arXiv subjects

Matthew Fahrbach

Publications and source records attributed to Matthew Fahrbach.

27 records · Page 2Linked to original sources

arXiv2020

A Fast Minimum Degree Algorithm and Matching Lower Bound

The minimum degree algorithm is one of the most widely-used heuristics for reducing the cost of solving large sparse systems of linear equations. It has been studied for nearly half a century and has a rich history of bridging techniques from data structures, graph algorithms, and scientific computing. In this paper, we present a simple but novel combinatorial algorithm for computing an exact minimum degree elimination ordering in $O(nm)$ time, which improves on the best known time complexity of $O(n^3)$ and offers practical improvements for sparse systems with small values of $m$. Our approach leverages a careful amortized analysis, which also allows us to derive output-sensitive bounds for the running time of $O(\min\{m\sqrt{m^+}, Δm^+\} \log n)$, where $m^+$ is the number of unique fill edges and original edges that the algorithm encounters and $Δ$ is the maximum degree of the input graph. Furthermore, we show there cannot exist an exact minimum degree algorithm that runs in $O(nm^{1-\varepsilon})$ time, for any $\varepsilon > 0$, assuming the strong exponential time hypothesis. This fine-grained reduction goes through the orthogonal vectors problem and uses a new low-degree graph construction called $U$-fillers, which act as pathological inputs and cause any minimum degree algorithm to exhibit nearly worst-case performance. With these two results, we nearly characterize the time complexity of computing an exact minimum degree ordering.

cs.DS↗

arXiv2020

Edge-Weighted Online Bipartite Matching

Online bipartite matching and its variants are among the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) introduced an elegant algorithm for the unweighted problem that achieves an optimal competitive ratio of $1-1/e$. Later, Aggarwal et al. (SODA 2011) generalized their algorithm and analysis to the vertex-weighted case. Little is known, however, about the most general edge-weighted problem aside from the trivial $1/2$-competitive greedy algorithm. In this paper, we present the first online algorithm that breaks the long-standing $1/2$ barrier and achieves a competitive ratio of at least $0.5086$. In light of the hardness result of Kapralov, Post, and Vondrák (SODA 2013) that restricts beating a $1/2$ competitive ratio for the more general problem of monotone submodular welfare maximization, our result can be seen as strong evidence that edge-weighted bipartite matching is strictly easier than submodular welfare maximization in the online setting. The main ingredient in our online matching algorithm is a novel subroutine called online correlated selection (OCS), which takes a sequence of pairs of vertices as input and selects one vertex from each pair. Instead of using a fresh random bit to choose a vertex from each pair, the OCS negatively correlates decisions across different pairs and provides a quantitative measure on the level of correlation. We believe our OCS technique is of independent interest and will find further applications in other online optimization problems.

cs.DS↗

arXiv2019

Online Weighted Matching: Breaking the $\frac{1}{2}$ Barrier

Online matching and its variants are some of the most fundamental problems in the online algorithms literature. In this paper, we study the online weighted bipartite matching problem. Karp et al. (STOC 1990) gave an elegant algorithm in the unweighted case that achieves a tight competitive ratio of $1-1/e$. In the weighted case, however, we can easily show that no competitive ratio is obtainable without the commonly accepted free disposal assumption. Under this assumption, it is not hard to prove that the greedy algorithm is $1/2$ competitive, and that this is tight for deterministic algorithms. We present the first randomized algorithm that breaks this long-standing $1/2$ barrier and achieves a competitive ratio of at least $0.501$. In light of the hardness result of Kapralov et al. (SODA 2013) that restricts beating a $1/2$ competitive ratio for the monotone submodular welfare maximization problem, our result can be seen as strong evidence that solving the weighted bipartite matching problem is strictly easier than submodular welfare maximization in the online setting. Our approach relies on a very controlled use of randomness, which allows our algorithm to safely make adaptive decisions based on its previous assignments.

cs.DS↗

arXiv2018

Graph Sketching Against Adaptive Adversaries Applied to the Minimum Degree Algorithm

Motivated by the study of matrix elimination orderings in combinatorial scientific computing, we utilize graph sketching and local sampling to give a data structure that provides access to approximate fill degrees of a matrix undergoing elimination in $O(\text{polylog}(n))$ time per elimination and query. We then study the problem of using this data structure in the minimum degree algorithm, which is a widely-used heuristic for producing elimination orderings for sparse matrices by repeatedly eliminating the vertex with (approximate) minimum fill degree. This leads to a nearly-linear time algorithm for generating approximate greedy minimum degree orderings. Despite extensive studies of algorithms for elimination orderings in combinatorial scientific computing, our result is the first rigorous incorporation of randomized tools in this setting, as well as the first nearly-linear time algorithm for producing elimination orderings with provable approximation guarantees. While our sketching data structure readily works in the oblivious adversary model, by repeatedly querying and greedily updating itself, it enters the adaptive adversarial model where the underlying sketches become prone to failure due to dependency issues with their internal randomness. We show how to use an additional sampling procedure to circumvent this problem and to create an independent access sequence. Our technique for decorrelating the interleaved queries and updates to this randomized data structure may be of independent interest.

