Lukasz Kowalik

Oracles for bounded-length shortest paths in planar graphs

We present a new approach for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G = (V, E), one can build in O(|V|) time a data structure, which allows to check in O(1) time whether two given vertices are at distance at most k in G and if so a shortest path between them is returned. Graph G can be undirected as well as directed.Our data structure works in fully dynamic environment. It can be updated in O(1) time after removing an edge or a vertex while updating after an edge insertion takes polylogarithmic amortized time. Besides deleting elements one can also disable ones for some time. It is motivated by a practical situation where nodes or links of a network may be temporarily out of service.Our results can be easily generalized to other wide classes of graphs---for instance we can take any minor-closed family of graphs.

Short path queries in planar graphs in constant time

We present a new algorithm for answering short path queries in planar graphs. For any fixed constant k and a given unweighted planar graph G=(V,E) one can build in O(|V|) time a data structure, which allows to check in O(1) time whether two given vertices are distant by at most k in G and if so a shortest path between them is returned. This significantly improves the previous result of D. Eppstein [5] where after a linear preprocessing the queries are answered in O(log |V|) time. Our approach can be applied to compute the girth of a planar graph and a corresponding shortest cycle in O(|V|) time provided that the constant bound on the girth is known.Our results can be easily generalized to other wide classes of graphs~--~for instance we can take graphs embeddable in a surface of bounded genus or graphs of bounded tree-width.

Fast 3-coloring Triangle-Free Planar Graphs

AbstractAlthough deciding whether the vertices of a planar graph can be colored with three colors is NP-hard, the widely known Grötzsch’s theorem states that every triangle-free planar graph is 3-colorable. We show the first o(n2) algorithm for 3-coloring vertices of triangle-free planar graphs. The time complexity of the algorithm is

Fast Witness Extraction Using a Decision Oracle

\mathcal{O}(n\log n)

A New 3-Color Criterion for Planar Graphs

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Spotting Trees with Few Leaves

The gist of many (NP-)hard combinatorial problems is to decide whether a universe of n elements contains a witness consisting of k elements that match some prescribed pattern. For some of these problems there are known advanced algebra-based FPT algorithms which solve the decision problem but do not return the witness. We investigate techniques for turning such a YES/NO-decision oracle into an algorithm for extracting a single witness, with an objective to obtain practical scalability for large values of n. By relying on techniques from combinatorial group testing, we demonstrate that a witness may be extracted with O(klogn) queries to either a deterministic or a randomized set inclusion oracle with one-sided probability of error. Furthermore, we demonstrate through implementation and experiments that the algebra-based FPT algorithms are practical, in particular in the setting of the k-path problem. Also discussed are engineering issues such as optimizing finite field arithmetic.

On Directed Feedback Vertex Set Parameterized by Treewidth

We present a new general 3-color criterion for planar graphs. Applying this criterion we characterize a broad class of 3-colorable planar graphs and provide a corresponding linear time 3-coloring algorithm. We also characterize fully infinite 3-colorable planar triangulations.

Tight Lower Bounds for List Edge Coloring

We show two results related to the Hamiltonicity and \(k\) -Path algorithms in undirected graphs by Bjorklund [FOCS’10], and Bjorklund et al., [arXiv’10]. First, we demonstrate that the technique used can be generalized to finding some \(k\)-vertex tree with \(l\) leaves in an \(n\)-vertex undirected graph in \(O^*(1.657^k2^{l/2})\) time. It can be applied as a subroutine to solve the \(k\) -Internal Spanning Tree (\(k\)-IST) problem in \(O^*({\text {min}}(3.455^k, 1.946^n))\) time using polynomial space, improving upon previous algorithms for this problem. In particular, for the first time, we break the natural barrier of \(O^*(2^n)\). Second, we show that the iterated random bipartition employed by the algorithm can be improved whenever the host graph admits a vertex coloring with few colors; it can be an ordinary proper vertex coloring, a fractional vertex coloring, or a vector coloring. In effect, we show improved bounds for \(k\) -Path and Hamiltonicity in any graph of maximum degree \(\Delta =4,\ldots ,12\) or with vector chromatic number at most \(8\).

Tight Lower Bounds for the Complexity of Multicoloring

We study the Directed Feedback Vertex Set problem parameterized by the treewidth of the input graph. We prove that unless the Exponential Time Hypothesis fails, the problem cannot be solved in time

On the Fine-Grained Complexity of Rainbow Coloring

2^{o(t\log t)}\cdot n^{\mathcal{O}(1)}