Tina Novak | University of Ljubljana (original) (raw)

Papers by Tina Novak

Operations Research Forum, Sep 2, 2022

A set D of vertices in a graph G is a k-fair dominating set if every vertex not in D is adjacent ... more A set D of vertices in a graph G is a k-fair dominating set if every vertex not in D is adjacent to exactly k vertices in D. The weighted k-fair domination number wfd k (G) of a vertex-weighted graph G is the minimum weight w(D) among all k-fair dominating sets D. In addition to the weighted k-fair domination number, some auxiliary parameters are defined. It is shown that for a cactus graph, the weighted k-fair domination number and auxiliary parameters can be calculated in linear time.

PloS one, 2017

The rapid worldwide evolution of LEDs as light sources has brought new challenges, which means th... more The rapid worldwide evolution of LEDs as light sources has brought new challenges, which means that new methods are needed and new algorithms have to be developed. Since the majority of LED luminaries are of the multi-source type, established methods for the design of light engines cannot be used in the design of LED light engines. This is because in the latter case what is involved is not just the design of a good reflector or projector lens, but the design of several lenses which have to work together in order to achieve satisfactory results. Since lenses can also be bought off the shelf from several manufacturers, it should be possible to combine together different off the shelf lenses in order to design a good light engine. However, with so many different lenses to choose from, it is almost impossible to find an optimal combination by hand, which means that some optimization algorithms need to be applied. In order for them to work properly, it is first necessary to describe the ...

The topology of Liouville sets of the real forms of the complex generic Neumann system depends in... more The topology of Liouville sets of the real forms of the complex generic Neumann system depends indirectly on the roots of the special polynomial U_ S(λ). For certain polynomials, the existence and positions of the real roots, according to the suitable parameters of the system, is not obvious. In the paper, a novel method for checking the existence and positions of the real roots of the polynomials U_ S(λ) is given. The method and algorithm are based on searching of a positive solution of a system of linear equations. We provide a complete solution to the problem of existence of real roots for all special polynomials in case n=2. This is a step closer to determining the topology of the Liouville sets.

J. Comput. Appl. Math., 2021

The topology of Liouville sets of the real forms of the complex generic Neumann system depends in... more The topology of Liouville sets of the real forms of the complex generic Neumann system depends indirectly on the roots of the special polynomial US(λ). For certain polynomials, the existence and positions of the real roots, according to the suitable parameters of the system, is not obvious. In the paper, a novel method for checking the existence and positions of the real roots of the polynomials US(λ) is given. The method and algorithm are based on searching of a positive solution of a system of linear equations. We provide a complete solution to the problem of existence of real roots for all special polynomials in case n = 2. This is a step closer to determining the topology of the Liouville sets.

Abstract: In this paper we propose a linear algorithm for calculating the weighted domination num... more Abstract: In this paper we propose a linear algorithm for calculating the weighted domination number of a vertex-weighted cactus. The algorithm is based on the well known depth first search (DFS) structure. Our algorithm needs less than 12n + 5b additions and 9n + 2b min-operations where n is the number of vertices and b is the number of blocks in the cactus.

International Journal of Apllied Mathematics, Aug 9, 2016

In this paper we propose a linear algorithm for calculating the weighted domination number of a v... more In this paper we propose a linear algorithm for calculating the weighted domination number of a vertex-weighted cactus. The algorithm is based on the well known depth first search (DFS) structure. Our algorithm needs less than 12n + 5b additions and 9n + 2b min-operations where n is the number of vertices and b is the number of blocks in the cactus.

Mathematics

The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem f... more The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k≥3. In this paper we introduce five new heuristics. Two algorithms, Bm and Cm, arise as natural improvements of Algorithm Am from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004). The other three algorithms, Dm, Em, and Fm, incorporate randomization. Algorithm Dm can be considered as a greedy version of Bm, whereas Em and Fm are versions of local search algorithm, specialized for the k-matching problem. The algorithms are implemented in Python and are run on three datasets. On the datasets available, all the algorithms clearly outperform Algorithm Am in terms of solution quality. On the first dataset with known optimal values the average relative error ranges from 1.47% over optimum (algorithm Am) to 0.08% over optimum (algorithm Em). On the second dataset with known optimal values the average relative error ranges from 4.41% over optimum (algorithm Am) to...

