arXiv subjects

Miroslav Rada

Publications and source records attributed to Miroslav Rada.

10 recordsLinked to original sources

arXiv2026

Heuristics for the Worst Optimal Value of Interval Transportation Problems

An interval transportation problem represents a model for a transportation problem in which the values of supply, demand, and transportation costs are affected by uncertainty and can vary independently within given interval ranges. One of the main tasks of solving interval programming models is computing the best and worst optimal value over all possible choices of the interval data. Although the best optimal value of an interval transportation problem can be computed in polynomial time, computing the worst (finite) optimal value was proved to be NP-hard. In this paper, we strengthen a previous result showing a quasi-extreme decomposition for finding the worst optimal value, and building on the result, we design heuristics for efficiently approximating the value. Using a simplified encoding of the scenarios, we first derive a local search method and a genetic algorithm for approximating the worst optimal value. Then, we integrate the two methods into a memetic algorithm, which combines the evolutionary improvement of a genetic algorithm with individual learning implemented via local search. Moreover, we include numerical experiments for a practical comparison of the three different approaches. We also show that the proposed memetic algorithm is competitive with the available state-of-the-art methods for approximating the worst optimal value of interval transportation problems, this is demonstrated by finding the new best solutions for several instances, among others.

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arXiv2026

Minimum cost network flow with interval capacities: The worst-case scenario

We study the problem of determining the worst optimal value and characterizing the corresponding worst-case scenarios in minimum cost network flow problems with interval uncertainty in arc capacities. In this setting, each capacity can take any value within its specified lower and upper bounds. We prove that computing the worst optimal value is a strongly NP-hard problem and remains NP-hard even when restricted to series-parallel graphs. Further, we propose a mixed-integer linear programming formulation that computes the exact worst optimal value, as well as a pseudopolynomial-time algorithm designed for the special case of series-parallel graphs. We also examine the structural properties of the most extremal worst-case scenarios and show that the arcs whose capacities are not fixed at their interval bounds form a forest. This result establishes an upper bound on the number of such arcs, which we show to be tight by constructing a class of instances in which the bound is attained. Finally, we investigate the more-for-less paradox in minimum cost network flow problems with interval capacities, which occurs in instances where increasing the required flow leads to a decrease in the worst-case optimal cost. We provide a general characterization of this phenomenon using augmenting paths and establish a stronger characterization for complete graphs. In addition, we discuss the properties of the cost matrices immune against the paradox and prove that deciding whether a given cost matrix has this property is a strongly co-NP-hard problem.

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arXiv2024

Dispersion of personal spaces

There are many entities that disseminate in the physical space - information, gossip, mood, innovation etc. Personal spaces are also entities that disperse and interplay. In this work we study the emergence of configurations formed by participants when choosing a place to sit in a rectangular auditorium. Based on experimental questionnaire data we design several models and assess their relevancy to a real time-lapse footage of lecture hall being filled up. The main focus is to compare the evolution of entropy of occupied seat configurations in time. Even though the process of choosing a seat is complex and could depend on various properties of participants or environment, some of the developed models can capture at least basic essence of the real processes. After introducing the problem of seat selection and related results in close research areas, we introduce preliminary collected data and build models of seat selection based on them. We compare the resulting models to the real observational data and discuss areas of future research directions.

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arXiv2023

Interval Transportation Problem: Feasibility, Optimality and the Worst Optimal Value

We consider the model of a transportation problem with the objective of finding a minimum-cost transportation plan for shipping a given commodity from a set of supply centers to the customers. Since the exact values of supply and demand and the exact transportation costs are not always available for real-world problems, we adopt the approach of interval programming to represent such uncertainty, resulting in the model of an interval transportation problem. The interval model assumes that lower and upper bounds on the data are given and the values can be independently perturbed within these bounds. In this paper, we provide an overview of conditions for checking basic properties of the interval transportation problems commonly studied in interval programming, such as weak and strong feasibility or optimality. We derive a condition for testing weak optimality of a solution in polynomial time by finding a suitable scenario of the problem. Further, we formulate a similar condition for testing strong optimality of a solution for transportation problems with interval supply and demand (and exact costs). Moreover, we also survey the results on computing the best and the worst optimal value. We build on an exact method for solving the NP-hard problem of computing the worst (finite) optimal value of the interval transportation problem based on a decomposition of the optimal solution set by complementary slackness. Finally, we conduct computational experiments to show that the method can be competitive with the state-of-the-art heuristic algorithms.

