114,783 research outputs found
Search Problems in Trees with Symmetries: Near Optimal Traversal Strategies for Individualization-Refinement Algorithms
We define a search problem on trees that closely captures the backtracking behavior of all current practical graph isomorphism algorithms. Given two trees with colored leaves, the goal is to find two leaves of matching color, one in each of the trees. The trees are subject to an invariance property which promises that for every pair of leaves of equal color there must be a symmetry (or an isomorphism) that maps one leaf to the other.
We describe a randomized algorithm with errors for which the number of visited nodes is quasilinear in the square root of the size of the smaller of the two trees. For inputs of bounded degree, we develop a Las Vegas algorithm with a similar running time.
We prove that these results are optimal up to logarithmic factors. For this, we show a lower bound for randomized algorithms on inputs of bounded degree that is the square root of the tree sizes. For inputs of unbounded degree, we show a linear lower bound for Las Vegas algorithms. For deterministic algorithms we can prove a linear bound even for inputs of bounded degree. This shows why randomized algorithms outperform deterministic ones.
Our results explain why the randomized "breadth-first with intermixed experimental path" search strategy of the isomorphism tool Traces (Piperno 2008) is often superior to the depth-first search strategy of other tools such as nauty (McKay 1977) or bliss (Junttila, Kaski 2007). However, our algorithm also provides a new traversal strategy, which is theoretically near optimal and which has better worst case behavior than traversal strategies that have previously been used
Real-time Planning as Decision-making Under Uncertainty
In real-time planning, an agent must select the next action to take within a fixed time bound.
Many popular real-time heuristic search methods approach this by expanding nodes using time-limited A* and selecting the action leading toward the frontier node with the lowest f value. In this thesis, we reconsider real-time planning as a problem of decision-making under uncertainty. We treat heuristic values as uncertain evidence and we explore several backup methods for aggregating this evidence. We then propose a novel lookahead strategy that expands nodes to minimize risk, the expected regret in case a non-optimal action is chosen. We evaluate these methods in a simple synthetic benchmark and the sliding tile puzzle and find that they outperform previous methods. This work illustrates how uncertainty can arise even when solving deterministic planning problems, due to the inherent ignorance of time-limited search algorithms about those portions of the state space that they have not computed, and how an agent can benefit from explicitly meta-reasoning about this uncertainty
Search for an Immobile Hider on a Stochastic Network
Harry hides on an edge of a graph and does not move from there. Sally,
starting from a known origin, tries to find him as soon as she can. Harry's
goal is to be found as late as possible. At any given time, each edge of the
graph is either active or inactive, independently of the other edges, with a
known probability of being active. This situation can be modeled as a zero-sum
two-person stochastic game. We show that the game has a value and we provide
upper and lower bounds for this value. Finally, by generalizing optimal
strategies of the deterministic case, we provide more refined results for trees
and Eulerian graphs.Comment: 28 pages, 9 figure
Testing systems of identical components
We consider the problem of testing sequentially the components of a multi-component reliability system in order to figure out the state of the system via costly tests. In particular, systems with identical components are considered. The notion of lexicographically large binary decision trees is introduced and a heuristic algorithm based on that notion is proposed. The performance of the heuristic algorithm is demonstrated by computational results, for various classes of functions. In particular, in all 200 random cases where the underlying function is a threshold function, the proposed heuristic produces optimal solutions
Optimal sampling strategies for multiscale stochastic processes
In this paper, we determine which non-random sampling of fixed size gives the
best linear predictor of the sum of a finite spatial population. We employ
different multiscale superpopulation models and use the minimum mean-squared
error as our optimality criterion. In multiscale superpopulation tree models,
the leaves represent the units of the population, interior nodes represent
partial sums of the population, and the root node represents the total sum of
the population. We prove that the optimal sampling pattern varies dramatically
with the correlation structure of the tree nodes. While uniform sampling is
optimal for trees with ``positive correlation progression'', it provides the
worst possible sampling with ``negative correlation progression.'' As an
analysis tool, we introduce and study a class of independent innovations trees
that are of interest in their own right. We derive a fast water-filling
algorithm to determine the optimal sampling of the leaves to estimate the root
of an independent innovations tree.Comment: Published at http://dx.doi.org/10.1214/074921706000000509 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
Learning optimization models in the presence of unknown relations
In a sequential auction with multiple bidding agents, it is highly
challenging to determine the ordering of the items to sell in order to maximize
the revenue due to the fact that the autonomy and private information of the
agents heavily influence the outcome of the auction.
The main contribution of this paper is two-fold. First, we demonstrate how to
apply machine learning techniques to solve the optimal ordering problem in
sequential auctions. We learn regression models from historical auctions, which
are subsequently used to predict the expected value of orderings for new
auctions. Given the learned models, we propose two types of optimization
methods: a black-box best-first search approach, and a novel white-box approach
that maps learned models to integer linear programs (ILP) which can then be
solved by any ILP-solver. Although the studied auction design problem is hard,
our proposed optimization methods obtain good orderings with high revenues.
Our second main contribution is the insight that the internal structure of
regression models can be efficiently evaluated inside an ILP solver for
optimization purposes. To this end, we provide efficient encodings of
regression trees and linear regression models as ILP constraints. This new way
of using learned models for optimization is promising. As the experimental
results show, it significantly outperforms the black-box best-first search in
nearly all settings.Comment: 37 pages. Working pape
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