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Let $G$ be a multigraph with $n$ vertices and $e>4n$ edges, drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior. Pach and T\'oth (A Crossing Lemma for Multigraphs, SoCG 2018) extended the Crossing Lemma of Ajtai et al. (Crossing-free subgraphs, North-Holland Mathematics Studies, 1982) and Leighton (Complexity issues in VLSI, Foundations of computing series, 1983) by showing that if no two adjacent edges cross and every pair of nonadjacent edges cross at most once, then the number of edge crossings in $G$ is at least $\alpha e^3/n^2$, for a suitable constant $\alpha>0$. The situation turns out to be quite different if nonparallel edges are allowed to cross any number of times. It is proved that in this case the number of crossings in $G$ is at least $\alpha e^{2.5}/n^{1.5}$. The order of magnitude of this bound cannot be improved.

We consider the termination problem for triangular weakly non-linear loops (twn-loops) over some ring $\mathcal{S}$ like $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{R}$. Essentially, the guard of such a loop is an arbitrary quantifier-free Boolean formula over (possibly non-linear) polynomial inequations, and the body is a single assignment of the form $(x_1, \ldots, x_d) \longleftarrow (c_1 \cdot x_1 + p_1, \ldots, c_d \cdot x_d + p_d)$ where each $x_i$ is a variable, $c_i \in \mathcal{S}$, and each $p_i$ is a (possibly non-linear) polynomial over $\mathcal{S}$ and the variables $x_{i+1},\ldots,x_{d}$. We show that the question of termination can be reduced to the existential fragment of the first-order theory of $\mathcal{S}$ and $\mathbb{R}$. For loops over $\mathbb{R}$, our reduction implies decidability of termination. For loops over $\mathbb{Z}$ and $\mathbb{Q}$, it proves semi-decidability of non-termination. Furthermore, we present a transformation to convert certain non-twn-loops into twn-form. Then the original loop terminates iff the transformed loop terminates over a specific subset of $\mathbb{R}$, which can also be checked via our reduction. Moreover, we formalize a technique to linearize twn-loops in our setting and analyze its complexity. Based on these results, we prove complexity bounds for the termination problem of twn-loops as well as tight bounds for two important classes of loops which can always be transformed into twn-loops. Finally, we show that there is an important class of linear loops where our decision procedure results in an efficient procedure for termination analysis, i.e., where the parameterized complexity of deciding termination is polynomial.

The approximate uniform sampling of graphs with a given degree sequence is a well-known, extensively studied problem in theoretical computer science and has significant applications, e.g., in the analysis of social networks. In this work we study an extension of the problem, where degree intervals are specified rather than a single degree sequence. We are interested in sampling and counting graphs whose degree sequences satisfy the degree interval constraints. A natural scenario where this problem arises is in hypothesis testing on social networks that are only partially observed. In this work, we provide the first fully polynomial almost uniform sampler (FPAUS) as well as the first fully polynomial randomized approximation scheme (FPRAS) for sampling and counting, respectively, graphs with near-regular degree intervals, in which every node $i$ has a degree from an interval not too far away from a given $d \in \N$. In order to design our FPAUS, we rely on various state-of-the-art tools from Markov chain theory and combinatorics. In particular, we provide the first non-trivial algorithmic application of a breakthrough result of Liebenau and Wormald (2017) regarding an asymptotic formula for the number of graphs with a given near-regular degree sequence. Furthermore, we also make use of the recent breakthrough of Anari et al. (2019) on sampling a base of a matroid under a strongly log-concave probability distribution. As a more direct approach, we also study a natural Markov chain recently introduced by Rechner, Strowick and M\"uller-Hannemann (2018), based on three simple local operations: Switches, hinge flips, and additions/deletions of a single edge. We obtain the first theoretical results for this Markov chain by showing it is rapidly mixing for the case of near-regular degree intervals of size at most one.

We study the problem of locating the source of a stochastic epidemic diffusion process from a sparse set of sensors. In a graph $G=(V,E)$, an unknown source node $v^* \in V$ is drawn uniformly at random, and unknown edge weights $w(e)$ for $e\in E$, representing the propagation delays along the edges, are drawn independently from a Gaussian distribution of mean $1$ and variance $\sigma^2$. An algorithm then attempts to locate $v^*$ by picking sensor (also called query) nodes $s \in V$ and being told the length of the shortest path between $s$ and $v^*$ in graph $G$ weighted by $w$. We consider two settings: \emph{static}, in which all query nodes must be decided in advance, and \emph{sequential}, in which each query can depend on the results of the previous ones. We characterize the query complexity when $G$ is an $n$-node path. In the static setting, $\Theta(n\sigma^2)$ queries are needed for $\sigma^2 \leq 1$, and $\Theta(n)$ for $\sigma^2 \geq 1$. In the sequential setting, somewhat surprisingly, only $\Theta(\log\log_{1/\sigma}n)$ are needed when $\sigma^2 \leq 1/2$, and $\Theta(\log \log n)+O_\sigma(1)$ when $\sigma^2 \geq 1/2$. This is the first mathematical study of sensor-based source location in a non-deterministic epidemic process.

In this paper, we propose a reduced-bias estimator of the EVI for Pareto-type tails (heavy-tailed) distributions. This is derived using the weighted least squares method. It is shown that the estimator is unbiased, consistent and asymptotically normal under the second-order conditions on the underlying distribution of the data. The finite sample properties of the proposed estimator are studied through a simulation study. The results show that it is competitive to the existing estimators of the extreme value index in terms of bias and Mean Square Error. In addition, it yields estimates of $\gamma>0$ that are less sensitive to the number of top-order statistics, and hence, can be used for selecting an optimal tail fraction. The proposed estimator is further illustrated using practical datasets from pedochemical and insurance.

