Nonnegative Weights Research Articles

Warm-Started QAOA with Custom Mixers Provably Converges and Computationally Beats Goemans-Williamson's Max-Cut at Low Circuit Depths

We generalize the Quantum Approximate Optimization Algorithm (QAOA) of Farhi et al. (2014) to allow for arbitrary separable initial states with corresponding mixers such that the starting state is the most excited state of the mixing Hamiltonian. We demonstrate this version of QAOA, which we call QAOA−warmest, by simulating Max-Cut on weighted graphs. We initialize the starting state as a warm−start using 2 and 3-dimensional approximations obtained using randomized projections of solutions to Max-Cut's semi-definite program, and define a warm-start dependent custommixer. We show that these warm-starts initialize the QAOA circuit with constant-factor approximations of 0.658 for 2-dimensional and 0.585 for 3-dimensional warm-starts for graphs with non-negative edge weights, improving upon previously known trivial (i.e., 0.5 for standard initialization) worst-case bounds at p=0. These factors in fact lower bound the approximation achieved for Max-Cut at higher circuit depths, since we also show that QAOA-warmest with any separable initial state converges to Max-Cut under the adiabatic limit as p→∞. However, the choice of warm-starts significantly impacts the rate of convergence to Max-Cut, and we show empirically that our warm-starts achieve a faster convergence compared to existing approaches. Additionally, our numerical simulations show higher quality cuts compared to standard QAOA, the classical Goemans-Williamson algorithm, and a warm-started QAOA without custom mixers for an instance library of 1148 graphs (upto 11 nodes) and depth p=8. We further show that QAOA-warmest outperforms the standard QAOA of Farhi et al. in experiments on current IBM-Q and Quantinuum hardware.

On a class of singular double phase problems with nonnegative weights whose sum can be zero

We study the existence, nonexistence and multiplicity of positive solutions for a class of one-dimensional double phase problems with nonnegative weights whose sum can be zero, singular nonlinearities and nonlinear boundary conditions. Such a weight condition is interesting in that it weakens the regularities of solutions, making it challenging to construct solution operators corresponding to the problems. We use a fixed point theorem to establish the existence and multiplicity of positive solutions according to the behaviors of the nonlinearity near 0 and ∞. In particular, we find intervals that guarantee at least three positive solutions for the problems with positive nonlinearities and at least two positive solutions for the problems with sign-changing nonlinearities.

Heuristic approach for bandwidth consecutive multi-coloring problem

Bandwidth consecutive multicoloring problem is also known as a b-coloring problem. Let G = (V, E) be a graph where V is the set of vertices and E is the set of edges. Let each vertex v of V has a positive integer weight b (v) and each edge (v, w) of E has a non-negative integer weight b (v, w). A bandwidth consecutive multicoloring of a graph is a problem of assigning b (v) consecutive positive integers to each vertex v of V in such a manner that the difference between all the integers of vertex v and all the integers of vertex w is greater than b (v, w). The maximum integer assigned in this coloring is called the span of the coloring. The b-coloring is the problem of minimizing this span. No metaheuristic is proposed for general graphs so far for this problem till date as it is strongly NP-hard. In this paper, we proposed three heuristics for the problem including a greedy randomized adaptive search procedure (GRASP). The efficiency of these algorithms is tested on benchmark graphs and their performance is compared among themselves. Experimental results showed that among all three proposed heuristics, GRASP performed well for the given problem.

Rates of convergence for Gibbs sampling in the analysis of almost exchangeable data

Motivated by de Finetti’s representation theorem for almost exchangeable arrays, we want to sample p∈[0,1]d from a distribution with density proportional to exp(−A2∑i<jcij(pi−pj)2), where A is large and cij’s are non-negative weights. We analyze the rate of convergence of a coordinate Gibbs sampler used to simulate from these measures. We show that for every non-zero fixed matrix C=(cij), and large enough A, mixing happens in Θ(A2) steps in a suitable Wasserstein distance. The upper and lower bounds are explicit and depend on the matrix C through few relevant spectral parameters.

