Erik D. Demaine

Fixed-parameter algorithms for ( k , r )-center in planar graphs and map graphs

The <i>(<i>k</i>, <i>r</i>)-center problem</i> asks whether an input graph <i>G</i> has ≤<i>k</i> vertices (called <i>centers</i>) such that every vertex of <i>G</i> is within distance ≤<i>r</i> from some center. In this article, we prove that the (<i>k</i>, <i>r</i>)-center problem, parameterized by <i>k</i> and <i>R</i>, is fixed-parameter tractable (FPT) on planar graphs, i.e., it admits an algorithm of complexity <i>f</i>(<i>k</i>, <i>r</i>)<i>n</i>^<i>O</i>(1) where the function <i>f</i> is independent of <i>n</i>. In particular, we show that <i>f</i>(<i>k,r</i>) = 2^{<i>O</i>(<i>r</i> log <i>r</i>) &ksqrt;}, where the exponent of the exponential term grows sublinearly in the number of centers. Moreover, we prove that the same type of FPT algorithms can be designed for the more general class of <i>map graphs</i> introduced by Chen, Grigni, and Papadimitriou. Our results combine dynamic-programming algorithms for graphs of small branchwidth and a graph-theoretic result bounding this parameter in terms of <i>k</i> and <i>r</i>. Finally, a byproduct of our algorithm is the existence of a PTAS for the <i>r</i>-domination problem in both planar graphs and map graphs.Our approach builds on the seminal results of Robertson and Seymour on Graph Minors, and as a result is much more powerful than the previous machinery of Alber et al. for exponential speedup on planar graphs. To demonstrate the versatility of our results, we show how our algorithms can be extended to general parameters that are “large” on grids. In addition, our use of branchwidth instead of the usual treewidth allows us to obtain much faster algorithms, and requires more complicated dynamic programming than the standard leaf/introduce/forget/join structure of nice tree decompositions. Our results are also unique in that they apply to classes of graphs that are not minor-closed, namely, constant powers of planar graphs and map graphs.

Approximation algorithms for classes of graphs excluding single-crossing graphs as minors

Many problems that are intractable for general graphs allow polynomial-time solutions for structured classes of graphs, such as planar graphs and graphs of bounded treewidth. In this paper, we demonstrate structural properties of larger classes of graphs and show how to exploit the properties to obtain algorithms. The classes considered are those formed by excluding as a minor a graph that can be embedded in the plane with at most one crossing. We show that graphs in these classes can be decomposed into planar graphs and graphs of small treewidth; we use the decomposition to show that all such graphs have locally bounded treewidth (all subgraphs of a certain form are graphs of bounded treewidth). Finally, we make use of the structural properties to derive polynomial-time algorithms for approximating treewidth within a factor of 1.5 and branchwidth within a factor of 2.25 as well as polynomial-time approximation schemes for both minimization and maximization problems and fixed-parameter algorithms for problems such as vertex cover, edge-dominating set, feedback vertex set, and others.

Diameter and Treewidth in Minor-Closed Graph Families, Revisited

Abstract nEppstein [5] characterized the minor-closed graph families for which the treewidth is boundednby a function of the diameter, which includes, e.g., planar graphs. This characterization has been used as thenbasis for several (approximation) algorithms on such graphs (e.g., [2] and [5]–[8]). The proof of Eppsteinnis complicated. In this short paper we obtain the same characterization with a simple proof. In addition, thenrelation between treewidth and diameter is slightly better and explicit.

Exponential Speedup of Fixed-Parameter Algorithms for Classes of Graphs Excluding Single-Crossing Graphs as Minors

AbstractnWe present a fixed-parameter algorithm that constructively solvesnthe

Fast algorithms for hard graph problems: bidimensionality, minors, and local treewidth

k

The Bidimensional Theory of Bounded-Genus Graphs

-dominating set problem on any class of graphs excluding ansingle-crossing graph (a graph that can be drawn in the plane withnat most one crossing) as a minor in

Bidimensional Parameters and Local Treewidth

O(4^{9.55sqrt{k}}n^{O(1)})

Algorithmic graph minor theory: Decomposition, approximation, and coloring

time.nExamples of such graph classes are the

Bidimensionality: new connections between FPT algorithms and PTASs

K_{3,3}

Subexponential parameterized algorithms on graphs of bounded genus and H-minor-free graphs

-minor-free graphs and then