In this paper, we address the problem of constructing $C^2$ cubic spline functions on a given arbitrary triangulation $\mathcal{T}$. To this end, we endow every triangle of $\mathcal{T}$ with a Wang-Shi macro-structure. The $C^2$ cubic space on such a refined triangulation has a stable dimension and optimal approximation power. Moreover, any spline function in such space can be locally built on each of the macro-triangles independently via Hermite interpolation. We provide a simplex spline basis for the space of $C^2$ cubics defined on a single macro-triangle which behaves like a Bernstein/B-spline basis over the triangle. The basis functions inherit recurrence relations and differentiation formulas from the simplex spline construction, they form a nonnegative partition of unity, they admit simple conditions for $C^2$ joins across the edges of neighboring triangles, and they enjoy a Marsden-like identity. Also, there is a single control net to facilitate control and early visualization of a spline function over the macro-triangle. Thanks to these properties, the complex geometry of the Wang-Shi macro-structure is transparent to the user. Stable global bases for the full space of $C^2$ cubics on the Wang-Shi refined triangulation $\mathcal{T}$ are deduced from the local simplex spline basis by extending the concept of minimal determining sets.

This short note present a "proof" of $P\neq NP$. The "proof" with double quotation marks is to indicate that we do not know whether the proof is correct or not (We're confused because we do know in which we make the mistakes).

We show that it is provable in PA that there is an arithmetically definable sequence $\{\phi_{n}:n \in \omega\}$ of $\Pi^{0}_{2}$-sentences, such that - PRA+$\{\phi_{n}:n \in \omega\}$ is $\Pi^{0}_{2}$-sound and $\Pi^{0}_{1}$-complete - the length of $\phi_{n}$ is bounded above by a polynomial function of $n$ with positive leading coefficient - PRA+$\phi_{n+1}$ always proves 1-consistency of PRA+$\phi_{n}$. One has that the growth in logical strength is in some sense "as fast as possible", manifested in the fact that the total general recursive functions whose totality is asserted by the true $\Pi^{0}_{2}$-sentences in the sequence are cofinal growth-rate-wise in the set of all total general recursive functions. We then develop an argument which makes use of a sequence of sentences constructed by an application of the diagonal lemma, which are generalisations in a broad sense of Hugh Woodin's "Tower of Hanoi" construction as outlined in his essay "Tower of Hanoi" in Chapter 18 of the anthology "Truth in Mathematics". The argument establishes the result that it is provable in PA that $P \neq NP$. We indicate how to pull the argument all the way down into EFA.

A connected partition is a partition of the vertices of a graph into sets that induce connected subgraphs. Such partitions naturally occur in many application areas such as road networks, and image processing. We consider Balanced Connected Partitions (BCP), where the two classical objectives for BCP are to maximize the weight of the smallest, or minimize the weight of the largest component. We study BCP on c-claw-free graphs, the class of graphs that do not have $K_{1,c}$ as an induced subgraph, and present efficient (c-1)-approximation algorithms for both objectives. In particular, due to the (3-)claw-freeness of line graphs, this also implies a 2-approximations for the edge-partition version of BCP in general graphs. In the 1970s Gy\H{o}ri and Lov\'{a}sz showed for natural numbers $w_1,\dots,w_k$ where $\sum_i w_i$ is the vertex size, that if $G$ is k-connected, then there exist a connected k-partition with part sizes $w_1,\dots,w_k$. However, to this day no polynomial algorithm to compute such partitions exists for k>4. Towards finding such a partition $T_1,\dots, T_k$, we show how to efficiently compute connected partitions that at least approximately meet the target weights, subject to the mild assumption that each $w_i$ is greater than the weight of the heaviest vertex. In particular, we give a 3-approximation for both the lower and the upper bounded version i.e. we guarantee that each $T_i$ has weight at least $\frac{w_i}{3}$ or that each $T_i$ has weight most $3w_i$, respectively. Also, we present a both-side bounded version that produces a connected partition where each $T_i$ has size at least $\frac{w_i}{3}$ and at most $\max(\{r,3\}) w_i$, where $r \geq 1$ is the ratio between the largest and smallest value in $w_1, \dots, w_k$. In particular for the balanced version, i.e.~$w_1=w_2=, \dots,=w_k$, this gives a partition with $\frac{1}{3}w_i \leq w(T_i) \leq 3w_i$.

We consider the task of learning the parameters of a {\em single} component of a mixture model, for the case when we are given {\em side information} about that component, we call this the "search problem" in mixture models. We would like to solve this with computational and sample complexity lower than solving the overall original problem, where one learns parameters of all components. Our main contributions are the development of a simple but general model for the notion of side information, and a corresponding simple matrix-based algorithm for solving the search problem in this general setting. We then specialize this model and algorithm to four common scenarios: Gaussian mixture models, LDA topic models, subspace clustering, and mixed linear regression. For each one of these we show that if (and only if) the side information is informative, we obtain parameter estimates with greater accuracy, and also improved computation complexity than existing moment based mixture model algorithms (e.g. tensor methods). We also illustrate several natural ways one can obtain such side information, for specific problem instances. Our experiments on real data sets (NY Times, Yelp, BSDS500) further demonstrate the practicality of our algorithms showing significant improvement in runtime and accuracy.