A Range Space with Constant VC Dimension for All-pairs Shortest Paths in Graphs

Let $G$ be an undirected graph with non-negative edge weights and let $S$ be a subset of its shortest paths such that, for every pair $(u,v)$ of distinct vertices, $S$ contains exactly one shortest path between $u$ and $v$. In this paper we define a range space associated with $S$ and prove that its VC dimension is 2. As a consequence, we show a bound for the number of shortest paths trees required to be sampled in order to solve a relaxed version of the All-pairs Shortest Paths problem (APSP) in $G$. In this version of the problem we are interested in computing all shortest paths with a certain "importance" at least $\varepsilon$. Given any $0 &lt; \varepsilon$, $ \delta &lt; 1$, we propose a $\mathcal{O}(m + n \log n + (\textrm{diam}_{V(G)})^2)$ sampling algorithm that outputs with probability $1 - \delta$ the (exact) distance and the shortest path between every pair of vertices $(u, v)$ that appears as subpath of at least a proportion $\varepsilon$ of all shortest paths in the set $S$, where $\textrm{diam}_{V(G)}$ is the vertex-diameter of $G$. The bound that we obtain for the sample size depends only on $\varepsilon$ and $\delta$, and do not depend on the size of the graph.

Gauss-type quadrature rules for variable-sign weight functions

When the Gauss quadrature formula Gn is applied, it is often assumed that the weight function (or the measure) is non-negative on the integration interval [a,b]. In the present paper, we introduce a Gauss-type quadrature formula Qn for weight functions that change the sign in the interior of [a,b]. Construction of Qn is based on the idea to transform the given integral into a sum of one integral that does not cause a quadrature error and the other integral with a property that the points from the interior of [a,b] at which the weight function changes sign are the zeros of its integrand. It proves that all nodes of Qn are pairwise distinct and contained in the interior of [a,b]. Moreover, Gn (with a non-negative weight function) turns out to be a special case of Qn. Obtained results on the remainder term of Qn suggest that the application of Qn makes sense both when the points from the interior of [a,b] at which the weight function changes sign are known exactly, as well as when those points are known approximately. The accuracy of Qn is confirmed by numerical examples.

THE MINIMAL DOMINATING SETS IN A DIRECTED GRAPH AND THE KEY INDICATORS SET OF SOCIO–ECONOMIC SYSTEM

The paper deals with a digraph with non-negative vertex weights. A subset \(W\) of the set of vertices is called dominating if any vertex that not belongs to it is reachable from the set \(W\) within precisely one step. A dominating set is called minimal if it ceases to be dominating when removing any vertex from it. The paper investigates the problem of searching for a minimal dominating set of maximum weight in a vertex-weighted digraph. An integer linear programming model is proposed for this problem. The model is tested on random instances and the real problem of choosing a family of key indicators in a specific socio-economic system. The paper compares this model with the problem of choosing a dominating set with a fixed number of vertices.

A reference free non-negative adaptive learning system for health care monitoring and adaptive physiological artifact elimination in brain waves

Electroencephalogram (EEG), also referred to as brain wave (BW), is a physiological phenomenon that depicts how the human brain functions. Brain wave analysis is fundamental in applications like brain-computer interference (BCI), beamforming, sleep analysis, epilepsy detection, and emotion recognition. In real-time applications, the brain wave encounters many physiological and non-physiological artifacts during acquisition. Due to this phenomenon, the analysis method is complicated and obscures the brain wave’s tiny features. This study proposes an intelligent signal enhancement unit (SEU) for processing EEG signals to enable decision-making under certainty. The proposed SEU enables healthcare professionals to analyze high-resolution brain wave components for various applications. A new singular spectrum decomposition (SSD) based on score reconstruction (score RC) is used in the first phase of the SEU to generate the artifact nature, which is then used as a reference signal in the adaptive artifact cancellation (AAC) method. The SSD performs the embedding, decomposition, grouping, and reconstruction procedures to provide the reference signal. A modified Logarithmic Non-Negative Adaptive Learning Algorithm (MLNNAL) is employed in the second stage of the SEU to improve the EEG signal. With the help of this proposed adaptive learning, a system with lower computing complexity which is stable and has non-negative weights can be realized. The adaptive learning algorithm’s weight recursion continually reweights the weight vector for each iteration to eliminate artifacts from contaminated brain waves. Excess mean square error (EMSE), signal to noise ratio improvement (SNRI) and computational cost of the adaptive learning algorithm are used to evaluate how the proposed SEU performs.

