465 research outputs found
Computational problems without computation
Problemen uit de discrete wiskunde lijken op het eerste gezicht vaak erg simpel. Ze kunnen meestal gemakkelijk en zonder gebruik te maken van wiskundige begrippen worden geformuleerd. Toch komt het vaak voor dat zoân ogenschijnlijk eenvoudig probleem nog open is of dat er, zoals bij het handelsreizigersprobleem, wel een oplossing gegeven kan worden,maar alleen een die onbruikbaar is omdat de rekentijd bij grotere getallen te snel groeit. In dit artikel, gebaseerd op zijn voordracht op het NMC 2002, kijkt Gerhard Woeginger naar de tegenovergestelde situatie. Hij introduceert allerlei discrete\ud
problemen die onoplosbaar lijken, maar waarvoor er een simpele oplossing bestaat
Tight bounds for break minimization
We consider round-robin sports tournaments with n teams and n â 1 rounds. We construct an infinite family of opponent schedules for which every home-away assignment induces at least 1/4 n(nâ2) breaks. This construction establishes a matching lower bound for a corresponding upper bound from the literature
The Open Shop Scheduling Problem
We discuss the computational complexity, the approximability, the algorithmics and the combinatorics of the open shop scheduling problem. We summarize the most important results from the literature and explain their main ideas, we sketch the most beautiful proofs, and we also list a number of open problems
Are there any nicely structured preference~profiles~nearby?
We investigate the problem of deciding whether a given preference profile is
close to having a certain nice structure, as for instance single-peaked,
single-caved, single-crossing, value-restricted, best-restricted,
worst-restricted, medium-restricted, or group-separable profiles. We measure
this distance by the number of voters or alternatives that have to be deleted
to make the profile a nicely structured one. Our results classify the problem
variants with respect to their computational complexity, and draw a clear line
between computationally tractable (polynomial-time solvable) and
computationally intractable (NP-hard) questions
Geometric versions of the 3-dimensional assignment problem under general norms
We discuss the computational complexity of special cases of the 3-dimensional
(axial) assignment problem where the elements are points in a Cartesian space
and where the cost coefficients are the perimeters of the corresponding
triangles measured according to a certain norm. (All our results also carry
over to the corresponding special cases of the 3-dimensional matching problem.)
The minimization version is NP-hard for every norm, even if the underlying
Cartesian space is 2-dimensional. The maximization version is polynomially
solvable, if the dimension of the Cartesian space is fixed and if the
considered norm has a polyhedral unit ball. If the dimension of the Cartesian
space is part of the input, the maximization version is NP-hard for every
norm; in particular the problem is NP-hard for the Manhattan norm and the
Maximum norm which both have polyhedral unit balls.Comment: 21 pages, 9 figure
Planar 3-dimensional assignment problems with Monge-like cost arrays
Given an cost array we consider the problem -P3AP
which consists in finding pairwise disjoint permutations
of such that
is minimized. For the case
the planar 3-dimensional assignment problem P3AP results.
Our main result concerns the -P3AP on cost arrays that are layered
Monge arrays. In a layered Monge array all matrices that result
from fixing the third index are Monge matrices. We prove that the -P3AP
and the P3AP remain NP-hard for layered Monge arrays. Furthermore, we show that
in the layered Monge case there always exists an optimal solution of the
-3PAP which can be represented as matrix with bandwidth . This
structural result allows us to provide a dynamic programming algorithm that
solves the -P3AP in polynomial time on layered Monge arrays when is
fixed.Comment: 16 pages, appendix will follow in v
The dynamics of power laws: Fitness and aging in preferential attachment trees
Continuous-time branching processes describe the evolution of a population
whose individuals generate a random number of children according to a birth
process. Such branching processes can be used to understand preferential
attachment models in which the birth rates are linear functions. We are
motivated by citation networks, where power-law citation counts are observed as
well as aging in the citation patterns. To model this, we introduce fitness and
age-dependence in these birth processes. The multiplicative fitness moderates
the rate at which children are born, while the aging is integrable, so that
individuals receives a finite number of children in their lifetime. We show the
existence of a limiting degree distribution for such processes. In the
preferential attachment case, where fitness and aging are absent, this limiting
degree distribution is known to have power-law tails. We show that the limiting
degree distribution has exponential tails for bounded fitnesses in the presence
of integrable aging, while the power-law tail is restored when integrable aging
is combined with fitness with unbounded support with at most exponential tails.
In the absence of integrable aging, such processes are explosive.Comment: 41 pages, 10 figure
Charlemagne's challenge: the periodic latency problem.
Latency problems are characterized by their focus on minimizing the waiting time for all clients. We study periodic latency problems, a non-trivial extension of standard latency problems. In a periodic latency problem each client has to be visited regularly: there is a server traveling at unit speed, and there is a set of n clients with given positions. The server must visit the clients over and over again, subject to the constraint that successive visits to client i are at most qi time units away from each other. We investigate two main problems. In problem PLPP the goal is to find a repeatable route for the server visiting as many clients as possible, without violating their qi's. In problem PLP the goal is to minimize the number of servers needed to serve all clients. In dependence on the topol- ogy of the underlying network, we derive polynomial-time algorithms or hardness results for these two problems. Our results draw sharp separation lines between easy and hard cases.Latency problem; Periodicity; Complexity;
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