arXiv subjects

P. M. Pardalos

Publications and source records attributed to P. M. Pardalos.

9 recordsLinked to original sources

arXiv2025

Delta-modular ILP Problems of Bounded Codimension, Discrepancy, and Convolution (new version)

For integers $k,n \geq 0$ and a cost vector $c \in Z^n$, we study two fundamental integer linear programming (ILP) problems: \[ \text{(Standard Form)} \quad \max\bigl\{c^\top x \colon Ax = b,\ x \in Z^n_{\geq 0}\bigr\} \text{ with } A \in Z^{k \times n}, \text{rank}(A) = k, b \in Z^k, \] \[ \text{(Canonical Form)} \quad \max\bigl\{c^\top x \colon Ax \leq b,\ x \in Z^n\bigr\} \text{ with } A \in Z^{(n+k) \times n}, \text{rank}(A) = n, b \in Z^{n+k}. \] We present improved algorithms for both problems and their feasibility versions, parameterized by $k$ and $Δ$, where $Δ$ denotes the maximum absolute value of $\text{rank}(A) \times \text{rank}(A)$ subdeterminants of $A$. Our main complexity results, stated in terms of required arithmetic operations, are: \[ \text{Optimization:}\quad O(\log k)^{2k} \cdot Δ^2 / 2^{Ω(\sqrt{\log Δ})} + 2^{O(k)} \cdot \text{poly}(φ), \] \[ \text{Feasibility:} \quad O(\log k)^k \cdot Δ\cdot (\log Δ)^3 + 2^{O(k)} \cdot \text{poly}(φ), \] where $φ$ represents the input size measured by the bit-encoding length of $(A,b,c)$. We also examine several special cases when $k \in \{0,1\}$, which have important applications in: expected computational complexity of ILP with varying right-hand side $b$, ILP problems with generic constraint matrices, ILP problems on simplices. Our results yield improved complexity bounds for these specific scenarios. As independent contributions, we present: An $n^2/2^{Ω(\sqrt{\log n})}$-time algorithm for the tropical convolution problem on sequences indexed by elements of a finite Abelian group of order $n$; A complete and self-contained error analysis of the generalized DFT over Abelian groups in the Word-RAM model.

cs.CC↗

arXiv2022

On $Δ$-Modular Integer Linear Problems In The Canonical Form And Equivalent Problems

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 $Δ(A)$, where $Δ(A)$ 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 $Δ(A)$, 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.

cs.CC↗

arXiv2021

Network diffusion capacity unveiled by dynamical paths

Improving the understanding of diffusive processes in networks with complex topologies is one of the main challenges of today's complexity science. Each network possesses an intrinsic diffusive potential that depends on its structural connectivity. However, the diffusion of a process depends not only on this topological potential but also on the dynamical process itself. Quantifying this potential will allow the design of more efficient systems in which it is necessary either to weaken or to enhance diffusion. Here we introduce a measure, the {\em diffusion capacity}, that quantifies, through the concept of dynamical paths, the potential of an element of the system, and also, of the system itself, to propagate information. Among other examples, we study a heat diffusion model and SIR model to demonstrate the value of the proposed measure. We found, in the last case, that diffusion capacity can be used as a predictor of the evolution of the spreading process. In general, we show that the diffusion capacity provides an efficient tool to evaluate the performance of systems, and also, to identify and quantify structural modifications that could improve diffusion mechanisms.

physics.data-an↗

arXiv2019

On the complexity of quasiconvex integer minimization problem

In this paper, we consider the class of quasiconvex functions and its proper subclass of conic functions. The integer minimization problem of these functions is considered in the paper, assuming that an optimized function is defined by the comparison oracle. We will show that there is no a polynomial algorithm on $\log R$ to optimize quasiconvex functions in the ball of integer radius $R$ using only the comparison oracle. On the other hand, if an optimized function is conic, then we show that there is a polynomial on $\log R$ algorithm. We also present an exponential on the dimension lower bound for the oracle complexity of the conic function integer optimization problem. Additionally, we give examples of known problems that can be polynomially reduced to the minimization problem of functions in our classes.

