28 research outputs found
Connection between conjunctive capacity and structural properties of graphs
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The complexity of combinatorial optimization problems on d-dimensional boxes
The Maximum Independent Set problem in d-box graphs, i.e., in intersection graphs of axis-parallel rectangles in R-d, is known to be NP-hard for any fixed d >= 2. A challenging open problem is that of how closely the solution can be approximated by a polynomial time algorithm. For the restricted case of d-boxes with bounded aspect ratio a PTAS exists [T. Erlebach, K. Jansen, and E. Seidel, SIAM J. Comput., 34 (2005), pp. 1302-1323]. In the general case no polynomial time algorithm with approximation ratio o(log(d-1)n) for a set of n d-boxes is known. In this paper we prove APX-hardness of the MAXIMUM INDEPENDENT SET problem in d-box graphs for any fixed d >= 3. We give an explicit lower bound 245/244 on efficient approximability for this problem unless P = NP. Additionally, we provide a generic method how to prove APX-hardness for other graph optimization problems in d-box graphs for any fixed d >= 3
The d-precoloring problem for k-degenerate graphs
AbstractIn this paper we deal with the d-PRECOLORING EXTENSION (d-PREXT) problem in various classes of graphs. The d-PREXTproblem is the special case of PRECOLORING EXTENSION problem where, for a fixed constant d, input instances are restricted to contain at most d precolored vertices for every available color. The goal is to decide if there exists an extension of given precoloring using only available colors or to find it.We present a linear time algorithm for both, the decision and the search version of d-PREXT, in the following cases: (i) restricted to the class of k-degenerate graphs (hence also planar graphs) and with sufficiently large set S of available colors, and (ii) restricted to the class of partial k-trees (without any size restriction on S). We also study the following problem related to d-PREXT: given an instance of the d-PREXT problem which is extendable by colors of S, what is the minimum number of colors of S sufficient to use for precolorless vertices over all such extensions? We establish lower and upper bounds on this value for k-degenerate graphs and its various subclasses (e.g., planar graphs, outerplanar graphs) and prove tight results for the class of trees
Colourful components in k-caterpillars and planar graphs
A connected component of a vertex-coloured graph is said to be colourful if
all its vertices have different colours. By extension, a graph is colourful if
all its connected components are colourful. Given a vertex-coloured graph
and an integer , the Colourful Components problem asks whether there exist
at most edges whose removal makes colourful and the Colourful Partition
problem asks whether there exists a partition of into at most colourful
components. In order to refine our understanding of the complexity of the
problems on trees, we study both problems on -caterpillars, which are trees
with a central path such that every vertex not in is within distance
from a vertex in . We prove that Colourful Components and Colourful
Partition are NP-complete on -caterpillars with maximum degree ,
-caterpillars with maximum degree and -caterpillars with maximum
degree . On the other hand, we show that the problems are linear-time
solvable on -caterpillars. Hence, our results imply two complexity
dichotomies on trees: Colourful Components and Colourful Partition are
linear-time solvable on trees with maximum degree if (that is,
on paths), and NP-complete otherwise; Colourful Components and Colourful
Partition are linear-time solvable on -caterpillars if , and
NP-complete otherwise. We leave three open cases which, if solved, would
provide a complexity dichotomy for both problems on -caterpillars, for every
non-negative integer , with respect to the maximum degree. We also show that
Colourful Components is NP-complete on -coloured planar graphs with maximum
degree and on -coloured planar graphs with maximum degree . Our
results answer two open questions of Bulteau et al. mentioned in [30th Annual
Symposium on Combinatorial Pattern Matching, 2019]
Inapproximability results for orthogonal rectangle packing problems with rotations
Recently Bansal and Sviridenko [4] proved that there is no asymptotic PTAS for 2-DIMENSIONAL ORTHOGONAL RECTANGLE BIN PACKING without rotations allowed, unless P = NP. We show that similar approximation hardness results hold for several rectangle packing problems even if rotations by ninety degrees around the axes are allowed. Moreover, for some of these problems we provide explicit lower bounds on asymptotic approximation ratio of any polynomial time approximation algorithm
Hardness of asymptotic approximation for orthogonal rectangle packing and covering problems
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