163 research outputs found
On the dynamic width of the 3-colorability problem
A graph is 3-colorable if and only if it maps homomorphically to the
complete 3-vertex graph . The last condition can be checked by a
-consistency algorithm where the parameter has to be chosen large
enough, dependent on . Let denote the minimum sufficient for this
purpose. For a non-3-colorable graph , is equal to the minimum
such that can be distinguished from in the -variable
existential-positive first-order logic. We define the dynamic width of the
3-colorability problem as the function , where the maximum is
taken over all non-3-colorable with vertices.
The assumption implies that is unbounded.
Indeed, a lower bound follows
unconditionally from the work of Nesetril and Zhu on bounded treewidth duality.
The Exponential Time Hypothesis implies a much stronger bound
and indeed we unconditionally prove that
. In fact, an even stronger statement is true: A first-order
sentence distinguishing any 3-colorable graph on vertices from any
non-3-colorable graph on vertices must have variables.
On the other hand, we observe that and for every non-3-colorable graph with vertices, where
denotes the independence number of . This implies that
, improving on the trivial upper bound .
We also show that for every non-3-colorable graph
, where denotes the girth of .
Finally, we consider the function over planar graphs and prove that
in the case.Comment: 18 pages, 2 figure
The role of planarity in connectivity problems parameterized by treewidth
For some years it was believed that for "connectivity" problems such as
Hamiltonian Cycle, algorithms running in time 2^{O(tw)}n^{O(1)} -called
single-exponential- existed only on planar and other sparse graph classes,
where tw stands for the treewidth of the n-vertex input graph. This was
recently disproved by Cygan et al. [FOCS 2011], Bodlaender et al. [ICALP 2013],
and Fomin et al. [SODA 2014], who provided single-exponential algorithms on
general graphs for essentially all connectivity problems that were known to be
solvable in single-exponential time on sparse graphs. In this article we
further investigate the role of planarity in connectivity problems
parameterized by treewidth, and convey that several problems can indeed be
distinguished according to their behavior on planar graphs. Known results from
the literature imply that there exist problems, like Cycle Packing, that cannot
be solved in time 2^{o(tw logtw)}n^{O(1)} on general graphs but that can be
solved in time 2^{O(tw)}n^{O(1)} when restricted to planar graphs. Our main
contribution is to show that there exist problems that can be solved in time
2^{O(tw logtw)}n^{O(1)} on general graphs but that cannot be solved in time
2^{o(tw logtw)}n^{O(1)} even when restricted to planar graphs. Furthermore, we
prove that Planar Cycle Packing and Planar Disjoint Paths cannot be solved in
time 2^{o(tw)}n^{O(1)}. The mentioned negative results hold unless the ETH
fails. We feel that our results constitute a first step in a subject that can
be further exploited.Comment: 23 page
List-coloring embedded graphs
For any fixed surface Sigma of genus g, we give an algorithm to decide
whether a graph G of girth at least five embedded in Sigma is colorable from an
assignment of lists of size three in time O(|V(G)|). Furthermore, we can allow
a subgraph (of any size) with at most s components to be precolored, at the
expense of increasing the time complexity of the algorithm to
O(|V(G)|^{K(g+s)+1}) for some absolute constant K; in both cases, the
multiplicative constant hidden in the O-notation depends on g and s. This also
enables us to find such a coloring when it exists. The idea of the algorithm
can be applied to other similar problems, e.g., 5-list-coloring of graphs on
surfaces.Comment: 14 pages, 0 figures, accepted to SODA'1
Measurement-based quantum computation and undecidable logic
We establish a connection between measurement-based quantum computation and
the field of mathematical logic. We show that the computational power of an
important class of quantum states called graph states, representing resources
for measurement-based quantum computation, is reflected in the expressive power
of (classical) formal logic languages defined on the underlying mathematical
graphs. In particular, we show that for all graph state resources which can
yield a computational speed-up with respect to classical computation, the
underlying graphs--describing the quantum correlations of the states--are
associated with undecidable logic theories. Here undecidability is to be
interpreted in a sense similar to Goedel's incompleteness results, meaning that
there exist propositions, expressible in the above classical formal logic,
which cannot be proven or disproven.Comment: 10 pages. Presentation improved. Paper to appear in Found. Phys.;
currently published onlin
Assigning tasks to agents under time conflicts: a parameterized complexity approach
We consider the problem of assigning tasks to agents under time conflicts,
with applications also to frequency allocations in point-to-point wireless
networks. In particular, we are given a set of agents, a set of
tasks, and different time slots. Each task can be carried out in one of the
predefined time slots, and can be represented by the subset
of the involved agents. Since each agent cannot participate to more than one
task simultaneously, we must find an allocation that assigns non-overlapping
tasks to each time slot. Being the number of slots limited by , in general
