322 research outputs found
A faster tree-decomposition based algorithm for counting linear extensions
We consider the problem of counting the linear extensions of an n-element poset whose cover graph has treewidth at most t. We show that the problem can be solved in time Õ(nt+3), where Õ suppresses logarithmic factors. Our algorithm is based on fast multiplication of multivariate polynomials, and so differs radically from a previous Õ(nt+4)-time inclusion–exclusion algorithm. We also investigate the algorithm from a practical point of view. We observe that the running time is not well characterized by the parameters n and t alone, fixing of which leaves large variance in running times due to uncontrolled features of the selected optimal-width tree decomposition. For selecting an efficient tree decomposition we adopt the method of empirical hardness models, and show that it typically enables picking a tree decomposition that is significantly more efficient than a random optimal-width tree decomposition. © Kustaa Kangas, Mikko Koivisto, and Sami Salonen; licensed under Creative Commons License CC-BY.Peer reviewe
Finding the bandit in a graph: Sequential search-and-stop
We consider the problem where an agent wants to find a hidden object that is
randomly located in some vertex of a directed acyclic graph (DAG) according to
a fixed but possibly unknown distribution. The agent can only examine vertices
whose in-neighbors have already been examined. In this paper, we address a
learning setting where we allow the agent to stop before having found the
object and restart searching on a new independent instance of the same problem.
Our goal is to maximize the total number of hidden objects found given a time
budget. The agent can thus skip an instance after realizing that it would spend
too much time on it. Our contributions are both to the search theory and
multi-armed bandits. If the distribution is known, we provide a quasi-optimal
and efficient stationary strategy. If the distribution is unknown, we
additionally show how to sequentially approximate it and, at the same time, act
near-optimally in order to collect as many hidden objects as possible.Comment: in International Conference on Artificial Intelligence and Statistics
(AISTATS 2019), April 2019, Naha, Okinawa, Japa
Discrete Convex Functions on Graphs and Their Algorithmic Applications
The present article is an exposition of a theory of discrete convex functions
on certain graph structures, developed by the author in recent years. This
theory is a spin-off of discrete convex analysis by Murota, and is motivated by
combinatorial dualities in multiflow problems and the complexity classification
of facility location problems on graphs. We outline the theory and algorithmic
applications in combinatorial optimization problems
Hardness and Approximation of Submodular Minimum Linear Ordering Problems
The minimum linear ordering problem (MLOP) generalizes well-known
combinatorial optimization problems such as minimum linear arrangement and
minimum sum set cover. MLOP seeks to minimize an aggregated cost due
to an ordering of the items (say ), i.e., , where is the set of items
mapped by to indices . Despite an extensive literature on MLOP
variants and approximations for these, it was unclear whether the graphic
matroid MLOP was NP-hard. We settle this question through non-trivial
reductions from mininimum latency vertex cover and minimum sum vertex cover
problems. We further propose a new combinatorial algorithm for approximating
monotone submodular MLOP, using the theory of principal partitions. This is in
contrast to the rounding algorithm by Iwata, Tetali, and Tripathi [ITT2012],
using Lov\'asz extension of submodular functions. We show a
-approximation for monotone submodular MLOP where
satisfies . Our theory provides new approximation bounds for special cases of the
problem, in particular a -approximation for the
matroid MLOP, where is the rank function of a matroid. We further show
that minimum latency vertex cover (MLVC) is -approximable, by
which we also lower bound the integrality gap of its natural LP relaxation,
which might be of independent interest
Sorting and Selection in Posets
Classical problems of sorting and searching assume an underlying linear ordering of the objects being compared. In this paper, we study these problems in the context of partially ordered sets, in which some pairs of objects are incomparable. This generalization is interesting from a combinatorial perspective, and it has immediate applications in ranking scenarios where there is no underlying linear ordering, e.g., conference submissions. It also has applications in reconstructing certain types of networks, including biological networks. Our results represent significant progress over previous results from two decades ago by Faigle and Turán. In particular, we present the first algorithm that sorts a width-w poset of size n with query complexity O(n(w+\log n)) and prove that this query complexity is asymptotically optimal. We also describe a variant of Mergesort with query complexity O(wn log n/w) and total complexity O(w2n log n/w); an algorithm with the same query complexity was given by Faigle and Turán, but no efficient implementation of that algorithm is known. Both our sorting algorithms can be applied with negligible overhead to the more general problem of reconstructing transitive relations. We also consider two related problems: finding the minimal elements, and its generalization to finding the bottom k “levels,” called the k-selection problem. We give efficient deterministic and randomized algorithms for finding the minimal elements with query complexity and total complexity O(wn). We provide matching lower bounds for the query complexity up to a factor of 2 and generalize the results to the k-selection problem. Finally, we present efficient algorithms for computing a linear extension of a poset and computing the heights of all elements
- …