68 research outputs found
Meta-Kernelization using Well-Structured Modulators
Kernelization investigates exact preprocessing algorithms with performance
guarantees. The most prevalent type of parameters used in kernelization is the
solution size for optimization problems; however, also structural parameters
have been successfully used to obtain polynomial kernels for a wide range of
problems. Many of these parameters can be defined as the size of a smallest
modulator of the given graph into a fixed graph class (i.e., a set of vertices
whose deletion puts the graph into the graph class). Such parameters admit the
construction of polynomial kernels even when the solution size is large or not
applicable. This work follows up on the research on meta-kernelization
frameworks in terms of structural parameters.
We develop a class of parameters which are based on a more general view on
modulators: instead of size, the parameters employ a combination of rank-width
and split decompositions to measure structure inside the modulator. This allows
us to lift kernelization results from modulator-size to more general
parameters, hence providing smaller kernels. We show (i) how such large but
well-structured modulators can be efficiently approximated, (ii) how they can
be used to obtain polynomial kernels for any graph problem expressible in
Monadic Second Order logic, and (iii) how they allow the extension of previous
results in the area of structural meta-kernelization
Bisection of Bounded Treewidth Graphs by Convolutions
In the Bisection problem, we are given as input an edge-weighted graph G. The task is to find a partition of V(G) into two parts A and B such that ||A| - |B|| <= 1 and the sum of the weights of the edges with one endpoint in A and the other in B is minimized. We show that the complexity of the Bisection problem on trees, and more generally on graphs of bounded treewidth, is intimately linked to the (min, +)-Convolution problem. Here the input consists of two sequences (a[i])^{n-1}_{i = 0} and (b[i])^{n-1}_{i = 0}, the task is to compute the sequence (c[i])^{n-1}_{i = 0}, where c[k] = min_{i=0,...,k}(a[i] + b[k - i]).
In particular, we prove that if (min, +)-Convolution can be solved in O(tau(n)) time, then Bisection of graphs of treewidth t can be solved in time O(8^t t^{O(1)} log n * tau(n)), assuming a tree decomposition of width t is provided as input. Plugging in the naive O(n^2) time algorithm for (min, +)-Convolution yields a O(8^t t^{O(1)} n^2 log n) time algorithm for Bisection. This improves over the (dependence on n of the) O(2^t n^3) time algorithm of Jansen et al. [SICOMP 2005] at the cost of a worse dependence on t. "Conversely", we show that if Bisection can be solved in time O(beta(n)) on edge weighted trees, then (min, +)-Convolution can be solved in O(beta(n)) time as well. Thus, obtaining a sub-quadratic algorithm for Bisection on trees is extremely challenging, and could even be impossible. On the other hand, for unweighted graphs of treewidth t, by making use of a recent algorithm for Bounded Difference (min, +)-Convolution of Chan and Lewenstein [STOC 2015], we obtain a sub-quadratic algorithm for Bisection with running time O(8^t t^{O(1)} n^{1.864} log n)
On Covering Segments with Unit Intervals
We study the problem of covering a set of segments on a line with the minimum number of unit-length intervals, where an interval covers a segment if at least one of the two endpoints of the segment falls in the unit interval. We also study several variants of this problem.
We show that the restrictions of the aforementioned problems to the set of instances in which all the segments have the same length are NP-hard. This result implies several NP-hardness results in the literature for variants and generalizations of the problems under consideration.
We then study the parameterized complexity of the aforementioned problems. We provide tight results for most of them by showing that they are fixed-parameter tractable for the restrictions in which all the segments have the same length, and are W[1]-complete otherwise
Complexity of the Steiner Network Problem with Respect to the Number of Terminals
In the Directed Steiner Network problem we are given an arc-weighted digraph
, a set of terminals , and an (unweighted) directed
request graph with . Our task is to output a subgraph of the minimum cost such that there is a directed path from to in
for all .
It is known that the problem can be solved in time
[Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time
even if is planar, unless Exponential-Time Hypothesis
(ETH) fails [Chitnis et al., SODA 2014]. However, as this reduction (and other
reductions showing hardness of the problem) only shows that the problem cannot
be solved in time unless ETH fails, there is a significant
gap in the complexity with respect to in the exponent.
We show that Directed Steiner Network is solvable in time , where is a constant depending solely on the
genus of and is a computable function. We complement this result by
showing that there is no algorithm for
any function for the problem on general graphs, unless ETH fails
How to Navigate Through Obstacles?
Given a set of obstacles and two points in the plane, is there a path between the two points that does not cross more than k different obstacles? This is a fundamental problem that has undergone a tremendous amount of work by researchers in various areas, including computational geometry, graph theory, wireless computing, and motion planning. It is known to be NP-hard, even when the obstacles are very simple geometric shapes (e.g., unit-length line segments). The problem can be formulated and generalized into the following graph problem: Given a planar graph G whose vertices are colored by color sets, two designated vertices s, t in V(G), and k in N, is there an s-t path in G that uses at most k colors? If each obstacle is connected, the resulting graph satisfies the color-connectivity property, namely that each color induces a connected subgraph.
We study the complexity and design algorithms for the above graph problem with an eye on its geometric applications. We prove a set of hardness results, among which a result showing that the color-connectivity property is crucial for any hope for fixed-parameter tractable (FPT) algorithms, as without it, the problem is W[SAT]-hard parameterized by k. Previous results only implied that the problem is W[2]-hard. A corollary of this result is that, unless W[2] = FPT, the problem cannot be approximated in FPT time to within a factor that is a function of k. By describing a generic plane embedding of the graph instances, we show that our hardness results translate to the geometric instances of the problem.
We then focus on graphs satisfying the color-connectivity property. By exploiting the planarity of the graph and the connectivity of the colors, we develop topological results that allow us to prove that, for any vertex v, there exists a set of paths whose cardinality is upper bounded by a function of k, that "represents" the valid s-t paths containing subsets of colors from v. We employ these structural results to design an FPT algorithm for the problem parameterized by both k and the treewidth of the graph, and extend this result further to obtain an FPT algorithm for the parameterization by both k and the length of the path. The latter result generalizes and explains previous FPT results for various obstacle shapes, such as unit disks and fat regions
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