1,455 research outputs found
Bounds on the maximum multiplicity of some common geometric graphs
We obtain new lower and upper bounds for the maximum multiplicity of some
weighted and, respectively, non-weighted common geometric graphs drawn on n
points in the plane in general position (with no three points collinear):
perfect matchings, spanning trees, spanning cycles (tours), and triangulations.
(i) We present a new lower bound construction for the maximum number of
triangulations a set of n points in general position can have. In particular,
we show that a generalized double chain formed by two almost convex chains
admits {\Omega}(8.65^n) different triangulations. This improves the bound
{\Omega}(8.48^n) achieved by the double zig-zag chain configuration studied by
Aichholzer et al.
(ii) We present a new lower bound of {\Omega}(12.00^n) for the number of
non-crossing spanning trees of the double chain composed of two convex chains.
The previous bound, {\Omega}(10.42^n), stood unchanged for more than 10 years.
(iii) Using a recent upper bound of 30^n for the number of triangulations,
due to Sharir and Sheffer, we show that n points in the plane in general
position admit at most O(68.62^n) non-crossing spanning cycles.
(iv) We derive lower bounds for the number of maximum and minimum weighted
geometric graphs (matchings, spanning trees, and tours). We show that the
number of shortest non-crossing tours can be exponential in n. Likewise, we
show that both the number of longest non-crossing tours and the number of
longest non-crossing perfect matchings can be exponential in n. Moreover, we
show that there are sets of n points in convex position with an exponential
number of longest non-crossing spanning trees. For points in convex position we
obtain tight bounds for the number of longest and shortest tours. We give a
combinatorial characterization of the longest tours, which leads to an O(nlog
n) time algorithm for computing them
An ETH-Tight Exact Algorithm for Euclidean TSP
We study exact algorithms for {\sc Euclidean TSP} in . In the
early 1990s algorithms with running time were presented for
the planar case, and some years later an algorithm with
running time was presented for any . Despite significant interest in
subexponential exact algorithms over the past decade, there has been no
progress on {\sc Euclidean TSP}, except for a lower bound stating that the
problem admits no algorithm unless ETH fails. Up to
constant factors in the exponent, we settle the complexity of {\sc Euclidean
TSP} by giving a algorithm and by showing that a
algorithm does not exist unless ETH fails.Comment: To appear in FOCS 201
Stable marriage and roommates problems with restricted edges: complexity and approximability
In the Stable Marriage and Roommates problems, a set of agents is given, each of them having a strictly ordered preference list over some or all of the other agents. A matching is a set of disjoint pairs of mutually acceptable agents. If any two agents mutually prefer each other to their partner, then they block the matching, otherwise, the matching is said to be stable. We investigate the complexity of finding a solution satisfying additional constraints on restricted pairs of agents. Restricted pairs can be either forced or forbidden. A stable solution must contain all of the forced pairs, while it must contain none of the forbidden pairs.
Dias et al. (2003) gave a polynomial-time algorithm to decide whether such a solution exists in the presence of restricted edges. If the answer is no, one might look for a solution close to optimal. Since optimality in this context means that the matching is stable and satisfies all constraints on restricted pairs, there are two ways of relaxing the constraints by permitting a solution to: (1) be blocked by as few as possible pairs, or (2) violate as few as possible constraints n restricted pairs.
Our main theorems prove that for the (bipartite) Stable Marriage problem, case (1) leads to View the MathML source-hardness and inapproximability results, whilst case (2) can be solved in polynomial time. For non-bipartite Stable Roommates instances, case (2) yields an View the MathML source-hard but (under some cardinality assumptions) 2-approximable problem. In the case of View the MathML source-hard problems, we also discuss polynomially solvable special cases, arising from restrictions on the lengths of the preference lists, or upper bounds on the numbers of restricted pairs
Dimers, Tilings and Trees
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others
we describe a natural equivalence between three planar objects: weighted
bipartite planar graphs; planar Markov chains; and tilings with convex
polygons. This equivalence provides a measure-preserving bijection between
dimer coverings of a weighted bipartite planar graph and spanning trees on the
corresponding Markov chain. The tilings correspond to harmonic functions on the
Markov chain and to ``discrete analytic functions'' on the bipartite graph.
The equivalence is extended to infinite periodic graphs, and we classify the
resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure
Ising Model Observables and Non-Backtracking Walks
This paper presents an alternative proof of the connection between the
partition function of the Ising model on a finite graph and the set of
non-backtracking walks on . The techniques used also give formulas for
spin-spin correlation functions in terms of non-backtracking walks. The main
tools used are Viennot's theory of heaps of pieces and turning numbers on
surfaces.Comment: 33 pages, 11 figures. Typos and errors corrected, exposition
improved, results unchange
Quadri-tilings of the plane
We introduce {\em quadri-tilings} and show that they are in bijection with
dimer models on a {\em family} of graphs arising from rhombus
tilings. Using two height functions, we interpret a sub-family of all
quadri-tilings, called {\em triangular quadri-tilings}, as an interface model
in dimension 2+2. Assigning "critical" weights to edges of , we prove an
explicit expression, only depending on the local geometry of the graph ,
for the minimal free energy per fundamental domain Gibbs measure; this solves a
conjecture of \cite{Kenyon1}. We also show that when edges of are
asymptotically far apart, the probability of their occurrence only depends on
this set of edges. Finally, we give an expression for a Gibbs measure on the
set of {\em all} triangular quadri-tilings whose marginals are the above Gibbs
measures, and conjecture it to be that of minimal free energy per fundamental
domain.Comment: Revised version, minor changes. 30 pages, 13 figure
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