arXiv subjects

Janez Povh

Publications and source records attributed to Janez Povh.

15 recordsLinked to original sources

arXiv2026

Improving Upper Bounds for the Maximum Clique Problem using Reduction Rules

We study the interaction between reduction rules and upper-bound functions for the Maximum Clique Problem (MCP). We show how MCP upper-bound functions can strengthen classical core and truss reductions by replacing local size conditions with upper-bound tests. This leads to the $(k,ω^u)$-core, the $(k,ω^u)$-truss, and the more general $(k,d,ω^u)$-truss, where the parameter $d$ controls the trade-off between stronger reductions and additional computational cost. For each of these notions, we prove clique-preservation properties, correctness of the corresponding peeling algorithm, and running-time bounds. Based on these reductions, we introduce a general framework for improving upper-bound values for MCP. We give two concrete instantiations of the framework: one that uses only the combined truss and core reductions, and one that combines the truss and core reductions with repeated applications of structions. Computational experiments on 73 benchmark graphs show that the proposed reductions can substantially improve several standard upper-bound functions and that combining multiple reduction methods can be beneficial in practice. In particular, the combination of structions, truss and core reductions with a DSatur-based bound often reached SDP-level upper-bound values faster than direct SDP computation; on the tested graphs with edge density below $0.7$, it did so in every case. Using the truss and core reduction with the Lovász theta upper-bound function, we also improve the previously best certified integer upper-bound values for three difficult DIMACS instances whose exact clique numbers are not known. In particular, we improve upper-bound values for graph \texttt{C500.9} from 83 to 73, for graph \texttt{C1000.9} from 122 to 115, and for graph \texttt{C2000.9} from 177 to 168.

math.CO↗

arXiv2026

On solving symmetric multi-type orthogonal non-negative matrix tri-factorization problem

We study the symmetric multi-type orthogonal non-negative matrix tri-factorization problem, where several symmetric non-negative matrices are simultaneously approximated by factors of the form $GS_{i}G^{\top}$, with a shared non-negative and orthogonal factor $G$. This model is motivated by clustering and network analysis, where non-negativity improves interpretability and orthogonality gives a natural assignment-type structure to the latent factor. Since the resulting optimization problem is highly non-convex, we develop two heuristic algorithms for computing high-quality local solutions. The first one is a fixed point method derived from the Karush-Kuhn-Tucker conditions after adding a penalty term for the orthogonality constraint. The second one is a three-stage ADAM-based method that combines non-negativity-preserving optimization, orthogonalization, and restricted ADAM refinement on the feasible set. We evaluate both methods on synthetic data, including noisy instances, and on citation network benchmarks. The synthetic experiments show that both algorithms recover factorizations close to the optimum and remain stable under noise. On real networks, the learned embeddings are competitive with or better than standard baselines such as SVD, node2vec, and classical link prediction heuristics in link prediction, node clustering, and node classification tasks.

cs.LG↗

arXiv2025

Quantum computing and the stable set problem

Given an undirected graph, the stable set problem asks to determine the cardinality of the largest subset of pairwise non-adjacent vertices. This value is called the stability number of the graph, and its computation is an NP-hard problem. In this paper, we solve the stable set problem using the D-Wave quantum annealer. By formulating the problem as a quadratic unconstrained binary optimization problem with the penalty method, we show its optimal value equals the graph's stability number for specific penalty values. However, D-Wave's quantum annealer is a heuristic, so the solutions may be far from the optimum and may not represent stable sets. To address these, we introduce a post-processing procedure that identifies samples that could lead to improved solutions. Additionally, we propose a partitioning method to handle larger instances that cannot be embedded on D-Wave's quantum processing unit. Finally, we investigate how different penalty parameter values affect the solutions' quality. Extensive computational results show that the post-processing procedure significantly improves the solution quality, while the partitioning method successfully extends our approach to medium-size instances.