cs.DS↗

arXiv2017

On Computing Min-Degree Elimination Orderings

We study faster algorithms for producing the minimum degree ordering used to speed up Gaussian elimination. This ordering is based on viewing the non-zero elements of a symmetric positive definite matrix as edges of an undirected graph, and aims at reducing the additional non-zeros (fill) in the matrix by repeatedly removing the vertex of minimum degree. It is one of the most widely used primitives for pre-processing sparse matrices in scientific computing. Our result is in part motivated by the observation that sub-quadratic time algorithms for finding min-degree orderings are unlikely, assuming the strong exponential time hypothesis (SETH). This provides justification for the lack of provably efficient algorithms for generating such orderings, and leads us to study speedups via degree-restricted algorithms as well as approximations. Our two main results are: (1) an algorithm that produces a min-degree ordering whose maximum degree is bounded by $Δ$ in $O(m Δ\log^3{n})$ time, and (2) an algorithm that finds an $(1 + ε)$-approximate marginal min-degree ordering in $O(m \log^{5}n ε^{-2})$ time. Both of our algorithms rely on a host of randomization tools related to the $\ell_0$-estimator by [Cohen `97]. A key technical issue for the final nearly-linear time algorithm are the dependencies of the vertex removed on the randomness in the data structures. To address this, we provide a method for generating a pseudo-deterministic access sequence, which then allows the incorporation of data structures that only work under the oblivious adversary model.

cs.DS↗

arXiv2017

Nearly Tight Bounds for Sandpile Transience on the Grid

We use techniques from the theory of electrical networks to give nearly tight bounds for the transience class of the Abelian sandpile model on the two-dimensional grid up to polylogarithmic factors. The Abelian sandpile model is a discrete process on graphs that is intimately related to the phenomenon of self-organized criticality. In this process, vertices receive grains of sand, and once the number of grains exceeds their degree, they topple by sending grains to their neighbors. The transience class of a model is the maximum number of grains that can be added to the system before it necessarily reaches its steady-state behavior or, equivalently, a recurrent state. Through a more refined and global analysis of electrical potentials and random walks, we give an $O(n^4\log^4{n})$ upper bound and an $Ω(n^4)$ lower bound for the transience class of the $n \times n$ grid. Our methods naturally extend to $n^d$-sized $d$-dimensional grids to give $O(n^{3d - 2}\log^{d+2}{n})$ upper bounds and $Ω(n^{3d -2})$ lower bounds.

cs.DS↗

arXiv2017

Analyzing Boltzmann Samplers for Bose-Einstein Condensates with Dirichlet Generating Functions

Boltzmann sampling is commonly used to uniformly sample objects of a particular size from large combinatorial sets. For this technique to be effective, one needs to prove that (1) the sampling procedure is efficient and (2) objects of the desired size are generated with sufficiently high probability. We use this approach to give a provably efficient sampling algorithm for a class of weighted integer partitions related to Bose-Einstein condensation from statistical physics. Our sampling algorithm is a probabilistic interpretation of the ordinary generating function for these objects, derived from the symbolic method of analytic combinatorics. Using the Khintchine-Meinardus probabilistic method to bound the rejection rate of our Boltzmann sampler through singularity analysis of Dirichlet generating functions, we offer an alternative approach to analyze Boltzmann samplers for objects with multiplicative structure.

cs.DS↗

arXiv2016

Approximately Sampling Elements with Fixed Rank in Graded Posets

Graded posets frequently arise throughout combinatorics, where it is natural to try to count the number of elements of a fixed rank. These counting problems are often $\#\textbf{P}$-complete, so we consider approximation algorithms for counting and uniform sampling. We show that for certain classes of posets, biased Markov chains that walk along edges of their Hasse diagrams allow us to approximately generate samples with any fixed rank in expected polynomial time. Our arguments do not rely on the typical proofs of log-concavity, which are used to construct a stationary distribution with a specific mode in order to give a lower bound on the probability of outputting an element of the desired rank. Instead, we infer this directly from bounds on the mixing time of the chains through a method we call $\textit{balanced bias}$. A noteworthy application of our method is sampling restricted classes of integer partitions of $n$. We give the first provably efficient Markov chain algorithm to uniformly sample integer partitions of $n$ from general restricted classes. Several observations allow us to improve the efficiency of this chain to require $O(n^{1/2}\log(n))$ space, and for unrestricted integer partitions, expected $O(n^{9/4})$ time. Related applications include sampling permutations with a fixed number of inversions and lozenge tilings on the triangular lattice with a fixed average height.

cs.DS↗

arXiv2016

Coefficients and roots of peak polynomials

Given a permutation $π=π_1π_2\cdots π_n \in \mathfrak{S}_n$, we say an index $i$ is a peak if $π_{i-1} < π_i > π_{i+1}$. Let $P(π)$ denote the set of peaks of $π$. Given any set $S$ of positive integers, define ${\mathcal{P}_S(n)=\{π\in \mathfrak{S}_n:P(π)=S\}}$. Billey-Burdzy-Sagan showed that for all fixed subsets of positive integers $S$ and sufficiently large $n$, $|\mathcal{P}_S(n)|=p_S(n)2^{n-|S|-1}$ for some polynomial $p_S(x)$ depending on $S$. They conjectured that the coefficients of $p_S(x)$ expanded in a binomial coefficient basis centered at $\max(S)$ are all positive. We show that this is a consequence of a stronger conjecture that bounds the modulus of the roots of $p_S(x)$. Furthermore, we give an efficient explicit formula for peak polynomials in the binomial basis centered at $0$, which we use to identify many integer roots of peak polynomials along with certain inequalities and identities.

math.CO↗