We develop a framework for graph sparsification and sketching, based on a new tool, short cycle decomposition, which is a decomposition of an unweighted graph into an edge-disjoint collection of short cycles, plus a small number of extra edges. A simple observation shows that every graph $G$ on $n$ vertices with $m$ edges can be decomposed in $O(mn)$ time into cycles of length at most $2 \log n,$ and at most $2n$ extra edges. We give an $(m^{1+o(1)})$-time algorithm for constructing a short cycle decomposition, with cycles of length $n^{o(1)},$ and $n^{1+o(1)}$ extra edges. Both the existential and algorithmic variants of this decomposition enable us to make the following progress on several open problems in randomized graph algorithms: (1) We present an algorithm that runs in time $m^{1+o(1)}\varepsilon^{-1.5}$ and returns $(1\pm\varepsilon)$-approximations to effective resistances of all edges, improving over the previous best runtime of $\widetilde{{O}}(\min\{m\varepsilon^{-2}, n^{2} \varepsilon^{-1}\})$. This routine in turn gives an algorithm for approximating the determinant of a graph Laplacian up to a factor of $(1\pm \varepsilon)$ in $m^{1 + o(1)} + n^{\nicefrac{15}{8}+o(1)}\varepsilon^{-\nicefrac{7}{4}}$ time. (2) We show the existence of graphical spectral sketches with about $n\varepsilon^{-1}$ edges, and also give efficient algorithms to construct them. A graphical spectral sketch is a distribution over sparse graphs $H$ such that for a fixed vector ${\mathit{x}}$, we have ${{x}}^{\top} {L}_H {{x}} = (1\pm\varepsilon) {{x}}^{\top} {L}_G {{x}}$ and ${{x}}^{\top} {L}^{+}_H {{x}} = (1\pm\varepsilon) {{x}}^{\top} {L}^{+}_G {{x}}$ with high probability, where ${L}$ is the graph Laplacian and ${L}^{+}$ is its pseudoinverse. This implies the existence of resistance sparsifiers with about $n \varepsilon^{-1}$ edges that preserve the effective resistance between every pair of vertices up to $(1\pm\varepsilon)$. (3) By combining short cycle decompositions with known tools in graph sparsification, we show the existence of nearly linear sized degree-preserving spectral sparsifiers, as well as significantly sparser approximations of Eulerian directed graphs. The latter is critical to recent breakthroughs on faster algorithms for solving linear systems in directed Laplacians. The running time and output qualities of our spectral sketch and degree-preserving (directed) sparsification algorithms are limited by the efficiency of our routines for constructing short cycle decompositions. Improved algorithms for short cycle decompositions will lead to improvement in each of these algorithms.

Commonsense Question Answering is an important natural language processing (NLP) task that aims to predict the correct answer to a question through commonsense reasoning.Previous studies utilize pre-trained models on large-scale corpora such as BERT, or perform reasoning on knowledge graphs.However, these methods do not explicitly model the relations that connect entities, which are informational and can be used to enhance reasoning.To address this issue, we propose a relationaware reasoning method.Our method uses a relation-aware graph neural network to capture the rich contextual information from both entities and relations.Compared with methods that use fixed relation embeddings from pre-trained models, our model dynamically updates relations with contextual information from a multi-source subgraph, built from multiple external knowledge sources.The enhanced representations of relations are then fed to a bidirectional reasoning module.A bidirectional attention mechanism is applied between the question sequence and the paths that connect entities, which provides us with transparent interpretability.Experimental results on the CommonsenseQA dataset illustrate that our method results in significant improvements over the baselines while also providing clear reasoning paths.

We give the first fully dynamic algorithm which maintains a (1−є)-approximate densest subgraph in worst-case time poly(logn, є−1) per update. Dense subgraph discovery is an important primitive for many real-world applications such as community detection, link spam detection, distance query indexing, and computational biology. We approach the densest subgraph problem by framing its dual as a graph orientation problem, which we solve using an augmenting path-like adjustment technique. Our result improves upon the previous best approximation factor of (1/4 − є) for fully dynamic densest subgraph [Bhattacharya et. al., STOC ‘15]. We also extend our techniques to solving the problem on vertex-weighted graphs with similar runtimes.

The problem of finding dense components of a graph is a major primitive in graph mining and data analysis. The densest subgraph problem (DSP) that asks to find a subgraph with maximum average degree forms a basic primitive in dense subgraph discovery with applications ranging from community detection to unsupervised discovery of biological network modules [16]. The DSP is exactly solvable in polynomial time using maximum flows [14, 17, 22]. Due to the high computational cost of maximum flows, Charikar’s greedy approximation algorithm is usually preferred in practice due to its linear time and linear space complexity [3, 8]. It constitutes a key algorithmic idea in scalable solutions for large-scale dynamic graphs [5, 7]. However, its output density can be a factor 2 off the optimal solution.

