157 research outputs found
On the extension complexity of combinatorial polytopes
In this paper we extend recent results of Fiorini et al. on the extension
complexity of the cut polytope and related polyhedra. We first describe a
lifting argument to show exponential extension complexity for a number of
NP-complete problems including subset-sum and three dimensional matching. We
then obtain a relationship between the extension complexity of the cut polytope
of a graph and that of its graph minors. Using this we are able to show
exponential extension complexity for the cut polytope of a large number of
graphs, including those used in quantum information and suspensions of cubic
planar graphs.Comment: 15 pages, 3 figures, 2 table
Geometric versions of the 3-dimensional assignment problem under general norms
We discuss the computational complexity of special cases of the 3-dimensional
(axial) assignment problem where the elements are points in a Cartesian space
and where the cost coefficients are the perimeters of the corresponding
triangles measured according to a certain norm. (All our results also carry
over to the corresponding special cases of the 3-dimensional matching problem.)
The minimization version is NP-hard for every norm, even if the underlying
Cartesian space is 2-dimensional. The maximization version is polynomially
solvable, if the dimension of the Cartesian space is fixed and if the
considered norm has a polyhedral unit ball. If the dimension of the Cartesian
space is part of the input, the maximization version is NP-hard for every
norm; in particular the problem is NP-hard for the Manhattan norm and the
Maximum norm which both have polyhedral unit balls.Comment: 21 pages, 9 figure
Path Puzzles: Discrete Tomography with a Path Constraint is Hard
We prove that path puzzles with complete row and column information--or
equivalently, 2D orthogonal discrete tomography with Hamiltonicity
constraint--are strongly NP-complete, ASP-complete, and #P-complete. Along the
way, we newly establish ASP-completeness and #P-completeness for 3-Dimensional
Matching and Numerical 3-Dimensional Matching.Comment: 16 pages, 8 figures. Revised proof of Theorem 2.4. 2-page abstract
appeared in Abstracts from the 20th Japan Conference on Discrete and
Computational Geometry, Graphs, and Games (JCDCGGG 2017
On the complexity of computing the -restricted edge-connectivity of a graph
The \emph{-restricted edge-connectivity} of a graph , denoted by
, is defined as the minimum size of an edge set whose removal
leaves exactly two connected components each containing at least vertices.
This graph invariant, which can be seen as a generalization of a minimum
edge-cut, has been extensively studied from a combinatorial point of view.
However, very little is known about the complexity of computing .
Very recently, in the parameterized complexity community the notion of
\emph{good edge separation} of a graph has been defined, which happens to be
essentially the same as the -restricted edge-connectivity. Motivated by the
relevance of this invariant from both combinatorial and algorithmic points of
view, in this article we initiate a systematic study of its computational
complexity, with special emphasis on its parameterized complexity for several
choices of the parameters. We provide a number of NP-hardness and W[1]-hardness
results, as well as FPT-algorithms.Comment: 16 pages, 4 figure
Some results on triangle partitions
We show that there exist efficient algorithms for the triangle packing
problem in colored permutation graphs, complete multipartite graphs,
distance-hereditary graphs, k-modular permutation graphs and complements of
k-partite graphs (when k is fixed). We show that there is an efficient
algorithm for C_4-packing on bipartite permutation graphs and we show that
C_4-packing on bipartite graphs is NP-complete. We characterize the cobipartite
graphs that have a triangle partition
Learning Character Strings via Mastermind Queries, with a Case Study Involving mtDNA
We study the degree to which a character string, , leaks details about
itself any time it engages in comparison protocols with a strings provided by a
querier, Bob, even if those protocols are cryptographically guaranteed to
produce no additional information other than the scores that assess the degree
to which matches strings offered by Bob. We show that such scenarios allow
Bob to play variants of the game of Mastermind with so as to learn the
complete identity of . We show that there are a number of efficient
implementations for Bob to employ in these Mastermind attacks, depending on
knowledge he has about the structure of , which show how quickly he can
determine . Indeed, we show that Bob can discover using a number of
rounds of test comparisons that is much smaller than the length of , under
reasonable assumptions regarding the types of scores that are returned by the
cryptographic protocols and whether he can use knowledge about the distribution
that comes from. We also provide the results of a case study we performed
on a database of mitochondrial DNA, showing the vulnerability of existing
real-world DNA data to the Mastermind attack.Comment: Full version of related paper appearing in IEEE Symposium on Security
and Privacy 2009, "The Mastermind Attack on Genomic Data." This version
corrects the proofs of what are now Theorems 2 and 4
Partitioning Perfect Graphs into Stars
The partition of graphs into "nice" subgraphs is a central algorithmic
problem with strong ties to matching theory. We study the partitioning of
undirected graphs into same-size stars, a problem known to be NP-complete even
for the case of stars on three vertices. We perform a thorough computational
complexity study of the problem on subclasses of perfect graphs and identify
several polynomial-time solvable cases, for example, on interval graphs and
bipartite permutation graphs, and also NP-complete cases, for example, on grid
graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor
The complexity of the Pk partition problem and related problems in bipartite graphs
International audienceIn this paper, we continue the investigation made in [MT05] about the approximability of Pk partition problems, but focusing here on their complexity. Precisely, we aim at designing the frontier between polynomial and NP-complete versions of the Pk partition problem in bipartite graphs, according to both the constant k and the maximum degree of the input graph. We actually extend the obtained results to more general classes of problems, namely, the minimum k-path partition problem and the maximum Pk packing problem. Moreover, we propose some simple approximation algorithms for those problems
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