math.OC↗

arXiv2025

Quantum and Simulated Annealing-Based Iterative Algorithms for QUBO Relaxations of the Sparsest $k$-Subgraph Problem

In this paper, we introduce three QUBO (Quadratic Unconstrained Binary Optimization) relaxations for the sparsest $k$-subgraph (SkS) problem: a quadratic penalty relaxation, a Lagrangian relaxation, and an augmented Lagrangian relaxation. The effectiveness of these approaches strongly depends on the choice of penalty parameters. We establish theoretical results characterizing the parameter values for which the QUBO relaxations are exact. For practical implementation, we propose three iterative algorithms, which have in their kernel the QUBO relaxations, that update the penalty parameters at each iteration while approximately solving the internal QUBO problems with simulated annealing and quantum processing units. Extensive numerical experiments validate our theoretical findings on exact relaxations and demonstrate the efficiency of the proposed iterative algorithms.

math.OC↗

arXiv2024

Accuracy and Performance Evaluation of Quantum, Classical and Hybrid Solvers for the Max-Cut Problem

This paper investigates the performance of quantum, classical, and hybrid solvers on the NP-hard Max-Cut and QUBO problems, examining their solution quality relative to the global optima and their computational efficiency. We benchmark the new fast annealing D-Wave quantum processing unit (QPU) and D-Wave Hybrid solver against the state-of-the-art classical simulated annealing algorithm (SA) and Toshiba's simulated bifurcation machine (SBM). Our study leverages three datasets encompassing 139 instances of the Max-Cut problem with sizes ranging from 100 to 10,000 nodes. For instances below 251 nodes, global optima are known and reported, while for larger instances, we utilize the best-known solutions from the literature. Our findings reveal that for the smaller instances where the global optimum is known, the Hybrid solver and SA algorithm consistently achieve the global optimum, outperforming the QPU. For larger instances where global optima are unknown, we observe that the SBM and the slower variant of SA deliver competitive solution quality, while the Hybrid solver and the faster variant of SA performed noticeably worse. Although computing time varies due to differing underlying hardware, the Hybrid solver and the SBM demonstrate both efficient computation times, while for SA reduction in computation time can be achieved at the expense of solution quality.

math.OC↗

arXiv2023

Advancing stable set problem solutions through quantum annealers

We assess the performance of D-wave quantum solvers for solving the stable set problem in a graph, one of the most studied NP-hard problems. We perform computations on some instances from the literature with up to 125 vertices and compare the quality of the obtained solutions with known optimum solutions. It turns out that the hybrid solver gives very good results, while the Quantum Processing Unit solver shows rather modest performance overall.

math.OC↗

arXiv2023

Ground truth clustering is not the optimum clustering

The clustering of data is one of the most important and challenging topics in data science. The minimum sum-of-squares clustering (MSSC) problem asks to cluster the data points into $k$ clusters such that the sum of squared distances between the data points and their cluster centers (centroids) is minimized. This problem is NP-hard, but there exist exact solvers that can solve such problem to optimality for small or medium size instances. In this paper, we use a branch-and-bound solver based on semidefinite programming relaxations called SOS-SDP to compute the optimum solutions of the MSSC problem for various $k$ and for multiple datasets, with real and artificial data, for which the data provider has provided ground truth clustering. Next, we use several extrinsic and intrinsic measures to evaluate how the optimum clustering and ground truth clustering matches, and how well these clusterings perform with respect to the criteria underlying the intrinsic measures. Our calculations show that the ground truth clusterings are generally far from the optimum solution to the MSSC problem. Moreover, the intrinsic measures evaluated on the ground truth clusterings are generally significantly worse compared to the optimum clusterings. However, when the ground truth clustering is in the form of convex sets, e.g., ellipsoids, that are well separated from each other, the ground truth clustering comes very close to the optimum clustering.

math.OC↗

arXiv2021

BiqBin: a parallel branch-and-bound solver for binary quadratic problems with linear constraints

