A parameterized problem is xed parameter tractable if it admits a solving algorithm whose running time on input instance (I; k) is f(k) jIj , where f is an arbitrary function depending only on k. Typically, f is some exponential function, e.g., f(k) = c k for constant c. We describe general techniques to obtain growth of the form f(k) = c p k for a large variety of planar graph problems. The key to this type of algorithm is what we call the "Layerwise Separation Property" of a planar graph problem. Problems having this property include planar vertex cover, planar independent set, and planar dominating set.

Graph minors theory, developed by Robertson & Seymour, provides a list of powerful theoretical results and tools. However, the wide spread opinion in Graph Algorithms community about this theory is that it is mainly of theoretical importance.

Liming Cai School of EE & CS Ohio University Athens, OH 45701 USA Email:cai@leon.cs.ohiou.edu Fax:+1 740 593 0007 David Juedes School of EE & CS Ohio University Athens, OH 45701 USA Email:juedes@ohiou.edu Fax:+1 740 593 0007 Abstract It is shown that for essentially all MAX SNP-hard optimization problems finding exact solutions in subexponential time is not possible unless W [1] = FPT . In particular, we show that O(2 o(k) p(n)) parameterized algorithms do not exist for Vertex Cover, Max Cut, Max c-Sat, and a number of problems on bounded degree graphs such as Dominating Set and Independent Set, unless W [1] = FPT . Our results are derived via an approach that uses an extended parameterization of optimization problems and associated techniques to relate the parameterized complexity of problems in FPT to the parameterized complexity of extended versions that are W [1]-hard. Track: A Keywords: computational complexity, parameterized complexity, combinatorial optimization. # This work was supported by the National Science Foundation research grant CCR-000246 1

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Graph separation is a well-known tool to make (hard) graph problems accessible to a divide and conquer approach. We show how to use graph separator theorems in combination with (linear) problem kernels in order to develop xed parameter algorithms for many well-known NP-hard (planar) graph problems. We coin the key notion of glueable select&verify graph problems and derive from that a prospective way to easily check whether a planar graph problem will allow for a xed parameter algorithm of running time c p

We present an improved dynamic programming strategy for dominating set and related problems on graphs that are given together with a tree decomposition of width k. We obtain an O(4 n) algorithm for dominating set, where n is the number of nodes of the tree decomposition.

Recent times have seen quite some progress in the development of "efficient" exponential time algorithms for NP-hard problems. These results are also tightly related to the so-called theory of fixed parameter tractability. In this incomplete, personally biased survey, we reflect on some recent developments and prospects in the field of fixed parameter algorithms.

Dealing with the NP-complete Dominating Set problem on undirected graphs, we demonstrate the power of data reduction by preprocessing from a theoretical as well as a practical side. In particular, we prove that Dominating Set on planar graphs has a so-called problem kernel of linear size, achieved by two simple and easy to implement reduction rules. This answers an open question from previous work on the parameterized complexity of Dominating Set on planar graphs.

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.

We solve an open problem posed by Eppstein in 1995 [14, 15] and re-enforced by Grohe [16, 17] concerning locally bounded treewidth in minor-closed families of graphs. A graph has bounded local treewidth if the subgraph induced by vertices within distance r of any vertex has treewidth bounded by a function of r (not n). Eppstein characterized minor-closed families of graphs with bounded local treewidth as precisely minor-closed families that minor-exclude an apex graph, where an apex graph has one vertex whose removal leaves a planar graph. In particular, Eppstein showed that all apex-minor-free graphs have bounded local treewidth, but his bound is doubly exponential in r, leaving open whether a tighter bound could be obtained. We improve this doubly exponential bound to a linear bound, which is optimal. In particular, any minor-closed graph family with bounded local treewidth has linear local treewidth. Our bound generalizes previously known linear bounds for special classes of graphs proved by several authors. As a consequence of our result, we obtain substantially faster polynomial-time approximation schemes for a broad class of problems in apex-minor-free graphs, improving the running time from .