At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a clique-sum of pieces almost-embeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2-approximation to graph coloring, constant-factor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to half-integral multicommodity flow, subexponential fixed-parameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.

We demonstrate a new connection between fixed-parameter tractability and approximation algorithms for combinatorial optimization problems on planar graphs and their generalizations. Specifically, we extend the theory of so-called “bidimensional” problems to show that essentially all such problems have both subexponential fixed-parameter algorithms and PTASs. Bidimensional problems include e.g. feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertex-removal problems, dominating set, edge dominating set, r-dominating set, diameter, connected dominating set, connected edge dominating set, and connected r-dominating set. We obtain PTASs for all of these problems in planar graphs and certain generalizations; of particular interest are our results for the two well-known problems of connected dominating set and general feedback vertex set for planar graphs and their generalizations, for which PTASs were not known to exist. Our techniques generalize and in some sense unify the two main previous approaches for designing PTASs in planar graphs, namely, the Lipton-Tarjan separator approach [FOCS’77] and the Baker layerwise decomposition approach [FOCS’83]. In particular, we replace the notion of separators with a more powerful tool from the bidimensionality theory, enabling the first approach to apply to a much broader class of minimization problems than previously possible; and through the use of a structural backbone and thickening of layers we demonstrate how the second approach can be applied to problems with a “nonlocal” structure.

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 .

This paper surveys the theory of bidimensionality. This theory characterizes a broad range of graph problems (‘bidimensional’) that admit efficient approximate or fixed-parameter solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, bounded-genus graphs and graphs excluding any fixed minor. In particular, bidimensionality theory builds on the Graph Minor Theory of Robertson and Seymour by extending the mathematical results and building new algorithmic tools. Here, we summarize the known combinatorial and algorithmic results of bidimensionality theory with the high-level ideas involved in their proof; we describe the previous work on which the theory is based and/or extends; and we mention several remaining open problems. 1.

We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge deletions instead of contractions, obtained in [Bak94, Epp00, DDO + 04, DHK05], and it generalizes a similar result for “compression ” (a variant of contraction) in planar graphs [Kle05]. Our decomposition result is a powerful tool for obtaining PTASs for contraction-closed problems (whose optimal solution only improves under contraction), a much more general class than minor-closed problems. We prove that any contraction-closed problem satisfying just a few simple conditions has a PTAS in bounded-genus graphs. In particular, our framework yields PTASs for the weighted Traveling Salesman Problem and for minimum-weight c-edge-connected submultigraph on bounded-genus graphs, improving and generalizing previous algorithms of [GKP95, AGK + 98, Kle05, Gri00, CGSZ04, BCGZ05]. We also highlight the only main difficulty in extending our results to general H-minor-free graphs.

We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an Ω(w) × Ω(w) grid graph as a minor. Thus grid minors suffice to certify that H-minor-free graphs have large treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for general graphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on H-minor-free graphs for any fixed H. Our result has many combinatorial consequences on bidimensionality theory, parameter-treewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial results has several algorithmic consequences including subexponential fixed-parameter algorithms and approximation algorithms.

We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an \Omega (w) *\Omega ( w) grid graph as a minor. Thus grid minors suffice to certify that H-minor-free graphs havelarge treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for generalgraphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on H-minor-free graphs for any fixed H. Ourresult has many combinatorial consequences on bidimensionality theory, parameter-treewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial resultshas several algorithmic consequences including subexponential fixed-parameter algorithms and approximation algorithms.

We give a review of a series of techniques and results on the design of subexponential parameterized algorithms for graph problems. The design of such algorithms usually consists of two main steps: first find a branch- (or tree-) decomposition of the input graph whose width is bounded by a sublinear function of the parameter and, second, use this decomposition to solve the problem in time that is single exponential to this bound. The main tool for the first step is Bidimensionality Theory. Here we present the potential, but also the boundaries, of this theory. For the second step, we describe recent techniques, associating the analysis of sub-exponential algorithms to combinatorial bounds related to Catalan numbers. As a result, we have 2 O( √ k) · n O(1) time algorithms for a wide variety of parameterized problems on graphs, where n is the size of the graph and k is the parameter. 1

1 Introduction The recent theory of fixed-parameter algorithms and parameterized complex-ity [13] has attracted much attention in its less than 10 years of existence. In general the goal is to understand when NP-hard problems have algorithms thatare exponential only in a parameter k of the problem instead of the problemsize n. Fixed-parameter algorithms whose running time is polynomial for fixedparameter values--or more precisely

Abstract. This paper surveys the theory of bidimensional graph problems. We summarize the known combinatorial and algorithmic results of this theory, the foundational Graph Minor results on which this theory is based, and the remaining open problems. 1