Fast Subexponential Algorithm for Non-local Problems on Graphs of Bounded Genus

Abstract

Abstract. We give a general technique for designing fast subexponential algorithms for several graph problems whose instances are restricted to graphs of bounded genus. We use it to obtain time 2 O( √ n) algorithms for a wide family of problems such as Hamiltonian Cycle, Σ-embedded Graph Travelling Salesman Problem, Longest Cycle, and Max Leaf Tree. For our results, we combine planarizing techniques with dynamic programming on special type branch decompositions. Our techniques can also be used to solve parameterized problems. Thus, for example, we show how to find a cycle of length p (or to conclude that there is no such a cycle) on graphs of bounded genus in time 2 O( √ p) · n O(1) .

Graph Minors and Parameterized Algorithm Design

“…There is a quite extended bibliography on how to do fast dynamic programming on graphs of bounded treewidth; as a sample of this, we just mention [5,6,9,11,13,16,22,35,35,36,37,37,38,114,120,120]. t: Bounds are much better for the function t. For most natural graph parameters, it holds that t(k) = O(k) while for some of them, including tw and pw, it holds that t(k) = Θ(k).…”

Dynamic Programming for H-minor-free Graphs

“…Then, for every t ∈ V (T ) we consider the underlying graph G t 0 , and find a branch decomposition of it of width O h (k). This can be done in O(n 3 ) steps by using a standard planarization procedure that cuts the graph along minimum length non-contractible nooses (see [11,13]) and then using the approximation algorithm in [34]. Our next aim is to transform this branch decomposition into a new branch decomposition (T t , τ t ) of G t 0 with the property that for every e ∈ E(T ), there exists a set N of nooses of G t 0 in Σ t such that the vertices of mid(e) ∩ V (G t 0 ) are all met by the nooses in N , in a way that conditions 1 -3 of Definition 2 hold and, moreover, Σ t \ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪ ∪N contains exactly two connected components, such that the graph G e \ A is embedded in the closure of one of them.…”

“…Nevertheless, for such problems one can do better for several classes of sparse graphs. This line of research was initiated in [14] and occupied several researchers in parameterized algorithms design (see also [5,8,11,13,24,33]). The current technology of dynamic programming in graphs of bounded decomposability implies single-exponential parametric dependence for problems encodable by pairings in H-minor free graphs [13] and for problems encodable by packings in graphs embedded in surfaces [33].…”

Approximation algorithms via contraction decomposition

“…Proof: TSP can be solved in graphs of bounded treewidth via dynamic programming; see [DFT06] for a particularly fast running time on graphs of bounded genus. Spanners for TSP in boundedgenus graphs are developed in [Gri00].…”

“…Bounded-genus graphs have been studied extensively in the algorithms community; see, e.g., [CM05,DFHT05b,DHT06,DFT06,FT04,GHT84,Kel06,Moh99]. One attraction of this graph class is that it includes every graph, using a sufficiently large bound on the genus.…”