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arXiv2022

The NP-hard problem of computing the maximal sample variance over interval data is solvable in almost linear time with high probability

We consider the algorithm by Ferson et al. (Reliable computing 11(3), p. 207-233, 2005) designed for solving the NP-hard problem of computing the maximal sample variance over interval data, motivated by robust statistics (in fact, the formulation can be written as a nonconvex quadratic program with a specific structure). First, we propose a new version of the algorithm improving its original time bound $O(n^2 2^ω)$ to $O(n \log n+n\cdot 2^ω)$, where $n$ is number of input data and $ω$ is the clique number in a certain intersection graph. Then we treat input data as random variables as it is usual in statistics) and introduce a natural probabilistic data generating model. We get $2^ω= O(n^{1/\log\log n})$ and $ω= O(\log n / \log\log n)$ on average. This results in average computing time $O(n^{1+ε})$ for $ε> 0$ arbitrarily small, which may be considered as "surprisingly good" average time complexity for solving an NP-hard problem. Moreover, we prove the following tail bound on the distribution of computation time: hard instances, forcing the algorithm to compute in time $2^{Ω(n)}$, occur rarely, with probability tending to zero at the rate $e^{-n\log\log n}$.

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arXiv2019

Walks on hyperplane arrangements and optimization of piecewise linear functions

We propose an exact iterative algorithm for minimization of a class of continuous cell-wise linear convex functions on a hyperplane arrangement. Our particular setup is motivated by evaluation of so-called rank estimators used in robust regression, where every cell of the underlying arrangement corresponds to a permutation of residuals (and we also show that the class of function for which the method works is more general). The main obstacle in the construction of the algorithm is how to find an improving direction while standing in a point incident with exponentially many cells of the arrangement. We overcome this difficulty using Birkhoff Theorem which allows us to express the cone of improving directions in the exponential number of cells using a linear system with quadratic number of variables only.

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arXiv2019

A new polynomially solvable class of quadratic optimization problems with box constraints

We consider the quadratic optimization problem $\max_{x \in C}\ x^T Q x + q^T x$, where $C\subseteq\mathbb{R}^n$ is a box and $r := \mathrm{rank}(Q)$ is assumed to be $\mathcal{O}(1)$ (i.e., fixed). We show that this case can be solved in polynomial time for an arbitrary $Q$ and $q$. The idea is based on a reduction of the problem to enumeration of faces of a certain zonotope in dimension $O(r)$. This paper generalizes previous results where $Q$ had been assumed to be positive semidefinite and no linear term was allowed in the objective function. Positive definiteness was a strong restriction and it is now relaxed. Generally, the problem is NP-hard; this paper describes a new polynomially solvable class of instances, larger than those known previously.

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arXiv2019

A class of optimization problems motivated by rank estimators in robust regression

A rank estimator in robust regression is a minimizer of a function which depends (in addition to other factors) on the ordering of residuals but not on their values. Here we focus on the optimization aspects of rank estimators. We distinguish two classes of functions: the class with a continuous and convex objective function (CCC), which covers the class of rank estimators known from statistics, and also another class (GEN), which is far more general. We propose efficient algorithms for both classes. For GEN we propose an enumerative algorithm that works in polynomial time as long as the number of regressors is O(1). The proposed algorithm utilizes the special structure of arrangements of hyperplanes that occur in our problem and is superior to other known algorithms in this area. For the continuous and convex case, we propose an unconditionally polynomial algorithm finding the exact minimizer, unlike the heuristic or approximate methods implemented in statistical packages.

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arXiv2018

Interval Linear Programming under Transformations: Optimal Solutions and Optimal Value Range

Interval linear programming provides a tool for solving real-world optimization problems under interval-valued uncertainty. Instead of approximating or estimating crisp input data, the coefficients of an interval program may perturb independently within the given lower and upper bounds. However, contrarily to classical linear programming, an interval program cannot always be converted into a desired form without affecting its properties, due to the so-called dependency problem. In this paper, we discuss the common transformations used in linear programming, such as imposing non-negativity on free variables or splitting equations into inequalities, and their effects on interval programs. Specifically, we examine changes in the set of all optimal solutions, optimal values and the optimal value range. Since some of the considered properties do not holds in the general case, we also study a special class of interval programs, in which uncertainty only affects the objective function and the right-hand-side vector. For this class, we obtain stronger results.

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arXiv2017

Testing weak optimality of a given solution in interval linear programming revisited: NP-hardness proof, algorithm and some polynomial cases

We address the problem of testing weak optimality of a given solution of a given interval linear program. The problem was recently wrongly stated to be polynomially solvable. We disprove it. We show that the problem is NP-hard in general. We propose a new algorithm for the problem, based on orthant decomposition and solving linear systems. Running time of the algorithm is exponential in the number of equality constraints. Interval linear programs with inequality constraints only can be processed in polynomial time.

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