Ars Mathematica Contemporanea

The modified Hosoya polynomial of double weighted graphs, i.e. edge and vertex weighted graphs, i... more The modified Hosoya polynomial of double weighted graphs, i.e. edge and vertex weighted graphs, is introduced that enables derivation of closed expressions for Hosoya polynomial of some special graphs including unicyclic graphs. Furthermore, the Hosoya polynomial is given as a sum of edge contributions generalizing well known analogous results for the Wiener number. A linear algorithm for computing the Hosoya polynomial on cactus graphs is provided. Hosoya polynomial is extensively studied in chemical graph theory, and in particular its weighted versions have interesting applications in theory of communication networks.

Journal of Nonlinear Mathematical Physics, 2016

In the paper, we study real forms of the complex generic Neumann system. We prove that the real f... more In the paper, we study real forms of the complex generic Neumann system. We prove that the real forms are completely integrable Hamiltonian systems. The complex Neumann system is an example of the more general Mumford system. The Mumford system is characterized by the Lax pair (L C (λ), M C (λ)) of 2 × 2 matrices, where L C (λ) = V C (λ) W C (λ) U C (λ) −V C (λ) and U C (λ), V C (λ), W C (λ) are suitable polynomials. The topology of a regular level set of the moment map of a real form is determined by the positions of the roots of the suitable real form of U C (λ), with respect to the position of the values of suitable parameters of the system. For two families of the real forms of the complex Neumann system, we describe the topology of the regular level set of the moment map. For one of these two families the level sets are noncompact. In the paper, we also give the formula which provides the relation between two systems of the first integrals in involution of the Neumann system. One of these systems is obtained from the Lax pair of the Mumford type, while the second is obtained from the Lax pair whose matrices are of dimension (n + 1) × (n + 1).

Operations Research Forum, Sep 2, 2022

A set D of vertices in a graph G is a k-fair dominating set if every vertex not in D is adjacent ... more A set D of vertices in a graph G is a k-fair dominating set if every vertex not in D is adjacent to exactly k vertices in D. The weighted k-fair domination number wfd k (G) of a vertex-weighted graph G is the minimum weight w(D) among all k-fair dominating sets D. In addition to the weighted k-fair domination number, some auxiliary parameters are defined. It is shown that for a cactus graph, the weighted k-fair domination number and auxiliary parameters can be calculated in linear time.

PloS one, 2017

The rapid worldwide evolution of LEDs as light sources has brought new challenges, which means th... more The rapid worldwide evolution of LEDs as light sources has brought new challenges, which means that new methods are needed and new algorithms have to be developed. Since the majority of LED luminaries are of the multi-source type, established methods for the design of light engines cannot be used in the design of LED light engines. This is because in the latter case what is involved is not just the design of a good reflector or projector lens, but the design of several lenses which have to work together in order to achieve satisfactory results. Since lenses can also be bought off the shelf from several manufacturers, it should be possible to combine together different off the shelf lenses in order to design a good light engine. However, with so many different lenses to choose from, it is almost impossible to find an optimal combination by hand, which means that some optimization algorithms need to be applied. In order for them to work properly, it is first necessary to describe the ...

The topology of Liouville sets of the real forms of the complex generic Neumann system depends in... more The topology of Liouville sets of the real forms of the complex generic Neumann system depends indirectly on the roots of the special polynomial U_ S(λ). For certain polynomials, the existence and positions of the real roots, according to the suitable parameters of the system, is not obvious. In the paper, a novel method for checking the existence and positions of the real roots of the polynomials U_ S(λ) is given. The method and algorithm are based on searching of a positive solution of a system of linear equations. We provide a complete solution to the problem of existence of real roots for all special polynomials in case n=2. This is a step closer to determining the topology of the Liouville sets.

