150 research outputs found
Connection between conjunctive capacity and structural properties of graphs
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Evolutionary Design of Ultrasound Treatment Plans
Tato práce se zabývá studiem vybraných evolučních systémů pro jejich použití při návrhu plánu pro ultrazvukové operace. Tyto algoritmy statisticky analyzuje a dle vhodných kritérií je experimentálně srovnává a diskutuje přínos pro klinickou praxi.The thesis studies selected evolution systems to use in planning of high intensity focused ultrasound surgeries. Considered algorithms are statistically analyzed and compared by appropriate criteria to find the one that adds the most value to the potential real world medical problems.
Optimization Methods for SIMLIB/C++ Simulation Library
Práce se věnuje metodám optimalizace parametrů simulačních modelů. Seznamuje se základy matematické optimalizace a jejím využitím v operačním výzkumu. Dále navrhuje rozšíření knihovny SIMLIB/C++ modulem pro optimalizační metody. Několik vybraných metod teoreticky popisuje, implementuje v jazyce C++, demonstruje jejich použití na několika příkladech a zhodnocuje jejich úspěšnost.This thesis addresses the topic of parametric optimization of simulation models. It introduces theoretical foundation of optimization and its uses in simulation analysis. Furthermore, it suggests the extension of SIMBLI/C++ library by module for optimization methods. Some of the chosen methods are then theoretically described, implemented in C++ language, demonstrates its uses and evaluates their success.
Approximation hardness of Travelling Salesman via weighted amplifiers
The expander graph constructions and their variants are the main tool used in gap preserving reductions to prove approximation lower bounds of combinatorial optimisation problems. In this paper we introduce the weighted amplifiers and weighted low occurrence of Constraint Satisfaction problems as intermediate steps in the NP-hard gap reductions. Allowing the weights in intermediate problems is rather natural for the edge-weighted problems as Travelling Salesman or Steiner Tree. We demonstrate the technique for Travelling Salesman and use the parametrised weighted amplifiers in the gap reductions to allow more flexibility in fine-tuning their expanding parameters. The purpose of this paper is to point out effectiveness of these ideas, rather than to optimise the expander’s parameters. Nevertheless, we show that already slight improvement of known expander values modestly improve the current best approximation hardness value for TSP from 123/122 ([9]) to 117/116 . This provides a new motivation for study of expanding properties of random graphs in order to improve approximation lower bounds of TSP and other edge-weighted optimisation problems
Going beyond variation of sets
We study integralgeometric representations of variations of general sets A ⊂ Rn without any regularity assumptions. If we assume, for example, that just one partial derivative of its characteristic function χA is a signed Borel measure on R n with finite total variation, can we provide a nice integralgeometric representation of this variation? This is a delicate question, as the Gauss-Green type theorems of De Giorgi and Federer are not available in this generality. We will show that a ‘measure-theoretic boundary’ plays its role in such representations similarly as for the sets of finite variation. There is a variety of suitable notions of ‘measure theoretic boundary’ and one can address the question to find notions of measure-theoretic boundary that are as fine as possible
Tropical Dominating Sets in Vertex-Coloured Graphs
Given a vertex-coloured graph, a dominating set is said to be tropical if
every colour of the graph appears at least once in the set. Here, we study
minimum tropical dominating sets from structural and algorithmic points of
view. First, we prove that the tropical dominating set problem is NP-complete
even when restricted to a simple path. Then, we establish upper bounds related
to various parameters of the graph such as minimum degree and number of edges.
We also give upper bounds for random graphs. Last, we give approximability and
inapproximability results for general and restricted classes of graphs, and
establish a FPT algorithm for interval graphs.Comment: 19 pages, 4 figure
Algorithmic aspects of disjunctive domination in graphs
For a graph , a set is called a \emph{disjunctive
dominating set} of if for every vertex , is either
adjacent to a vertex of or has at least two vertices in at distance
from it. The cardinality of a minimum disjunctive dominating set of is
called the \emph{disjunctive domination number} of graph , and is denoted by
. The \textsc{Minimum Disjunctive Domination Problem} (MDDP)
is to find a disjunctive dominating set of cardinality .
