38 research outputs found
Candy Crush is NP-hard
We prove that playing Candy Crush to achieve a given score in a fixed number
of swaps is NP-hard
An Empirical Study of the Manipulability of Single Transferable Voting
Voting is a simple mechanism to combine together the preferences of multiple
agents. Agents may try to manipulate the result of voting by mis-reporting
their preferences. One barrier that might exist to such manipulation is
computational complexity. In particular, it has been shown that it is NP-hard
to compute how to manipulate a number of different voting rules. However,
NP-hardness only bounds the worst-case complexity. Recent theoretical results
suggest that manipulation may often be easy in practice. In this paper, we
study empirically the manipulability of single transferable voting (STV) to
determine if computational complexity is really a barrier to manipulation. STV
was one of the first voting rules shown to be NP-hard. It also appears one of
the harder voting rules to manipulate. We sample a number of distributions of
votes including uniform and real world elections. In almost every election in
our experiments, it was easy to compute how a single agent could manipulate the
election or to prove that manipulation by a single agent was impossible.Comment: To appear in Proceedings of the 19th European Conference on
Artificial Intelligence (ECAI 2010
Allocation in Practice
How do we allocate scarcere sources? How do we fairly allocate costs? These
are two pressing challenges facing society today. I discuss two recent projects
at NICTA concerning resource and cost allocation. In the first, we have been
working with FoodBank Local, a social startup working in collaboration with
food bank charities around the world to optimise the logistics of collecting
and distributing donated food. Before we can distribute this food, we must
decide how to allocate it to different charities and food kitchens. This gives
rise to a fair division problem with several new dimensions, rarely considered
in the literature. In the second, we have been looking at cost allocation
within the distribution network of a large multinational company. This also has
several new dimensions rarely considered in the literature.Comment: To appear in Proc. of 37th edition of the German Conference on
Artificial Intelligence (KI 2014), Springer LNC
Phase Transitions of the Typical Algorithmic Complexity of the Random Satisfiability Problem Studied with Linear Programming
Here we study the NP-complete -SAT problem. Although the worst-case
complexity of NP-complete problems is conjectured to be exponential, there
exist parametrized random ensembles of problems where solutions can typically
be found in polynomial time for suitable ranges of the parameter. In fact,
random -SAT, with as control parameter, can be solved quickly
for small enough values of . It shows a phase transition between a
satisfiable phase and an unsatisfiable phase. For branch and bound algorithms,
which operate in the space of feasible Boolean configurations, the empirically
hardest problems are located only close to this phase transition. Here we study
-SAT () and the related optimization problem MAX-SAT by a linear
programming approach, which is widely used for practical problems and allows
for polynomial run time. In contrast to branch and bound it operates outside
the space of feasible configurations. On the other hand, finding a solution
within polynomial time is not guaranteed. We investigated several variants like
including artificial objective functions, so called cutting-plane approaches,
and a mapping to the NP-complete vertex-cover problem. We observed several
easy-hard transitions, from where the problems are typically solvable (in
polynomial time) using the given algorithms, respectively, to where they are
not solvable in polynomial time. For the related vertex-cover problem on random
graphs these easy-hard transitions can be identified with structural properties
of the graphs, like percolation transitions. For the present random -SAT
problem we have investigated numerous structural properties also exhibiting
clear transitions, but they appear not be correlated to the here observed
easy-hard transitions. This renders the behaviour of random -SAT more
complex than, e.g., the vertex-cover problem.Comment: 11 pages, 5 figure
A multiobjective optimization approach to statistical mechanics
Optimization problems have been the subject of statistical physics
approximations. A specially relevant and general scenario is provided by
optimization methods considering tradeoffs between cost and efficiency, where
optimal solutions involve a compromise between both. The theory of Pareto (or
multi objective) optimization provides a general framework to explore these
problems and find the space of possible solutions compatible with the
underlying tradeoffs, known as the {\em Pareto front}. Conflicts between
constraints can lead to complex landscapes of Pareto optimal solutions with
interesting implications in economy, engineering, or evolutionary biology.
Despite their disparate nature, here we show how the structure of the Pareto
front uncovers profound universal features that can be understood in the
context of thermodynamics. In particular, our study reveals that different
fronts are connected to different classes of phase transitions, which we can
define robustly, along with critical points and thermodynamic potentials. These
equivalences are illustrated with classic thermodynamic examples.Comment: 14 pages, 8 figure
Phase transitions in project scheduling.
The analysis of the complexity of combinatorial optimization problems has led to the distinction between problems which are solvable in a polynomially bounded amount of time (classified in P) and problems which are not (classified in NP). This implies that the problems in NP are hard to solve whereas the problems in P are not. However, this analysis is based on worst-case scenarios. The fact that a decision problem is shown to be NP-complete or the fact that an optimization problem is shown to be NP-hard implies that, in the worst case, solving it is very hard. Recent computational results obtained with a well known NP-hard problem, namely the resource-constrained project scheduling problem, indicate that many instances are actually easy to solve. These results are in line with those recently obtained by researchers in the area of artificial intelligence, which show that many NP-complete problemsexhibit so-called phase transitions, resulting in a sudden and dramatic change of computational complexity based on one or more order parameters that are characteristic of the system as a whole. In this paper we provide evidence for the existence of phase transitions in various resource-constrained project scheduling problems. We discuss the use of network complexity measures and resource parameters as potential order parameters. We show that while the network complexity measures seem to reveal continuous easy-hard or hard-easy phase-transitions, the resource parameters exhibit an easy-hard-easy transition behaviour.Networks; Problems; Scheduling; Algorithms;
Phase Transitions and Backbones of the Asymmetric Traveling Salesman Problem
In recent years, there has been much interest in phase transitions of
combinatorial problems. Phase transitions have been successfully used to
analyze combinatorial optimization problems, characterize their typical-case
features and locate the hardest problem instances. In this paper, we study
phase transitions of the asymmetric Traveling Salesman Problem (ATSP), an
NP-hard combinatorial optimization problem that has many real-world
applications. Using random instances of up to 1,500 cities in which intercity
distances are uniformly distributed, we empirically show that many properties
of the problem, including the optimal tour cost and backbone size, experience
sharp transitions as the precision of intercity distances increases across a
critical value. Our experimental results on the costs of the ATSP tours and
assignment problem agree with the theoretical result that the asymptotic cost
of assignment problem is pi ^2 /6 the number of cities goes to infinity. In
addition, we show that the average computational cost of the well-known
branch-and-bound subtour elimination algorithm for the problem also exhibits a
thrashing behavior, transitioning from easy to difficult as the distance
precision increases. These results answer positively an open question regarding
the existence of phase transitions in the ATSP, and provide guidance on how
difficult ATSP problem instances should be generated