Trimming of Graphs, with Application to Point Labeling

For $t,g>0$, a vertex-weighted graph of total weight $W$ is $(t,g)$-trimmable if it contains a vertex-induced subgraph of total weight at least $(1-1/t)W$ and with no simple path of more than $g$ edges. A family of graphs is trimmable if for each constant $t>0$, there is a constant $g=g(t)$ such that every vertex-weighted graph in the family is $(t,g)$-trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. Based on this result, we derive a polynomial-time approximation scheme for the problem of labeling weighted points with nonoverlapping sliding labels of unit height and given lengths so as to maximize the total weight of the labeled points. This settles one of the last major open questions in the theory of map labeling

On the complexity of the Whitehead minimization problem

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem -- to decide whether a word is an element of some basis of the free group -- and the free factor problem can also be solved in polynomial time.Comment: v.2: Corrected minor typos and mistakes, improved the proof of the main technical lemma (Statement 2.4); added a section of open problems. 30 page

Finding branch-decompositions of matroids, hypergraphs, and more

Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field $\mathcal F$, we wish to find a "branch-decomposition" of these subspaces of width at most $k$, that is a subcubic tree $T$ with $n$ leaves mapped bijectively to the subspaces such that for every edge $e$ of $T$, the sum of subspaces associated with leaves in one component of $T-e$ and the sum of subspaces associated with leaves in the other component have the intersection of dimension at most $k$. This problem includes the problems of computing branch-width of $\mathcal F$-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs. We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most $k$, if it exists, for input subspaces of a finite-dimensional vector space over $\mathcal F$. Our algorithm is analogous to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. The only known previous fixed-parameter algorithm for branch-width of $\mathcal F$-represented matroids was due to Hlin\v{e}n\'y and Oum (2008) that runs in time $O(n^3)$ where $n$ is the number of elements of the input $\mathcal F$-represented matroid. But their method is highly indirect. Their algorithm uses the non-trivial fact by Geelen et al. (2003) that the number of forbidden minors is finite and uses the algorithm of Hlin\v{e}n\'y (2005) on checking monadic second-order formulas on $\mathcal F$-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed $k$.Comment: 73 pages, 10 figure

Sampling from Stochastic Finite Automata with Applications to CTC Decoding

Stochastic finite automata arise naturally in many language and speech processing tasks. They include stochastic acceptors, which represent certain probability distributions over random strings. We consider the problem of efficient sampling: drawing random string variates from the probability distribution represented by stochastic automata and transformations of those. We show that path-sampling is effective and can be efficient if the epsilon-graph of a finite automaton is acyclic. We provide an algorithm that ensures this by conflating epsilon-cycles within strongly connected components. Sampling is also effective in the presence of non-injective transformations of strings. We illustrate this in the context of decoding for Connectionist Temporal Classification (CTC), where the predictive probabilities yield auxiliary sequences which are transformed into shorter labeling strings. We can sample efficiently from the transformed labeling distribution and use this in two different strategies for finding the most probable CTC labeling

On the complexity of the Whitehead minimization problem

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem - to decide whether a word is an element of some basis of the free group - and the free factor problem can also be solved in polynomial time

Efficient Classification of Locally Checkable Problems in Regular Trees

We give practical, efficient algorithms that automatically determine the asymptotic distributed round complexity of a given locally checkable graph problem in the [?(log n), ?(n)] region, in two settings. We present one algorithm for unrooted regular trees and another algorithm for rooted regular trees. The algorithms take the description of a locally checkable labeling problem as input, and the running time is polynomial in the size of the problem description. The algorithms decide if the problem is solvable in O(log n) rounds. If not, it is known that the complexity has to be ?(n^{1/k}) for some k = 1, 2, ..., and in this case the algorithms also output the right value of the exponent k. In rooted trees in the O(log n) case we can then further determine the exact complexity class by using algorithms from prior work; for unrooted trees the more fine-grained classification in the O(log n) region remains an open question

Adversarial Data Programming: Using GANs to Relax the Bottleneck of Curated Labeled Data

Paucity of large curated hand-labeled training data for every domain-of-interest forms a major bottleneck in the deployment of machine learning models in computer vision and other fields. Recent work (Data Programming) has shown how distant supervision signals in the form of labeling functions can be used to obtain labels for given data in near-constant time. In this work, we present Adversarial Data Programming (ADP), which presents an adversarial methodology to generate data as well as a curated aggregated label has given a set of weak labeling functions. We validated our method on the MNIST, Fashion MNIST, CIFAR 10 and SVHN datasets, and it outperformed many state-of-the-art models. We conducted extensive experiments to study its usefulness, as well as showed how the proposed ADP framework can be used for transfer learning as well as multi-task learning, where data from two domains are generated simultaneously using the framework along with the label information. Our future work will involve understanding the theoretical implications of this new framework from a game-theoretic perspective, as well as explore the performance of the method on more complex datasets.Comment: CVPR 2018 main conference pape