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Many papers in the field of integer linear programming (ILP, for short) are devoted to problems of the type $\max\{c^\top x \colon A x = b,\, x \in \mathbb{Z}^n_{\geq 0}\}$, where all the entries of $A,b,c$ are integer, parameterized by the number of rows of $A$ and $\|A\|_{\max}$. This class of problems is known under the name of ILP problems in the standard form, adding the word "bounded" if $x \leq u$, for some integer vector $u$. Recently, many new sparsity, proximity, and complexity results were obtained for bounded and unbounded ILP problems in the standard form. In this paper, we consider ILP problems in the canonical form $$\max\{c^\top x \colon b_l \leq A x \leq b_r,\, x \in \mathbb{Z}^n\},$$ where $b_l$ and $b_r$ are integer vectors. We assume that the integer matrix $A$ has the rank $n$, $(n + m)$ rows, $n$ columns, and parameterize the problem by $m$ and $\Delta(A)$, where $\Delta(A)$ is the maximum of $n \times n$ sub-determinants of $A$, taken in the absolute value. We show that any ILP problem in the standard form can be polynomially reduced to some ILP problem in the canonical form, preserving $m$ and $\Delta(A)$, but the reverse reduction is not always possible. More precisely, we define the class of generalized ILP problems in the standard form, which includes an additional group constraint, and prove the equivalence to ILP problems in the canonical form. We generalize known sparsity, proximity, and complexity bounds for ILP problems in the canonical form. Additionally, sometimes, we strengthen previously known results for ILP problems in the canonical form, and, sometimes, we give shorter proofs. Finally, we consider the special cases of $m \in \{0,1\}$. By this way, we give specialised sparsity, proximity, and complexity bounds for the problems on simplices, Knapsack problems and Subset-Sum problems.

Triangle centrality is introduced for finding important vertices in a graph based on the concentration of triangles surrounding each vertex. An important vertex in triangle centrality is at the center of many triangles, and therefore it may be in many triangles or none at all. We give optimal algorithms that compute triangle centrality in $O(m\sqrt{m})$ time and $O(m+n)$ space. Using fast matrix multiplication it takes $n^{\omega+o(1)}$ time where $\omega$ is the matrix product exponent. On a Concurrent Read Exclusive Write (CREW) Parallel Random Access Memory (PRAM) machine, we give a near work-optimal algorithm that takes $O(\log n)$ time using $O(m\sqrt{m})$ CREW PRAM processors. In MapReduce, we show it takes four rounds using $O(m\sqrt{m})$ communication bits, and is therefore optimal. We also give a deterministic algorithm to find the triangle neighborhood and triangle count of each vertex in $O(m\sqrt{m})$ time and $O(m+n)$ space. Our empirical results demonstrate that triangle centrality uniquely identified central vertices thirty-percent of the time in comparison to five other well-known centrality measures, while being asymptotically faster to compute on sparse graphs than all but the most trivial of these other measures.

We present an algorithm for strongly refuting smoothed instances of all Boolean CSPs. The smoothed model is a hybrid between worst and average-case input models, where the input is an arbitrary instance of the CSP with only the negation patterns of the literals re-randomized with some small probability. For an $n$-variable smoothed instance of a $k$-arity CSP, our algorithm runs in $n^{O(\ell)}$ time, and succeeds with high probability in bounding the optimum fraction of satisfiable constraints away from $1$, provided that the number of constraints is at least $\tilde{O}(n) (\frac{n}{\ell})^{\frac{k}{2} - 1}$. This matches, up to polylogarithmic factors in $n$, the trade-off between running time and the number of constraints of the state-of-the-art algorithms for refuting fully random instances of CSPs [RRS17]. We also make a surprising new connection between our algorithm and even covers in hypergraphs, which we use to positively resolve Feige's 2008 conjecture, an extremal combinatorics conjecture on the existence of even covers in sufficiently dense hypergraphs that generalizes the well-known Moore bound for the girth of graphs. As a corollary, we show that polynomial-size refutation witnesses exist for arbitrary smoothed CSP instances with number of constraints a polynomial factor below the "spectral threshold" of $n^{k/2}$, extending the celebrated result for random 3-SAT of Feige, Kim and Ofek [FKO06].

A streaming algorithm is adversarially robust if it is guaranteed to perform correctly even in the presence of an adaptive adversary. Recently, several sophisticated frameworks for robustification of classical streaming algorithms have been developed. One of the main open questions in this area is whether efficient adversarially robust algorithms exist for moment estimation problems under the turnstile streaming model, where both insertions and deletions are allowed. So far, the best known space complexity for streams of length $m$, achieved using differential privacy (DP) based techniques, is of order $\tilde{O}(m^{1/2})$ for computing a constant-factor approximation with high constant probability. In this work, we propose a new simple approach to tracking moments by alternating between two different regimes: a sparse regime, in which we can explicitly maintain the current frequency vector and use standard sparse recovery techniques, and a dense regime, in which we make use of existing DP-based robustification frameworks. The results obtained using our technique break the previous $m^{1/2}$ barrier for any fixed $p$. More specifically, our space complexity for $F_2$-estimation is $\tilde{O}(m^{2/5})$ and for $F_0$-estimation, i.e., counting the number of distinct elements, it is $\tilde O(m^{1/3})$. All existing robustness frameworks have their space complexity depend multiplicatively on a parameter $\lambda$ called the \emph{flip number} of the streaming problem, where $\lambda = m$ in turnstile moment estimation. The best known dependence in these frameworks (for constant factor approximation) is of order $\tilde{O}(\lambda^{1/2})$, and it is known to be tight for certain problems. Again, our approach breaks this barrier, achieving a dependence of order $\tilde{O}(\lambda^{1/2 - c(p)})$ for $F_p$-estimation, where $c(p) > 0$ depends only on $p$.