We develop a framework for graph sparsification based on a new tool, short cycle decomposition for graphs - a decomposition of a graph into a collection of short cycles, plus a small number of extra edges. A simple observation gives that every graph G on n vertices with m edges can be decomposed in O(mn) time into cycles of length at most 2 log n, and at most 2n extra edges. We give an m1+o(1)time algorithm for constructing a short cycle decomposition of the graph, with cycles of length no(1), and n1+o(1)extra edges. Both the existential and algorithmic variants of this decomposition enable us to make progress on several open problems in randomized graph algorithms. 1. We present an algorithm that runs in time m1+o(1)ε-1.5and returns (1 ± ε)-approximations to effective resistances of all edges, improving over the previous best of Õ(min{mε-2, n2ε-1}) This gives an algorithm to approximate the determinant of a graph Laplacian up to a factor of (1 ± ε) in roughly m + n15/8ε-7/4. 2. We show existence and efficient algorithms for constructing graphical spectral sketches - a distribution over sparse graphs H with about nε-1edges such that for a fixed vector x, we have xTLHx = (1 ± eps) xTLGx and xTL+Hx = (1 ± ε) xTL+Gx with high probability, where L is the graph Laplacian and L+ is its pseudoinverse. This implies resistance-sparsifiers with about nε edges that preserve the effective resistances between every pair of vertices up to (1 + eps). 3. By combining short cycle decomposition with importance sampling, we show the existence of nearly-linear sized degree-preserving spectral sparsifiers, as well as significantly sparser approximations of directed graphs. The latter is critical to recent breakthroughs on faster algorithms for directed random walks and linear systems in directed Laplacian. The running time and output qualities of our spectral sketch and degree-preserving (directed) sparsification algorithms are limited by the efficiency of our routines for producing short cycle decompositions. Improved algorithms for short cycle decompositions will lead to improvements for each of these algorithms.

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 polylogarithmic 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 interleaved queries and updates to this randomized data structure may be of independent interest.

We consider the problem of fairly allocating indivisible goods, focusing on a recently introduced notion of fairness called maximin share guarantee : each player’s value for his allocation should be at least as high as what he can guarantee by dividing the items into as many bundles as there are players and receiving his least desirable bundle. Assuming additive valuation functions, we show that such allocations may not exist, but allocations guaranteeing each player 2/3 of the above value always exist. These theoretical results have direct practical implications.

We are interested in the problem of dividing a cake -- a heterogeneous divisible good -- among n players, in a way that is ε-equitable: every pair of players must have the same value for their own allocated pieces, up to a difference of at most ε. It is known that such allocations can be computed using O(n ln(1/ε)) operations in the standard Robertson-Webb Model. We establish a lower bound of Ω(ln(1/ε)/lnln(1/ε)) on the complexity of this problem, which is almost tight for a constant number of players. Importantly, our result implies that allocations that are exactly equitable cannot be computed.

The fairness notion of maximin share (MMS) guarantee underlies a deployed algorithm for allocating indivisible goods under additive valuations. Our goal is to understand when we can expect to be able to give each player his MMS guarantee. Previous work has shown that such an MMS allocation may not exist, but the counterexample requires a number of goods that is exponential in the number of players; we give a new construction that uses only a linear number of goods. On the positive side, we formalize the intuition that these counterexamples are very delicate by designing an algorithm that provably finds an MMS allocation with high probability when valuations are drawn at random.

The maximum Nash welfare (MNW) solution --- which selects an allocation that maximizes the product of utilities --- is known to provide outstanding fairness guarantees when allocating divisible goods. And while it seems to lose its luster when applied to indivisible goods, we show that, in fact, the MNW solution is unexpectedly, strikingly fair even in that setting. In particular, we prove that it selects allocations that are envy free up to one good --- a compelling notion that is quite elusive when coupled with economic efficiency. We also establish that the MNW solution provides a good approximation to another popular (yet possibly infeasible) fairness property, the maximin share guarantee, in theory and --- even more so --- in practice. While finding the MNW solution is computationally hard, we develop a nontrivial implementation, and demonstrate that it scales well on real data. These results lead us to believe that MNW is the ultimate solution for allocating indivisible goods, and underlie its deployment on a popular fair division website.

Wayside supercapacitor based energy storage system (ESS) for urban rail transit can reuse the braking energy and reduce energy consumption. Most ESSs switch over from charging to discharging states based on voltage of DC bus. But open circuit voltage of DC power supply in subway system always changes, which would influence the normal operation of ESS. This paper will show the output characteristic of substation and the basic structure of supercapacitor based ESS. At the same time, influences of substation output voltage fluctuation on ESS will be analyzed. And then, a control strategy for eliminating the influence will be come up with. Finally, this paper will present the experiment results of 200kW supercapacitor based ESS prototype field test in Beijing subway.

We consider the problem of fairly allocating indivisible goods, focusing on a recently-introduced notion of fairness called maximin share guarantee: Each player's value for his allocation should be at least as high as what he can guarantee by dividing the items into as many bundles as there are players and receiving his least desirable bundle. Assuming additive valuation functions, we show that such allocations may not exist, but allocations guaranteeing each player 2/3 of the above value always exist, and can be computed in polynomial time when the number of players is constant. These theoretical results have direct practical implications.