Given a positive integer and a graph , the \textsc{Disjunctive
Domination Decision Problem} (DDDP) is to decide whether has a disjunctive
dominating set of cardinality at most . In this article, we first propose a
linear time algorithm for MDDP in proper interval graphs. Next we tighten the
NP-completeness of DDDP by showing that it remains NP-complete even in chordal
graphs. We also propose a -approximation
algorithm for MDDP in general graphs and prove that MDDP can not be
approximated within for any unless NP
DTIME. Finally, we show that MDDP is
APX-complete for bipartite graphs with maximum degree
Approximating Clustering of Fingerprint Vectors with Missing Values
The problem of clustering fingerprint vectors is an interesting problem in
Computational Biology that has been proposed in (Figureroa et al. 2004). In
this paper we show some improvements in closing the gaps between the known
lower bounds and upper bounds on the approximability of some variants of the
biological problem. Namely we are able to prove that the problem is APX-hard
even when each fingerprint contains only two unknown position. Moreover we have
studied some variants of the orginal problem, and we give two 2-approximation
algorithm for the IECMV and OECMV problems when the number of unknown entries
for each vector is at most a constant.Comment: 13 pages, 4 figure
Inapproximability of maximal strip recovery
In comparative genomic, the first step of sequence analysis is usually to
decompose two or more genomes into syntenic blocks that are segments of
homologous chromosomes. For the reliable recovery of syntenic blocks, noise and
ambiguities in the genomic maps need to be removed first. Maximal Strip
Recovery (MSR) is an optimization problem proposed by Zheng, Zhu, and Sankoff
for reliably recovering syntenic blocks from genomic maps in the midst of noise
and ambiguities. Given genomic maps as sequences of gene markers, the
objective of \msr{d} is to find subsequences, one subsequence of each
genomic map, such that the total length of syntenic blocks in these
subsequences is maximized. For any constant , a polynomial-time
2d-approximation for \msr{d} was previously known. In this paper, we show that
for any , \msr{d} is APX-hard, even for the most basic version of the
problem in which all gene markers are distinct and appear in positive
orientation in each genomic map. Moreover, we provide the first explicit lower
bounds on approximating \msr{d} for all . In particular, we show that
\msr{d} is NP-hard to approximate within . From the other
direction, we show that the previous 2d-approximation for \msr{d} can be
optimized into a polynomial-time algorithm even if is not a constant but is
part of the input. We then extend our inapproximability results to several
related problems including \cmsr{d}, \gapmsr{\delta}{d}, and
\gapcmsr{\delta}{d}.Comment: A preliminary version of this paper appeared in two parts in the
Proceedings of the 20th International Symposium on Algorithms and Computation
(ISAAC 2009) and the Proceedings of the 4th International Frontiers of
Algorithmics Workshop (FAW 2010
Hypergraphic LP Relaxations for Steiner Trees
We investigate hypergraphic LP relaxations for the Steiner tree problem,
primarily the partition LP relaxation introduced by Koenemann et al. [Math.
Programming, 2009]. Specifically, we are interested in proving upper bounds on
the integrality gap of this LP, and studying its relation to other linear
relaxations. Our results are the following. Structural results: We extend the
technique of uncrossing, usually applied to families of sets, to families of
partitions. As a consequence we show that any basic feasible solution to the
partition LP formulation has sparse support. Although the number of variables
could be exponential, the number of positive variables is at most the number of
terminals. Relations with other relaxations: We show the equivalence of the
partition LP relaxation with other known hypergraphic relaxations. We also show
that these hypergraphic relaxations are equivalent to the well studied
bidirected cut relaxation, if the instance is quasibipartite. Integrality gap
upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap
of these hypergraph relaxations in general graphs. In the special case of
uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~
1.216. By our equivalence theorem, the latter result implies an improved upper
bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010
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