We give a strongly polynomial-time algorithm for integer linear programs defined by integer coefficient matrices whose subdeterminants are bounded by a constant and that contain at most two nonzero entries in each row. The core of our approach is the first polynomial-time algorithm for the weighted stable set problem on graphs that do not contain more than $k$ vertex-disjoint odd cycles, where $k$ is any constant. Previously, polynomial-time algorithms were only known for $k=0$ (bipartite graphs) and for $k=1$. We observe that integer linear programs defined by coefficient matrices with bounded subdeterminants and two nonzeros per column can be also solved in strongly polynomial-time, using a reduction to $b$-matching.
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In this paper, we give a spectral approximation result for the Laplacian on submanifolds of Euclidean spaces with singularities by the $\epsilon$-neighborhood graph constructed from random points on the submanifold. Our convergence rate for the eigenvalue of the Laplacian is $O\left(\left(\log n/n\right)^{1/(m+2)}\right)$, where $m$ and $n$ denote the dimension of the manifold and the sample size, respectively.
Given a property (graph class) $\Pi$, a graph $G$, and an integer $k$, the \emph{$\Pi$-completion} problem consists in deciding whether we can turn $G$ into a graph with the property $\Pi$ by adding at most $k$ edges to $G$. The $\Pi$-completion problem is known to be NP-hard for general graphs when $\Pi$ is the property of being a proper interval graph (PIG). In this work, we study the PIG-completion problem %when $\Pi$ is the class of proper interval graphs (PIG) within different subclasses of chordal graphs. We show that the problem remains NP-complete even when restricted to split graphs. We then turn our attention to positive results and present polynomial time algorithms to solve the PIG-completion problem when the input is restricted to caterpillar and threshold graphs. We also present an efficient algorithm for the minimum co-bipartite-completion for quasi-threshold graphs, which provides a lower bound for the PIG-completion problem within this graph class.
We study the relationship between the Quantum Approximate Optimization Algorithm (QAOA) and the underlying symmetries of the objective function to be optimized. Our approach formalizes the connection between quantum symmetry properties of the QAOA dynamics and the group of classical symmetries of the objective function. The connection is general and includes but is not limited to problems defined on graphs. We show a series of results exploring the connection and highlight examples of hard problem classes where a nontrivial symmetry subgroup can be obtained efficiently. In particular we show how classical objective function symmetries lead to invariant measurement outcome probabilities across states connected by such symmetries, independent of the choice of algorithm parameters or number of layers. To illustrate the power of the developed connection, we apply machine learning techniques towards predicting QAOA performance based on symmetry considerations. We provide numerical evidence that a small set of graph symmetry properties suffices to predict the minimum QAOA depth required to achieve a target approximation ratio on the MaxCut problem, in a practically important setting where QAOA parameter schedules are constrained to be linear and hence easier to optimize.
We investigate a family of bilevel imaging learning problems where the lower-level instance corresponds to a convex variational model involving first- and second-order nonsmooth regularizers. By using geometric properties of the primal-dual reformulation of the lower-level problem and introducing suitable changes of variables, we are able to reformulate the original bilevel problems as Mathematical Programs with Complementarity Constraints (MPCC). For the latter, we prove tight constraint qualification conditions (MPCC-MFCQ and partial MPCC-LICQ) and derive Mordukovich (M-) and Strong (S-) stationarity conditions. The S-stationarity system for the MPCC turns also into S-stationarity conditions for the original formulation. Second-order sufficient optimality conditions are derived as well. The proposed reformulation may be extended to problems in function spaces, leading to MPCC's with additional constraints on the gradient of the state. Finally, we report on some numerical results obtained by using the proposed MPCC reformulations together with available large-scale nonlinear programming solvers.
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.
The Infection Fatality Rate (IFR) of COVID-19 is difficult to estimate because the number of infections is unknown and there is a lag between each infection and the potentially subsequent death. We introduce a new approach for estimating the IFR by first estimating the entire sequence of daily infections. Unlike prior approaches, we incorporate existing data on the number of daily COVID-19 tests into our estimation; knowing the test rates helps us estimate the ratio between the number of cases and the number of infections. Also unlike prior approaches, rather than determining a constant lag from studying a group of patients, we treat the lag as a random variable, whose parameters we determine empirically by fitting our infections sequence to the sequence of deaths. Our approach allows us to narrow our estimation to smaller time intervals in order to observe how the IFR changes over time. We analyze a 250 day period starting on March 1, 2020. We estimate that the IFR in the U.S. decreases from a high of $0.68\%$ down to $0.24\%$ over the course of this time period. We also provide IFR and lag estimates for Italy, Denmark, and the Netherlands, all of which also exhibit decreasing IFRs but to different degrees.
Reversible computing basically means computation with less or not at all electrical power. Since the standard binary gates are not usually reversible we use the Fredkin gate in order to achieve reversibility. An algorithm for designing reversible digital circuits is described in this paper. The algorithm is based on Multi Expression Programming (MEP), a Genetic Programming variant with a linear representation of individuals. The case of digital circuits for the even-parity problem is investigated. Numerical experiments show that the MEP-based algorithm is able to easily design reversible digital circuits for up to the even-8-parity problem.
Death has long been overlooked in evolutionary algorithms. Recent research has shown that death (when applied properly) can benefit the overall fitness of a population and can outperform sub-sections of a population that are "immortal" when allowed to evolve together in an environment [1]. In this paper, we strive to experimentally determine whether death is an adapted trait and whether this adaptation can be used to enhance our implementations of conventional genetic algorithms. Using some of the most widely accepted evolutionary death and aging theories, we observed that senescent death (in various forms) can lower the total run-time of genetic algorithms, increase the optimality of a solution, and decrease the variance in an algorithm's performance. We believe that death-enhanced genetic algorithms can accomplish this through their unique ability to backtrack out of and/or avoid getting trapped in local optima altogether.
For a Hermitian matrix $H \in \mathbb C^{n,n}$ and symmetric matrices $S_0, S_1,\ldots,S_k \in \mathbb C^{n,n}$, we consider the problem of computing the supremum of $\left\{ \frac{v^*Hv}{v^*v}:~v\in \mathbb C^{n}\setminus \{0\},\,v^TS_iv=0~\text{for}~i=0,\ldots,k\right\}$. For this, we derive an estimation in the form of minimizing the second largest eigenvalue of a parameter depending Hermitian matrix, which is exact when the eigenvalue at the optimal is simple. The results are then applied to compute the eigenvalue backward errors of higher degree matrix polynomials with T-palindromic, T-antipalindromic, T-even, T-odd, and skew-symmetric structures. The results are illustrated by numerical experiments.
We demonstrate that many detection methods are designed to identify only a sufficently accurate bounding box, rather than the best available one. To address this issue we propose a simple and fast modification to the existing methods called Fitness NMS. This method is tested with the DeNet model and obtains a significantly improved MAP at greater localization accuracies without a loss in evaluation rate, and can be used in conjunction with Soft NMS for additional improvements. Next we derive a novel bounding box regression loss based on a set of IoU upper bounds that better matches the goal of IoU maximization while still providing good convergence properties. Following these novelties we investigate RoI clustering schemes for improving evaluation rates for the DeNet wide model variants and provide an analysis of localization performance at various input image dimensions. We obtain a MAP of 33.6%@79Hz and 41.8%@5Hz for MSCOCO and a Titan X (Maxwell). Source code available from: //github.com/lachlants/denet
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