78 research outputs found
Tree-width and dimension
Over the last 30 years, researchers have investigated connections between
dimension for posets and planarity for graphs. Here we extend this line of
research to the structural graph theory parameter tree-width by proving that
the dimension of a finite poset is bounded in terms of its height and the
tree-width of its cover graph.Comment: Updates on solutions of problems and on bibliograph
First-Fit is Linear on Posets Excluding Two Long Incomparable Chains
A poset is (r + s)-free if it does not contain two incomparable chains of
size r and s, respectively. We prove that when r and s are at least 2, the
First-Fit algorithm partitions every (r + s)-free poset P into at most
8(r-1)(s-1)w chains, where w is the width of P. This solves an open problem of
Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010).Comment: v3: fixed some typo
An Improved Bound for First-Fit on Posets Without Two Long Incomparable Chains
It is known that the First-Fit algorithm for partitioning a poset P into
chains uses relatively few chains when P does not have two incomparable chains
each of size k. In particular, if P has width w then Bosek, Krawczyk, and
Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010) proved an upper bound
of ckw^{2} on the number of chains used by First-Fit for some constant c, while
Joret and Milans (Order, 28(3):455--464, 2011) gave one of ck^{2}w. In this
paper we prove an upper bound of the form ckw. This is best possible up to the
value of c.Comment: v3: referees' comments incorporate
Treedepth vs circumference
The circumference of a graph is the length of a longest cycle in , or if has no cycle. Birmel\'e (2003) showed that the treewidth of agraph is at most its circumference minus one. We strengthen this result for-connected graphs as follows: If is -connected, then its treedepth isat most its circumference. The bound is best possible and improves on anearlier quadratic upper bound due to Marshall and Wood (2015).<br
Hitting all Maximal Independent Sets of a Bipartite Graph
We prove that given a bipartite graph G with vertex set V and an integer k,
deciding whether there exists a subset of V of size k hitting all maximal
independent sets of G is complete for the class Sigma_2^P.Comment: v3: minor chang
Nonrepetitive colourings of planar graphs with O(log n) colours
A vertex colouring of a graph is nonrepetitive if there is no path for which the first half of the path is assigned the same sequence of colours as the second half. The nonrepetitive chromatic number of a graph G is the minimum integer k such that G has a nonrepetitive k-colouring. Whether planar graphs have bounded nonrepetitive chromatic number is one of the most important open problems in the field. Despite this, the best known upper bound is O(n−−√) for n-vertex planar graphs. We prove a O(logn) upper bound
Parameterized vertex deletion problems for hereditary graph classes with a block property
For a class of graphs P, the Bounded P-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether there is a set S of at most k vertices such that each block of G − S has at most d vertices and is in P. We show that when P satisfies a natural hereditary property and is recognizable in polynomial time, Bounded P-Block Vertex Deletion can be solved in time 2O(k log d)nO(1), and this running time cannot be improved to 2o(k log d)nO(1), in general, unless the Exponential Time Hypothesis fails. On the other hand, if P consists of only complete graphs, or only K1,K2, and cycle graphs, then Bounded P-Block Vertex Deletion admits a cknO(1)-time algorithm for some constant c independent of d. We also show that Bounded P-Block Vertex Deletion admits a kernel with O(k2d7) vertices. © Springer-Verlag GmbH Germany 2016
Irreducible triangulations of surfaces with boundary
A triangulation of a surface is irreducible if no edge can be contracted to
produce a triangulation of the same surface. In this paper, we investigate
irreducible triangulations of surfaces with boundary. We prove that the number
of vertices of an irreducible triangulation of a (possibly non-orientable)
surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was
known only for surfaces without boundary (b=0). While our technique yields a
worse constant in the O(.) notation, the present proof is elementary, and
simpler than the previous ones in the case of surfaces without boundary
Assortment optimisation under a general discrete choice model: A tight analysis of revenue-ordered assortments
The assortment problem in revenue management is the problem of deciding which
subset of products to offer to consumers in order to maximise revenue. A simple
and natural strategy is to select the best assortment out of all those that are
constructed by fixing a threshold revenue and then choosing all products
with revenue at least . This is known as the revenue-ordered assortments
strategy. In this paper we study the approximation guarantees provided by
revenue-ordered assortments when customers are rational in the following sense:
the probability of selecting a specific product from the set being offered
cannot increase if the set is enlarged. This rationality assumption, known as
regularity, is satisfied by almost all discrete choice models considered in the
revenue management and choice theory literature, and in particular by random
utility models. The bounds we obtain are tight and improve on recent results in
that direction, such as for the Mixed Multinomial Logit model by
Rusmevichientong et al. (2014). An appealing feature of our analysis is its
simplicity, as it relies only on the regularity condition.
We also draw a connection between assortment optimisation and two pricing
problems called unit demand envy-free pricing and Stackelberg minimum spanning
tree: These problems can be restated as assortment problems under discrete
choice models satisfying the regularity condition, and moreover revenue-ordered
assortments correspond then to the well-studied uniform pricing heuristic. When
specialised to that setting, the general bounds we establish for
revenue-ordered assortments match and unify the best known results on uniform
pricing.Comment: Minor changes following referees' comment
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