Results 1  10
of
45
Approximation Algorithms and Online Mechanisms for Item Pricing
 IN ACM CONFERENCE ON ELECTRONIC COMMERCE
, 2005
"... We present approximation and online algorithms for a number of problems of pricing items for sale so as to maximize seller's revenue in an unlimited supply setting. Our first result is an O(k)approximation algorithm for pricing items to singleminded bidders who each want at most k items. This impr ..."
Abstract

Cited by 58 (9 self)
 Add to MetaCart
We present approximation and online algorithms for a number of problems of pricing items for sale so as to maximize seller's revenue in an unlimited supply setting. Our first result is an O(k)approximation algorithm for pricing items to singleminded bidders who each want at most k items. This improves over recent independent work of Briest and Krysta [6] who achieve an O(k ) bound. For the case k = 2, where we obtain a 4approximation, this can be viewed as the following graph vertex pricing problem: given a (multi) graph G with valuations w e on the edges, find prices p i 0 for the vertices to maximize (p i + p j ) .
Locally excluding a minor
, 2007
"... We introduce the concept of locally excluded minors. Graph classes locally excluding a minor generalise the concept of excluded minor classes but also of graph classes with bounded local treewidth and graph classes with bounded expansion. We show that firstorder modelchecking is fixedparameter t ..."
Abstract

Cited by 34 (13 self)
 Add to MetaCart
We introduce the concept of locally excluded minors. Graph classes locally excluding a minor generalise the concept of excluded minor classes but also of graph classes with bounded local treewidth and graph classes with bounded expansion. We show that firstorder modelchecking is fixedparameter tractable on any class of graphs locally excluding a minor. This strictly generalises analogous results by Flum and Grohe on excluded minor classes and Frick and Grohe on classes with bounded local treewidth. As an important consequence of the proof we obtain fixedparameter algorithms for problems such as dominating or independent set on graph classes excluding a minor, where now the parameter is the size of the dominating set and the excluded minor. We also study graph classes with excluded minors, where the minor may grow slowly with the size of the graphs and show that again, firstorder modelchecking is fixedparameter tractable on any such class of graphs.
The bidimensionality Theory and Its Algorithmic Applications
 Computer Journal
, 2005
"... This paper surveys the theory of bidimensionality. This theory characterizes a broad range of graph problems (‘bidimensional’) that admit efficient approximate or fixedparameter solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, boundedgenus graphs and gra ..."
Abstract

Cited by 29 (1 self)
 Add to MetaCart
This paper surveys the theory of bidimensionality. This theory characterizes a broad range of graph problems (‘bidimensional’) that admit efficient approximate or fixedparameter solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, boundedgenus 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 highlevel 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.
Parameterized complexity and approximation algorithms
 Comput. J
, 2006
"... Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We ..."
Abstract

Cited by 28 (1 self)
 Add to MetaCart
Approximation algorithms and parameterized complexity are usually considered to be two separate ways of dealing with hard algorithmic problems. In this paper, our aim is to investigate how these two fields can be combined to achieve better algorithms than what any of the two theories could offer. We discuss the different ways parameterized complexity can be extended to approximation algorithms, survey results of this type and propose directions for future research. 1.
Every minorclosed property of sparse graphs is testable
, 2007
"... Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a sim ..."
Abstract

Cited by 26 (3 self)
 Add to MetaCart
Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a similar result is proved for any minorclosed property of bounded degree graphs. As an immediate corollary of the above result we infer that many well studied graph properties, like being planar, outerplanar, seriesparallel, bounded genus, bounded treewidth and several others, are testable with a constant number of queries. None of these properties was previously known to be testable even with o(n) queries. 1
Linearity of Grid Minors in Treewidth with Applications through Bidimensionality
, 2005
"... We prove that any Hminorfree 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 Hminorfree graphs havelarge treewidth, up to constant factors. This strong relationship was previously known for the special cas ..."
Abstract