Some characterizations of dynamic weighted Hardy-type inequalities with applications

The pairs of non-negative weights (u,v) for which dynamic inequalities of Hardy's type on a time scale T hold are (p,q) characterized in two different time-scale spaces Lvp(T) and Luq(T). We consider two distinct scenarios of values of the exponents p and q. These results complement the classical strong (p,p) results and their generalizations considered by some authors. For applications, we provide the corresponding continuous and discrete results when T=R, T=N, and T is the quantum space ℓN0 for ℓ>1, respectively. As special cases, we capture some consequences for the results obtained by Anderson, Heinig, Kufner, Flett, and Saker.

Approximation Algorithms for the Maximum-Weight Cycle/Path Packing Problems

Given an undirected complete graph [Formula: see text] on [Formula: see text] vertices with a non-negative weight function on [Formula: see text], the maximum-weight [Formula: see text]-cycle ([Formula: see text]-path) packing problem aims to compute a set of [Formula: see text] vertex-disjoint cycles (paths) in [Formula: see text] containing [Formula: see text] vertices so that the total weight of the edges in these [Formula: see text] cycles (paths) is maximized. For the maximum-weight [Formula: see text]-cycle packing problem, we develop an algorithm achieving an approximation ratio of [Formula: see text], where [Formula: see text] is the approximation ratio for the maximum traveling salesman problem. For the case [Formula: see text], we design a better [Formula: see text]-approximation algorithm. When the weights of edges obey the triangle inequality, we propose a [Formula: see text]-approximation algorithm and a [Formula: see text]-approximation algorithm for the maximum-weight [Formula: see text]-cycle packing problem with [Formula: see text] and [Formula: see text], respectively. For the maximum-weight [Formula: see text]-path packing problem with [Formula: see text] (or [Formula: see text]) with the triangle inequality, we devise an algorithm with approximation ratio [Formula: see text] and give a tight example.

A general efficient neighborhood structure framework for the job-shop and flexible job-shop scheduling problems

This article introduces a framework that unifies and generalizes well-known literature results related to local search for the job-shop and flexible job-shop scheduling problems. In addition to the choice of the metaheuristic and the neighborhood structure, the success of most of the influential local search approaches relies on the ability to quickly and efficiently rule out infeasible moves and evaluate the quality of the feasible neighbors. Hence, the proposed framework focuses on the feasibility and quality evaluation of a general move when solving the job-shop and flexible job-shop scheduling problems for any regular objective function. The proposed framework is valid for any scheduling problem where the defined neighborhood structure is appropriate, and each solution to the problem can be modeled with a directed acyclic graph with non-negative weights on nodes and arcs. The feasibility conditions and quality estimation procedures proposed in the literature rely heavily on information on the existence of a path between two nodes. Thus, based on an original parameterized algorithm that asserts the existence of a path between two nodes, novel generic procedures to evaluate the feasibility of a move and estimate the value of any regular objective function of a neighbor solution are proposed. We show that many well-known literature results are special cases of our results, which can be applied to a wide range of shop scheduling problems.

New algorithms for a simple measure of network partitioning

Partitioning a network into k pieces is a fundamental problem in network science. A simple measure of partitioning a network is provided by the Maxk-Uncut problem. Given an n-vertex undirected graph G with nonnegative weights defined on edges, and a positive integer k, the Maxk-Uncut problem asks to find a k-partition of the vertices of G to maximize the total weight of edges that are not in the cut. This problem is the complement of the classic Mink-Cut problem, and has close relation to many combinatorial optimization problems, including the famous Densestk-Subgraph problem. In this paper, we propose a greedy approximation algorithm for the Maxk-Uncut problem with performance ratio 1−2(k−1)n. The algorithm is very simple, which consists of only k−1 min cut computations. The algorithm has fast running time O(kn2) and is hence implementable. The experimental results show that the algorithm has excellent practical performance.