math.OC↗

arXiv2019

FPT-algorithms for some problems related to integer programming

In this paper, we present FPT-algorithms for special cases of the shortest lattice vector, integer linear programming, and simplex width computation problems, when matrices included in the problems' formulations are near square. The parameter is the maximum absolute value of rank minors of the corresponding matrices. Additionally, we present FPT-algorithms with respect to the same parameter for the problems, when the matrices have no singular rank sub-matrices.

math.OC↗

arXiv2018

Assessing diversity in multiplex networks

Diversity, understood as the variety of different elements or configurations that an extensive system has, is a crucial property that allows maintaining the system's functionality in a changing environment, where failures, random events or malicious attacks are often unavoidable. Despite the relevance of preserving diversity in the context of ecology, biology, transport, finances, etc., the elements or configurations that more contribute to the diversity are often unknown, and thus, they can not be protected against failures or environmental crises. This is due to the fact that there is no generic framework that allows identifying which elements or configurations have crucial roles in preserving the diversity of the system. Existing methods treat the level of heterogeneity of a system as a measure of its diversity, being unsuitable when systems are composed of a large number of elements with different attributes and types of interactions. Besides, with limited resources, one needs to find the best preservation policy, i.e., one needs to solve an optimization problem. Here we aim to bridge this gap by developing a metric between labeled graphs to compute the diversity of the system, which allows identifying the most relevant components, based on their contribution to a global diversity value. The proposed framework is suitable for large multiplex structures, which are constituted by a set of elements represented as nodes, which have different types of interactions, represented as layers. The proposed method allows us to find, in a genetic network (HIV-1), the elements with the highest diversity values, while in a European airline network, we systematically identify the companies that maximize (and those that less compromise) the variety of options for routes connecting different airports.

physics.soc-ph↗

arXiv2015

Optimal decision for the market graph identification problem in sign similarity network

Investigation of the market graph attracts a growing attention in market network analysis. One of the important problem connected with market graph is to identify it from observations. Traditional way for the market graph identification is to use a simple procedure based on statistical estimations of Pearson correlations between pairs of stocks. Recently a new class of statistical procedures for the market graph identification was introduced and optimality of these procedures in Pearson correlation Gaussian network was proved. However the obtained procedures have a high reliability only for Gaussian multivariate distributions of stocks attributes. One of the way to correct this drawback is to consider a different networks generated by different measures of pairwise similarity of stocks. A new and promising model in this context is the sign similarity network. In the present paper the market graph identification problem in sign similarity network is considered. A new class of statistical procedures for the market graph identification is introduced and optimality of these procedures is proved. Numerical experiments detect essential difference in quality of optimal procedures in sign similarity and Pearson correlation networks. In particular it is observed that the quality of optimal identification procedure in sign similarity network is not sensitive to the assumptions on distribution of stocks attributes.

q-fin.CP↗

arXiv2013

Measures of uncertainty in market network analysis

Statistical uncertainty of different filtration techniques for market network analysis is studied. Two measures of statistical uncertainty are discussed. One is based on conditional risk for multiple decision statistical procedures and another one is based on average fraction of errors. It is shown that for some important cases the second measure is a particular case of the first one. Statistical uncertainty for some popular market network structures is analyzed. Results of numerical evaluation of statistical uncertainty for minimum spanning tree, market graph, maximum cliques and maximum independent sets are given. The most stable structures are derived.

q-fin.ST↗

arXiv2008

Simulating Protein Conformations through Global Optimization

Many researches have been working on the protein folding problem from more than half century. Protein folding is indeed one of the major unsolved problems in science. In this work, we discuss a model for the simulation of protein conformations. This simple model is based on the idea of imposing few geometric requirements on chains of atoms representing the backbone of a protein conformation. The model leads to the formulation of a global optimization problem, whose solutions correspond to conformations satisfying the desired requirements. The global optimization problem is solved by the recently proposed Monkey Search algorithm. The simplicity of the optimization problem and the effectiveness of the used meta-heuristic search allowed the simulation of a large set of high-quality conformations. We show that, even though only few geometric requirements are imposed, some of the simulated conformation results to be similar (in terms of RMSD) to conformations real proteins actually have in nature.

math.OC↗