
when no solution can be found, since none can provide a proof that there is no
admissible structure. This article presents an algorithm that introduces a
novel type of solution method to "efficiently" solve the graph 3-coloring
problem; an NP-complete problem. The proposed method provides certificates
(proofs) in both cases: present or absent, so it is possible to accept or
reject the hypothesis on the basis of a rigorous proof. It provides exact
solutions and is polynomial-time (i.e., efficient) however parametric. The only
requirement is sufficient computational power, which is controlled by the
parameter . Nevertheless, here it is proved that the
probability of requiring a value of to obtain a solution for a
random graph decreases exponentially: , making
tractable almost all problem instances. Thorough experimental analyses were
performed. The algorithm was tested on random graphs, planar graphs and
4-regular planar graphs. The obtained experimental results are in accordance
with the theoretical expected results.Comment: Working pape
Vertex colorings of graphs
Import 23/08/2017V této práci se zabývám vrcholovým barvením grafů a především algoritmy k nalezení vrcholového barvení. Vrcholovým dobrým m - barvením grafu rozumíme takové přiřazení m barev vrcholům, že sousední vrcholy jsou obarveny různě. O vrcholovém m - barvení grafu je známo, že je to NP - kompletní problém, pro m ≥ 3. Tedy zatím žádný algoritmus nedokáže tento problém vyřešit obecně v polynomiálním čase (předpokládáme, že P ≠ NP). V práci ukazuji heuristický algoritmus využívající hladové barvení i s jeho optimalizovanou verzí a uvádím příklad deterministického algoritmu. Algoritmy porovnávám a na jednoduchých příkladech ukazuji jejich princip a funkčnost.In this bachelor thesis I am dealing with vertex colorings of simple graphs with focus on vertex coloring algorithms. A proper vertex m - coloring of a graph is an assignment of m colors to the vertices so that, no two adjacent vertices share a color. The problem of the proper vertex m - coloring of a graph is well known to be an NP - complete problem for m ≥ 3. So far, no algorithm can solve this problem generaly in polynomial time (we assume that P ≠ NP). I present a heuristic algorithm based on the greedy coloring and it's optimized version, then example of deterministic algorithm is examined. I compare these algorithms and on simple examples show their principles and functionality.470 - Katedra aplikované matematikyvelmi dobř
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