We present BiqBin, an exact solver for linearly constrained binary quadratic problems. Our approach is based on an exact penalty method to first efficiently transform the original problem into an instance of Max-Cut, and then to solve the Max-Cut problem by a branch-and-bound algorithm. All the main ingredients are carefully developed using new semidefinite programming relaxations obtained by strengthening the existing relaxations with a set of hypermetric inequalities, applying the bundle method as the bounding routine and using new strategies for exploring the branch-and-bound tree. Furthermore, an efficient C implementation of a sequential and a parallel branch-and-bound algorithm is presented. The latter is based on a load coordinator-worker scheme using MPI for multi-node parallelization and is evaluated on a high-performance computer. The new solver is benchmarked against BiqCrunch, GUROBI, and SCIP on four families of (linearly constrained) binary quadratic problems. Numerical results demonstrate that BiqBin is a highly competitive solver. The serial version outperforms the other three solvers on the majority of the benchmark instances. We also evaluate the parallel solver and show that it has good scaling properties. The general audience can use it as an on-line service available at http://www.biqbin.eu.

math.OC↗

arXiv2021

MADAM: A parallel exact solver for Max-Cut based on semidefinite programming and ADMM

We present MADAM, a parallel semidefinite based exact solver for Max-Cut, a problem of finding the cut with maximum weight in a given graph. The algorithm uses branch and bound paradigm that applies alternating direction method of multipliers as the bounding routine to solve the basic semidefinite relaxation strengthened by a subset of hypermetric inequalities. The benefit of the new approach is less computationally expensive update rule for the dual variable with respect to the inequality constraints. We provide theoretical convergence of the algorithm, as well as extensive computational experiments with this method, to show that our algorithm outperformes current state-of-the-art approaches. Furthermore, by combining algorithmic ingredients from the serial algorithm we develop an efficient distributed parallel solver based on MPI.

math.OC↗

arXiv2020

Four algorithms to solve symmetric multi-type non-negative matrix tri-factorization problem

In this paper, we consider the symmetric multi-type non-negative matrix tri-factorization problem (SNMTF), which attempts to factorize several symmetric non-negative matrices simultaneously. This can be considered as a generalization of the classical non-negative matrix tri-factorization problem and includes a non-convex objective function which is a multivariate sixth degree polynomial and a has convex feasibility set. It has a special importance in data science, since it serves as a mathematical model for the fusion of different data sources in data clustering. We develop four methods to solve the SNMTF. They are based on four theoretical approaches known from the literature: the fixed point method (FPM), the block-coordinate descent with projected gradient (BCD), the gradient method with exact line search (GM-ELS) and the adaptive moment estimation method (ADAM). For each of these methods we offer a software implementation: for the former two methods we use Matlab and for the latter Python with the TensorFlow library. We test these methods on three data-sets: the synthetic data-set we generated, while the others represent real-life similarities between different objects. Extensive numerical results show that with sufficient computing time all four methods perform satisfactorily and ADAM most often yields the best mean square error ($\mathrm{MSE}$). However, if the computation time is limited, FPM gives the best $\mathrm{MSE}$ because it shows the fastest convergence at the beginning. All data-sets and codes are publicly available on our GitLab profile.

cs.DS↗

arXiv2020

Sparse Noncommutative Polynomial Optimization

This article focuses on optimization of polynomials in noncommuting variables, while taking into account sparsity in the input data. A converging hierarchy of semidefinite relaxations for eigenvalue and trace optimization is provided. This hierarchy is a noncommutative analogue of results due to Lasserre [SIAM J. Optim. 17(3) (2006), pp. 822--843] and Waki et al. [SIAM J. Optim. 17(1) (2006), pp. 218--242]. The Gelfand-Naimark-Segal (GNS) construction is applied to extract optimizers if flatness and irreducibility conditions are satisfied. Among the main techniques used are amalgamation results from operator algebra. The theoretical results are utilized to compute lower bounds on minimal eigenvalue and trace of noncommutative polynomials from the literature.