Graphs with $$G^p$$-connected medians

The median of a graph G with weighted vertices is the set of all vertices x minimizing the sum of weighted distances from x to the vertices of G. For any integer $$p\ge 2$$ , we characterize the graphs in which, with respect to any non-negative weights, median sets always induce connected subgraphs in the pth power $$G^p$$ of G. This extends some characterizations of graphs with connected medians (case $$p=1$$ ) provided by Bandelt and Chepoi (SIAM J Discrete Math 15(2):268–282, 2002. https://doi.org/10.1137/S089548019936360X ). The characteristic conditions can be tested in polynomial time for any p. We also show that several important classes of graphs in metric graph theory, including bridged graphs (and thus chordal graphs), graphs with convex balls, bucolic graphs, and bipartite absolute retracts, have $$G^2$$ -connected medians. Extending the result of Bandelt and Chepoi that basis graphs of matroids are graphs with connected medians, we characterize the isometric subgraphs of Johnson graphs and of halved-cubes with connected medians.

Optimal controller synthesis for timed systems

Weighted timed games are zero-sum games played by two players on a timed automaton equipped with weights, where one player wants to minimise the cumulative weight while reaching a target. Used in a reactive synthesis perspective, this quantitative extension of timed games allows one to measure the quality of controllers in real-time systems. Weighted timed games are notoriously difficult and quickly undecidable, even when restricted to non-negative weights. For non-negative weights, the largest class that can be analysed has been introduced by Bouyer, Jaziri and Markey in 2015. Though the value problem is undecidable, the authors show how to approximate the value by considering regions with a refined granularity. In this work, we extend this class to incorporate negative weights, allowing one to model energy for instance, and prove that the value can still be approximated, with the same complexity. A small restriction also allows us to obtain a class of decidable weighted timed games with negative weights and an arbitrary number of clocks. In addition, we show that a symbolic algorithm, relying on the paradigm of value iteration, can be used as an approximation/computation schema over these classes. We also consider the special case of untimed weighted games, where the same fragments are solvable in polynomial time: this contrasts with the pseudo-polynomial complexity, known so far, for weighted games without restrictions.

Packing Feedback Arc Sets in Tournaments Exactly

Let [Formula: see text] be a tournament with a nonnegative integral weight w(e) on each arc e. A subset F of arcs is called a feedback arc set (FAS) if [Formula: see text] contains no cycles (directed). A collection [Formula: see text] of FASs (with repetition allowed) is called an FAS packing if each arc e is used at most w(e) times by the members of [Formula: see text]. The purpose of this paper is to give a characterization of all tournaments [Formula: see text] with the property that, for every nonnegative integral weight function w defined on A, the minimum total weight of a cycle is equal to the maximum size of an FAS packing. Funding: This work was supported by the National Natural Science Foundation of China [Grants 11801266 and 11971228]; the Chinese Academy of Sciences [Grants XDA27000000 and ZDBS-LY-7008]; the Ministry of Science and Technology of China [Grant 2018AAA0101002]; the Research Grants Council of Hong Kong; the Fundamental Research Funds for the Central Universities [Grant 020314380035].

On the density of multivariate polynomials with varying weights

In this paper we consider multivariate approximation by weighted polynomials of the form w γ n ( x ) p n ( x ) w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x}) , where p n p_n is a multivariate polynomial of degree at most n n , w w is a given nonnegative weight with nonempty zero set, and γ n ↑ ∞ \gamma _n\uparrow \infty . We study the question if every continuous function vanishing on the zero set of w w is a uniform limit of weighted polynomials w γ n ( x ) p n ( x ) w^{\gamma _n}(\mathbf {x})p_n(\mathbf {x}) . It turns out that for various classes of weights in order for this approximation property to hold it is necessary and sufficient that γ n = o ( n ) . \gamma _n=o(n).

Subcubic Equivalences between Graph Centrality Problems, APSP, and Diameter

Measuring the importance of a node in a network is a major goal in the analysis of social networks, biological systems, transportation networks, and so forth. Different centrality measures have been proposed to capture the notion of node importance. For example, the center of a graph is a node that minimizes the maximum distance to any other node (the latter distance is the radius of the graph). The median of a graph is a node that minimizes the sum of the distances to all other nodes. Informally, the betweenness centrality of a node w measures the fraction of shortest paths that have w as an intermediate node. Finally, the reach centrality of a node w is the smallest distance r such that any s - t shortest path passing through w has either s or t in the ball of radius r around w . The fastest known algorithms to compute the center and the median of a graph and to compute the betweenness or reach centrality even of a single node take roughly cubic time in the number n of nodes in the input graph. It is open whether these problems admit truly subcubic algorithms, i.e., algorithms with running time Õ(n 3-δ ) for some constant δ &gt; 0. 1 We relate the complexity of the mentioned centrality problems to two classical problems for which no truly subcubic algorithm is known, namely All Pairs Shortest Paths (APSP) and Diameter. We show that Radius, Median, and Betweenness Centrality are equivalent under subcubic reductions to APSP, i.e., that a truly subcubic algorithm for any of these problems implies a truly subcubic algorithm for all of them. We then show that Reach Centrality is equivalent to Diameter under subcubic reductions. The same holds for the problem of approximating Betweenness Centrality within any finite factor. Thus, the latter two centrality problems could potentially be solved in truly subcubic time, even if APSP required essentially cubic time. On the positive side, our reductions for Reach Centrality imply an improved Õ(Mn ω )-time algorithm for this problem in case of non-negative integer weights upper bounded by M , where ω is a fast matrix multiplication exponent.

A new full-Newton step feasible interior point method for convex quadratic programming

In this paper, we propose and analyse a new full-Newton step feasible interior point method for convex quadratic programming. The basic idea of this method is to replace a complementarity condition by a non-negative variable weight vector. With a zero of weight vector, the limit of the weighted path exists and satisfies the complementarity condition, the limit yields an optimal solution of problem. In each main iteration of the new algorithm consisted of only full-Newton steps with a quadratic rate of convergence. The advantage of this method is the use of a full-Newton step, that is no calculation of the step size is required. Finally, some numerical results are reported to show the practical performance of the proposed algorithm with different parameters.

Weighted combinatorial Ricci flow and metrics defined by degenerate circle packings

Chow and Luo [1] in 2003 had shown that the combinatorial analogue of the Hamilton Ricci flow on surfaces under certain conditions converges to Thruston?s circle packing metric of constant curvature. The combinatorial setting includes weights defined for edges of a triangulation. Crucial assumption in the paper [1] was that the weights are nonnegative. Recently we have shown that same statement on convergence can be proved under weaker condition: some weights can be negative and should satisfy certain inequalities, [3]. Moreover, in [6] notions of degenerate circle packing and corresponding metric were introduced. In [6] theory of combinatorial Ricci flow for such metrics was developed, which includes Chow-Luo theory as a partial case for nondegenerate circle packing and nonnegative weights on edges. On the other hand, in [2] the combinatorial Yamabe flow was introduced and investigated. In [7, 8] we developed weighted modification of Yamabe flow. In this paper we merge ideas from these two theories and introduce weighted combinatorial Ricci flow on metrics defined by degenerate circle packings. We prove that under certain conditions for any initial metric the flow converges to a unique metric of constant curvature.

A chaotic discrete-time continuous-state Hopfield network with piecewise-affine activation functions

We construct a chaotic discrete-time continuous-state Hopfield networkwith piecewise-affine nonnegative activation functions and weight matrix with small positive entries. More precisely, there exists a Cantor set $ C $ in the state space such that the network has sensitive dependence on initial conditions at initial states in $ C $ and the network orbit of each initial state in $ C $ has $ C $ as its $ \omega $-limit set. The approach we use is based on tools developed and employed recently in the study of the topological dynamics of piecewise-contractions. The parameters of the chaotic network are explicitly given.