404 research outputs found
On the complexity of computing the capacity of codes that avoid forbidden difference patterns
We consider questions related to the computation of the capacity of codes
that avoid forbidden difference patterns. The maximal number of -bit
sequences whose pairwise differences do not contain some given forbidden
difference patterns increases exponentially with . The exponent is the
capacity of the forbidden patterns, which is given by the logarithm of the
joint spectral radius of a set of matrices constructed from the forbidden
difference patterns. We provide a new family of bounds that allows for the
approximation, in exponential time, of the capacity with arbitrary high degree
of accuracy. We also provide a polynomial time algorithm for the problem of
determining if the capacity of a set is positive, but we prove that the same
problem becomes NP-hard when the sets of forbidden patterns are defined over an
extended set of symbols. Finally, we prove the existence of extremal norms for
the sets of matrices arising in the capacity computation. This result makes it
possible to apply a specific (even though non polynomial) approximation
algorithm. We illustrate this fact by computing exactly the capacity of codes
that were only known approximately.Comment: 7 pages. Submitted to IEEE Trans. on Information Theor
A Spectral Approach to Consecutive Pattern-Avoiding Permutations
We consider the problem of enumerating permutations in the symmetric group on
elements which avoid a given set of consecutive pattern , and in
particular computing asymptotics as tends to infinity. We develop a general
method which solves this enumeration problem using the spectral theory of
integral operators on , where the patterns in has length
. Kre\u{\i}n and Rutman's generalization of the Perron--Frobenius theory
of non-negative matrices plays a central role. Our methods give detailed
asymptotic expansions and allow for explicit computation of leading terms in
many cases. As a corollary to our results, we settle a conjecture of Warlimont
on asymptotics for the number of permutations avoiding a consecutive pattern.Comment: a reference is added; corrected typos; to appear in Journal of
Combinatoric
The Hopf algebra of diagonal rectangulations
We define and study a combinatorial Hopf algebra dRec with basis elements
indexed by diagonal rectangulations of a square. This Hopf algebra provides an
intrinsic combinatorial realization of the Hopf algebra tBax of twisted Baxter
permutations, which previously had only been described extrinsically as a sub
Hopf algebra of the Malvenuto-Reutenauer Hopf algebra of permutations. We
describe the natural lattice structure on diagonal rectangulations, analogous
to the Tamari lattice on triangulations, and observe that diagonal
rectangulations index the vertices of a polytope analogous to the
associahedron. We give an explicit bijection between twisted Baxter
permutations and the better-known Baxter permutations, and describe the
resulting Hopf algebra structure on Baxter permutations.Comment: Very minor changes from version 1, in response to comments by
referees. This is the final version, to appear in JCTA. 43 pages, 17 figure
New and simple algorithms for stable flow problems
Stable flows generalize the well-known concept of stable matchings to markets
in which transactions may involve several agents, forwarding flow from one to
another. An instance of the problem consists of a capacitated directed network,
in which vertices express their preferences over their incident edges. A
network flow is stable if there is no group of vertices that all could benefit
from rerouting the flow along a walk.
Fleiner established that a stable flow always exists by reducing it to the
stable allocation problem. We present an augmenting-path algorithm for
computing a stable flow, the first algorithm that achieves polynomial running
time for this problem without using stable allocation as a black-box
subroutine. We further consider the problem of finding a stable flow such that
the flow value on every edge is within a given interval. For this problem, we
present an elegant graph transformation and based on this, we devise a simple
and fast algorithm, which also can be used to find a solution to the stable
marriage problem with forced and forbidden edges.
Finally, we study the stable multicommodity flow model introduced by
Kir\'{a}ly and Pap. The original model is highly involved and allows for
commodity-dependent preference lists at the vertices and commodity-specific
edge capacities. We present several graph-based reductions that show
equivalence to a significantly simpler model. We further show that it is
NP-complete to decide whether an integral solution exists
Polyhedral geometry of Phylogenetic Rogue Taxa
It is well known among phylogeneticists that adding an extra taxon (e.g.
species) to a data set can alter the structure of the optimal phylogenetic tree
in surprising ways. However, little is known about this "rogue taxon" effect.
In this paper we characterize the behavior of balanced minimum evolution (BME)
phylogenetics on data sets of this type using tools from polyhedral geometry.
First we show that for any distance matrix there exist distances to a "rogue
taxon" such that the BME-optimal tree for the data set with the new taxon does
not contain any nontrivial splits (bipartitions) of the optimal tree for the
original data. Second, we prove a theorem which restricts the topology of
BME-optimal trees for data sets of this type, thus showing that a rogue taxon
cannot have an arbitrary effect on the optimal tree. Third, we construct
polyhedral cones computationally which give complete answers for BME rogue
taxon behavior when our original data fits a tree on four, five, and six taxa.
We use these cones to derive sufficient conditions for rogue taxon behavior for
four taxa, and to understand the frequency of the rogue taxon effect via
simulation.Comment: In this version, we add quartet distances and fix Table 4
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
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