Cited by 21 (1 self)
 Add to MetaCart
We prove that any Hminorfree 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 Hminorfree graphs havelarge treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and boundedgenus 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 Hminorfree graphs for any fixed H. Ourresult has many combinatorial consequences on bidimensionality theory, parametertreewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial resultshas several algorithmic consequences including subexponential fixedparameter algorithms and approximation algorithms.
Digraph measures: Kelly decompositions, games and orderings
"... We consider various wellknown, equivalent complexity measures for graphs such as elimination orderings, ktrees and cops and robber games and study their natural translations to digraphs. We show that on digraphs all these measures are also equivalent and induce a natural connectivity measure. We i ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
We consider various wellknown, equivalent complexity measures for graphs such as elimination orderings, ktrees and cops and robber games and study their natural translations to digraphs. We show that on digraphs all these measures are also equivalent and induce a natural connectivity measure. We introduce a decomposition for digraphs and an associated width, Kellywidth, which is equivalent to the aforementioned measure. We demonstrate its usefulness by exhibiting a number of potential applications including polynomialtime algorithms for NPcomplete problems on graphs of bounded Kellywidth, and complexity analysis of asymmetric matrix factorization. Finally, we compare the new width to other known decompositions of digraphs.
Planar decompositions and the crossing number of graphs with an excluded minor
 IN GRAPH DRAWING 2006; LECTURE NOTES IN COMPUTER SCIENCE 4372
, 2007
"... Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar ..."
Abstract

Cited by 14 (1 self)
 Add to MetaCart
Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded treewidth has linear convex crossing number, and every K3,3minorfree graph with bounded degree has linear rectilinear crossing number.
Algorithmic MetaTheorems
 In M. Grohe and R. Neidermeier eds, International Workshop on Parameterized and Exact Computation (IWPEC), volume 5018 of LNCS
, 2008
"... Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of ..."
Abstract

Cited by 14 (2 self)
 Add to MetaCart
Algorithmic metatheorems are algorithmic results that apply to a whole range of problems, instead of addressing just one specific problem. This kind of theorems are often stated relative to a certain class of graphs, so the general form of a meta theorem reads “every problem in a certain class C of problems can be solved efficiently on every graph satisfying a certain property P”. A particularly well known example of a metatheorem is Courcelle’s theorem that every decision problem definable in monadic secondorder logic (MSO) can be decided in linear time on any class of graphs of bounded treewidth [1]. The class C of problems can be defined in a number of different ways. One option is to state combinatorial or algorithmic criteria of problems in C. For instance, Demaine, Hajiaghayi and Kawarabayashi [5] showed that every minimisation problem that can be solved efficiently on graph classes of bounded treewidth and for which approximate solutions can be computed efficiently from solutions of certain subinstances, have a PTAS on any class of graphs excluding a fixed minor. While this gives a strong unifying explanation for PTAS of many
Approximation schemes for firstorder definable optimisation problems
 In Proc. LICS’06
, 2006
"... Let ϕ(X) be a firstorder formula in the language of graphs that has a free set variable X, and assume that X only occurs positively in ϕ(X). Then a natural minimisation problem associated with ϕ(X) is to find, in a given graph G, a vertex set S of minimum size such that G satisfies ϕ(S). Similarly, ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
Let ϕ(X) be a firstorder formula in the language of graphs that has a free set variable X, and assume that X only occurs positively in ϕ(X). Then a natural minimisation problem associated with ϕ(X) is to find, in a given graph G, a vertex set S of minimum size such that G satisfies ϕ(S). Similarly, if X only occurs negatively in ϕ(X), then ϕ(X) defines a maximisation problem. Many wellknown optimisation problems are firstorder definable in this sense, for example, MINIMUM DOMINATING SET or MAXIMUM INDEPENDENT SET. We prove that for each class C of graphs with excluded minors, in particular for each class of planar graphs, the restriction of a firstorder definable optimisation problem to the class C has a polynomial time approximation scheme. A crucial building block of the proof of this approximability result is a version of Gaifman’s locality theorem for formulas positive in a set variable. This result may be of independent interest. 1.