it is not possible to executed all the possible tasks, and our aim is to
determine a solution maximizing the overall social welfare, that is the number
of executed tasks. We focus on the restriction of this problem in which the
number of time slots is fixed to be , and each task is performed by
exactly two agents, that is . In fact, even under this assumptions, the
problem is still challenging, as it remains computationally difficult. We
provide parameterized complexity results with respect to several reasonable
parameters, showing for the different cases that the problem is fixed-parameter
tractable or it is paraNP-hard.Comment: 31 pages, 3 figure
Complexity of fall coloring for restricted graph classes
We strengthen a result by Laskar and Lyle (Discrete Appl. Math. (2009),
330-338) by proving that it is NP-complete to decide whether a bipartite planar
graph can be partitioned into three independent dominating sets. In contrast,
we show that this is always possible for every maximal outerplanar graph with
at least three vertices. Moreover, we extend their previous result by proving
that deciding whether a bipartite graph can be partitioned into independent
dominating sets is NP-complete for every . We also strengthen a
result by Henning et al. (Discrete Math. (2009), 6451-6458) by showing that it
is NP-complete to determine if a graph has two disjoint independent dominating
sets, even when the problem is restricted to triangle-free planar graphs.
Finally, for every , we show that there is some constant
depending only on such that deciding whether a -regular graph can be
partitioned into independent dominating sets is NP-complete. We conclude by
deriving moderately exponential-time algorithms for the problem.Comment: To appear at the 30th International Workshop on Combinatorial
Algorithms (IWOCA 2019
Multi-Clique-Width
Multi-clique-width is obtained by a simple modification in the definition of
clique-width. It has the advantage of providing a natural extension of
tree-width. Unlike clique-width, it does not explode exponentially compared to
tree-width. Efficient algorithms based on multi-clique-width are still possible
for interesting tasks like computing the independent set polynomial or testing
-colorability. In particular, -colorability can be tested in time linear
in and singly exponential in and the width of a given
multi--expression. For these tasks, the running time as a function of the
multi-clique-width is the same as the running time of the fastest known
algorithm as a function of the clique-width. This results in an exponential
speed-up for some graphs, if the corresponding graph generating expressions are
given. The reason is that the multi-clique-width is never bigger, but is
exponentially smaller than the clique-width for many graphs. This gap shows up
when the tree-width is basically equal to the multi-clique width as well as
when the tree-width is not bounded by any function of the clique-width.Comment: 16 page
The Width and Integer Optimization on Simplices With Bounded Minors of the Constraint Matrices
In this paper, we will show that the width of simplices defined by systems of
linear inequalities can be computed in polynomial time if some minors of their
constraint matrices are bounded. Additionally, we present some
quasi-polynomial-time and polynomial-time algorithms to solve the integer
linear optimization problem defined on simplices minus all their integer
vertices assuming that some minors of the constraint matrices of the simplices
are bounded.Comment: 12 page
High speed all optical networks
An inherent problem of conventional point-to-point wide area network (WAN) architectures is that they cannot translate optical transmission bandwidth into comparable user available throughput due to the limiting electronic processing speed of the switching nodes. The first solution to wavelength division multiplexing (WDM) based WAN networks that overcomes this limitation is presented. The proposed Lightnet architecture takes into account the idiosyncrasies of WDM switching/transmission leading to an efficient and pragmatic solution. The Lightnet architecture trades the ample WDM bandwidth for a reduction in the number of processing stages and a simplification of each switching stage, leading to drastically increased effective network throughputs. The principle of the Lightnet architecture is the construction and use of virtual topology networks, embedded in the original network in the wavelength domain. For this construction Lightnets utilize the new concept of lightpaths which constitute the links of the virtual topology. Lightpaths are all-optical, multihop, paths in the network that allow data to be switched through intermediate nodes using high throughput passive optical switches. The use of the virtual topologies and the associated switching design introduce a number of new ideas, which are discussed in detail
FPT-algorithms for The Shortest Lattice Vector and Integer Linear Programming Problems
In this paper, we present FPT-algorithms for special cases of the shortest
vector problem (SVP) and the integer linear programming problem (ILP), when
matrices included to the problems' formulations are near square. The main
parameter is the maximal absolute value of rank minors of matrices included to
the problem formulation. Additionally, we present FPT-algorithms with respect
to the same main parameter for the problems, when the matrices have no singular
rank sub-matrices.Comment: 17 page
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