J. Comput. Appl. Math., 2021

The topology of Liouville sets of the real forms of the complex generic Neumann system depends in... more The topology of Liouville sets of the real forms of the complex generic Neumann system depends indirectly on the roots of the special polynomial US(λ). For certain polynomials, the existence and positions of the real roots, according to the suitable parameters of the system, is not obvious. In the paper, a novel method for checking the existence and positions of the real roots of the polynomials US(λ) is given. The method and algorithm are based on searching of a positive solution of a system of linear equations. We provide a complete solution to the problem of existence of real roots for all special polynomials in case n = 2. This is a step closer to determining the topology of the Liouville sets.

Abstract: In this paper we propose a linear algorithm for calculating the weighted domination num... more Abstract: In this paper we propose a linear algorithm for calculating the weighted domination number of a vertex-weighted cactus. The algorithm is based on the well known depth first search (DFS) structure. Our algorithm needs less than 12n + 5b additions and 9n + 2b min-operations where n is the number of vertices and b is the number of blocks in the cactus.

International Journal of Apllied Mathematics, Aug 9, 2016

In this paper we propose a linear algorithm for calculating the weighted domination number of a v... more In this paper we propose a linear algorithm for calculating the weighted domination number of a vertex-weighted cactus. The algorithm is based on the well known depth first search (DFS) structure. Our algorithm needs less than 12n + 5b additions and 9n + 2b min-operations where n is the number of vertices and b is the number of blocks in the cactus.

Mathematics

The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem f... more The k-assignment problem (or, the k-matching problem) on k-partite graphs is an NP-hard problem for k≥3. In this paper we introduce five new heuristics. Two algorithms, Bm and Cm, arise as natural improvements of Algorithm Am from (He et al., in: Graph Algorithms And Applications 2, World Scientific, 2004). The other three algorithms, Dm, Em, and Fm, incorporate randomization. Algorithm Dm can be considered as a greedy version of Bm, whereas Em and Fm are versions of local search algorithm, specialized for the k-matching problem. The algorithms are implemented in Python and are run on three datasets. On the datasets available, all the algorithms clearly outperform Algorithm Am in terms of solution quality. On the first dataset with known optimal values the average relative error ranges from 1.47% over optimum (algorithm Am) to 0.08% over optimum (algorithm Em). On the second dataset with known optimal values the average relative error ranges from 4.41% over optimum (algorithm Am) to...

Ars Mathematica Contemporanea

The modified Hosoya polynomial of double weighted graphs, i.e. edge and vertex weighted graphs, i... more The modified Hosoya polynomial of double weighted graphs, i.e. edge and vertex weighted graphs, is introduced that enables derivation of closed expressions for Hosoya polynomial of some special graphs including unicyclic graphs. Furthermore, the Hosoya polynomial is given as a sum of edge contributions generalizing well known analogous results for the Wiener number. A linear algorithm for computing the Hosoya polynomial on cactus graphs is provided. Hosoya polynomial is extensively studied in chemical graph theory, and in particular its weighted versions have interesting applications in theory of communication networks.

Journal of Nonlinear Mathematical Physics, 2016

In the paper, we study real forms of the complex generic Neumann system. We prove that the real f... more In the paper, we study real forms of the complex generic Neumann system. We prove that the real forms are completely integrable Hamiltonian systems. The complex Neumann system is an example of the more general Mumford system. The Mumford system is characterized by the Lax pair (L C (λ), M C (λ)) of 2 × 2 matrices, where L C (λ) = V C (λ) W C (λ) U C (λ) −V C (λ) and U C (λ), V C (λ), W C (λ) are suitable polynomials. The topology of a regular level set of the moment map of a real form is determined by the positions of the roots of the suitable real form of U C (λ), with respect to the position of the values of suitable parameters of the system. For two families of the real forms of the complex Neumann system, we describe the topology of the regular level set of the moment map. For one of these two families the level sets are noncompact. In the paper, we also give the formula which provides the relation between two systems of the first integrals in involution of the Neumann system. One of these systems is obtained from the Lax pair of the Mumford type, while the second is obtained from the Lax pair whose matrices are of dimension (n + 1) × (n + 1).