We show that for every fixed $k\geq 3$, the problem whether the termination/counter complexity of a given demonic VASS is $\mathcal{O}(n^k)$, $\Omega(n^{k})$, and $\Theta(n^{k})$ is coNP-complete, NP-complete, and DP-complete, respectively. We also classify the complexity of these problems for $k\leq 2$. This shows that the polynomial-time algorithm designed for strongly connected demonic VASS in previous works cannot be extended to the general case. Then, we prove that the same problems for VASS games are PSPACE-complete. Again, we classify the complexity also for $k\leq 2$. Interestingly, tractable subclasses of demonic VASS and VASS games are obtained by bounding certain structural parameters, which opens the way to applications in program analysis despite the presented lower complexity bounds.

Node classification is an important problem in graph data management. It is commonly solved by various label propagation methods that work iteratively starting from a few labeled seed nodes. For graphs with arbitrary compatibilities between classes, these methods crucially depend on knowing the compatibility matrix that must be provided by either domain experts or heuristics. Can we instead directly estimate the correct compatibilities from a sparsely labeled graph in a principled and scalable way? We answer this question affirmatively and suggest a method called distant compatibility estimation that works even on extremely sparsely labeled graphs (e.g., 1 in 10,000 nodes is labeled) in a fraction of the time it later takes to label the remaining nodes. Our approach first creates multiple factorized graph representations (with size independent of the graph) and then performs estimation on these smaller graph sketches. We define algebraic amplification as the more general idea of leveraging algebraic properties of an algorithm's update equations to amplify sparse signals. We show that our estimator is by orders of magnitude faster than an alternative approach and that the end-to-end classification accuracy is comparable to using gold standard compatibilities. This makes it a cheap preprocessing step for any existing label propagation method and removes the current dependence on heuristics.

The main contribution of this paper is a new submap joining based approach for solving large-scale Simultaneous Localization and Mapping (SLAM) problems. Each local submap is independently built using the local information through solving a small-scale SLAM; the joining of submaps mainly involves solving linear least squares and performing nonlinear coordinate transformations. Through approximating the local submap information as the state estimate and its corresponding information matrix, judiciously selecting the submap coordinate frames, and approximating the joining of a large number of submaps by joining only two maps at a time, either sequentially or in a more efficient Divide and Conquer manner, the nonlinear optimization process involved in most of the existing submap joining approaches is avoided. Thus the proposed submap joining algorithm does not require initial guess or iterations since linear least squares problems have closed-form solutions. The proposed Linear SLAM technique is applicable to feature-based SLAM, pose graph SLAM and D-SLAM, in both two and three dimensions, and does not require any assumption on the character of the covariance matrices. Simulations and experiments are performed to evaluate the proposed Linear SLAM algorithm. Results using publicly available datasets in 2D and 3D show that Linear SLAM produces results that are very close to the best solutions that can be obtained using full nonlinear optimization algorithm started from an accurate initial guess. The C/C++ and MATLAB source codes of Linear SLAM are available on OpenSLAM.

Representation learning for networks provides a new way to mine graphs. Although current researches in this area are able to generate reliable results of node embeddings, they are still limited to homogeneous networks in which all nodes and edges are of the same type. While, increasingly, graphs are heterogeneous with multiple node- and edge- types in the real world. Existing heterogeneous embedding methods are mostly task-based or only able to deal with limited types of node & edge. To tackle this challenge, in this paper, an edge2vec model is proposed to represent nodes in ways that incorporate edge semantics represented as different edge-types in heterogeneous networks. An edge-type transition matrix is optimized from an Expectation-Maximization (EM) framework as an extra criterion of a biased node random walk on networks, and a biased skip-gram model is leveraged to learn node embeddings based on the random walks afterwards. edge2vec is validated and evaluated using three medical domain problems on an ensemble of complex medical networks (more than 10 node- \& edge- types): medical entity classification, compound-gene binding prediction, and medical information searching cost. Results show that by considering edge semantics, edge2vec significantly outperforms other state-of-art models on all three tasks.

Embedding representation learning via neural networks is at the core foundation of modern similarity based search. While much effort has been put in developing algorithms for learning binary hamming code representations for search efficiency, this still requires a linear scan of the entire dataset per each query and trades off the search accuracy through binarization. To this end, we consider the problem of directly learning a quantizable embedding representation and the sparse binary hash code end-to-end which can be used to construct an efficient hash table not only providing significant search reduction in the number of data but also achieving the state of the art search accuracy outperforming previous state of the art deep metric learning methods. We also show that finding the optimal sparse binary hash code in a mini-batch can be computed exactly in polynomial time by solving a minimum cost flow problem. Our results on Cifar-100 and on ImageNet datasets show the state of the art search accuracy in precision@k and NMI metrics while providing up to 98X and 478X search speedup respectively over exhaustive linear search.