On the representability of totally unimodular matrices on bidirected graphs

Pitsoulis, