52 research outputs found
The Complexity of Escaping Labyrinths and Enchanted Forests
The board games The aMAZEing Labyrinth (or simply Labyrinth for short) and Enchanted Forest published by Ravensburger are seemingly simple family games.
In Labyrinth, the players move though a labyrinth in order to collect specific items. To do so, they shift the tiles making up the labyrinth in order to open up new paths (and, at the same time, close paths for their opponents). We show that, even without any opponents, determining a shortest path (i.e., a path using the minimum possible number of turns) to the next desired item in the labyrinth is strongly NP-hard. Moreover, we show that, when competing with another player, deciding whether there exists a strategy that guarantees to reach one\u27s next item faster than one\u27s opponent is PSPACE-hard.
In Enchanted Forest, items are hidden under specific trees and the objective of the players is to report their locations to the king in his castle. Movements are performed by rolling two dice, resulting in two numbers of fields one has to move, where each of the two movements must be executed consecutively in one direction (but the player can choose the order in which the two movements are performed). Here, we provide an efficient polynomial-time algorithm for computing a shortest path between two fields on the board for a given sequence of die rolls, which also has implications for the complexity of problems the players face in the game when future die rolls are unknown
Complexity of the Temporal Shortest Path Interdiction Problem
In the shortest path interdiction problem, an interdictor aims to remove arcs of total cost at most a given budget from a directed graph with given arc costs and traversal times such that the length of a shortest s-t-path is maximized. For static graphs, this problem is known to be strongly NP-hard, and it has received considerable attention in the literature.
While the shortest path problem is one of the most fundamental and well-studied problems also for temporal graphs, the shortest path interdiction problem has not yet been formally studied on temporal graphs, where common definitions of a "shortest path" include: latest start path (path with maximum start time), earliest arrival path (path with minimum arrival time), shortest duration path (path with minimum traveling time including waiting times at nodes), and shortest traversal path (path with minimum traveling time not including waiting times at nodes).
In this paper, we analyze the complexity of the shortest path interdiction problem on temporal graphs with respect to all four definitions of a shortest path mentioned above. Even though the shortest path interdiction problem on static graphs is known to be strongly NP-hard, we show that the latest start and the earliest arrival path interdiction problems on temporal graphs are polynomial-time solvable. For the shortest duration and shortest traversal path interdiction problems, however, we show strong NP-hardness, but we obtain polynomial-time algorithms for these problems on extension-parallel temporal graphs
Efficiently Constructing Convex Approximation Sets in Multiobjective Optimization Problems
Convex approximation sets for multiobjective optimization problems are a
well-studied relaxation of the common notion of approximation sets. Instead of
approximating each image of a feasible solution by the image of some solution
in the approximation set up to a multiplicative factor in each component, a
convex approximation set only requires this multiplicative approximation to be
achieved by some convex combination of finitely many images of solutions in the
set. This makes convex approximation sets efficiently computable for a wide
range of multiobjective problems - even for many problems for which (classic)
approximations sets are hard to compute.
In this article, we propose a polynomial-time algorithm to compute convex
approximation sets that builds upon an exact or approximate algorithm for the
weighted sum scalarization and is, therefore, applicable to a large variety of
multiobjective optimization problems. The provided convex approximation quality
is arbitrarily close to the approximation quality of the underlying algorithm
for the weighted sum scalarization. In essence, our algorithm can be
interpreted as an approximate variant of the dual variant of Benson's Outer
Approximation Algorithm. Thus, in contrast to existing convex approximation
algorithms from the literature, information on solutions obtained during the
approximation process is utilized to significantly reduce both the practical
running time and the cardinality of the returned solution sets while still
guaranteeing the same worst-case approximation quality. We underpin these
advantages by the first comparison of all existing convex approximation
algorithms on several instances of the triobjective knapsack problem and the
triobjective symmetric metric traveling salesman problem
Integrated Planning in Hospitals: A Review
Efficient planning of scarce resources in hospitals is a challenging task for
which a large variety of Operations Research and Management Science approaches
have been developed since the 1950s. While efficient planning of single
resources such as operating rooms, beds, or specific types of staff can already
lead to enormous efficiency gains, integrated planning of several resources has
been shown to hold even greater potential, and a large number of integrated
planning approaches have been presented in the literature over the past
decades.
This paper provides the first literature review that focuses specifically on
the Operations Research and Management Science literature related to integrated
planning of different resources in hospitals. We collect the relevant
literature and analyze it regarding different aspects such as uncertainty
modeling and the use of real-life data. Several cross comparisons reveal
interesting insights concerning, e.g., relations between the modeling and
solution methods used and the practical implementation of the approaches
developed. Moreover, we provide a high-level taxonomy for classifying different
resource-focused integration approaches and point out gaps in the literature as
well as promising directions for future research
Using Scalarizations for the Approximation of Multiobjective Optimization Problems: Towards a General Theory
We study the approximation of general multiobjective optimization problems
with the help of scalarizations. Existing results state that multiobjective
minimization problems can be approximated well by norm-based scalarizations.
However, for multiobjective maximization problems, only impossibility results
are known so far. Countering this, we show that all multiobjective optimization
problems can, in principle, be approximated equally well by scalarizations. In
this context, we introduce a transformation theory for scalarizations that
establishes the following: Suppose there exists a scalarization that yields an
approximation of a certain quality for arbitrary instances of multiobjective
optimization problems with a given decomposition specifying which objective
functions are to be minimized / maximized. Then, for each other decomposition,
our transformation yields another scalarization that yields the same
approximation quality for arbitrary instances of problems with this other
decomposition. In this sense, the existing results about the approximation via
scalarizations for minimization problems carry over to any other objective
decomposition -- in particular, to maximization problems -- when suitably
adapting the employed scalarization.
We further provide necessary and sufficient conditions on a scalarization
such that its optimal solutions achieve a constant approximation quality. We
give an upper bound on the best achievable approximation quality that applies
to general scalarizations and is tight for the majority of norm-based
scalarizations applied in the context of multiobjective optimization. As a
consequence, none of these norm-based scalarizations can induce approximation
sets for optimization problems with maximization objectives, which unifies and
generalizes the existing impossibility results concerning the approximation of
maximization problems
Approximating Multiobjective Optimization Problems: How exact can you be?
It is well known that, under very weak assumptions, multiobjective
optimization problems admit -approximation
sets (also called -Pareto sets) of polynomial cardinality (in the
size of the instance and in ). While an approximation
guarantee of for any is the best one can expect
for singleobjective problems (apart from solving the problem to optimality),
even better approximation guarantees than
can be considered in the multiobjective case since the approximation might be
exact in some of the objectives.
Hence, in this paper, we consider partially exact approximation sets that
require to approximate each feasible solution exactly, i.e., with an
approximation guarantee of , in some of the objectives while still obtaining
a guarantee of in all others. We characterize the types of
polynomial-cardinality, partially exact approximation sets that are guaranteed
to exist for general multiobjective optimization problems. Moreover, we study
minimum-cardinality partially exact approximation sets concerning (weak)
efficiency of the contained solutions and relate their cardinalities to the
minimum cardinality of a -approximation
set
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