math.OC↗

arXiv2020

A Block Coordinate Descent-based Projected Gradient Algorithm for Orthogonal Non-negative Matrix Factorization

This article utilizes the projected gradient method (PG) for a non-negative matrix factorization problem (NMF), where one or both matrix factors must have orthonormal columns or rows. We penalise the orthonormality constraints and apply the PG method via a block coordinate descent approach. This means that at a certain time one matrix factor is fixed and the other is updated by moving along the steepest descent direction computed from the penalised objective function and projecting onto the space of non-negative matrices. Our method is tested on two sets of synthetic data for various values of penalty parameters. The performance is compared to the well-known multiplicative update (MU) method from Ding (2006), and with a modified global convergent variant of the MU algorithm recently proposed by Mirzal (2014). We provide extensive numerical results coupled with appropriate visualizations, which demonstrate that our method is very competitive and usually outperforms the other two methods.

math.OC↗

arXiv2020

Collective dynamics of phase-repulsive oscillators solves graph coloring problem

We show how to couple phase-oscillators on a graph so that collective dynamics "searches" for the coloring of that graph as it relaxes toward the dynamical equilibrium. This translates a combinatorial optimization problem (graph coloring) into a functional optimization problem (finding and evaluating the global minimum of dynamical non-equilibrium potential, done by the natural system's evolution). Using a sample of graphs, we show that our method can serve as a viable alternative to the traditional combinatorial algorithms. Moreover, we show that, with the same computational cost, our method efficiently solves the harder problem of improper coloring of weighed graphs.

nlin.CD↗

arXiv2018

Is science driven by principal investigators?

In this paper we consider the scientific and career performance of principal investigators (PI's) of publicly funded research projects compared to scientific performance of all researchers. Our study is based on high quality data about (i) research projects awarded in Slovenia in the period 1994-2016 (7508 projects with 2725 PI's in total) and (ii) about scientific productivity of all researchers in Slovenia that were active in the period 1970-2016 - there are 19598 such researchers in total, including the PI's. We compare average productivity, collaboration, internationality and interdisciplinarity of PI's and of all active researchers. Our analysis shows that for all four indicators the average performance of PI's is much higher compared to average performance of all active researchers. Additionally, we analyze careers of both groups of researchers. The results show that the PI's have on average longer and more fruitful career compared to all active researchers, with regards to all career indicators. The PI's that have received a postdoc grant have at the beginning outstanding scientific performance, but later deviate towards average. On long run, the PI's leading the research programs (the most prestigious grants) on average demonstrate the best scientific performance. In the last part of the paper we study 23 co-authorship networks, spanned by all active researchers in the periods 1970-1994, ..., 1970-2016. We find out that they are well connected and that PI's are well distributed across these networks forming their backbones. Even more, PI's generate new PI's, since more than 90% of new PI's are connected (have at least one joint scientific publication) with existing PI's. We believe that our study sheds new light to the relations between the public funding of the science and the scientific output and can be considered as an affirmative answer to the question posed in the title.

cs.DL↗

arXiv2014

Community structure and the evolution of interdisciplinarity in Slovenia's scientific collaboration network

Interaction among the scientific disciplines is of vital importance in modern science. Focusing on the case of Slovenia, we study the dynamics of interdisciplinary sciences from 1960 to 2010. Our approach relies on quantifying the interdisciplinarity of research communities detected in the coauthorship network of Slovenian scientists over time. Examining the evolution of the community structure, we find that the frequency of interdisciplinary research is only proportional with the overall growth of the network. Although marginal improvements in favor of interdisciplinarity are inferable during the 70s and 80s, the overall trends during the past 20 years are constant and indicative of stalemate. We conclude that the flow of knowledge between different fields of research in Slovenia is in need of further stimulation.

physics.soc-ph↗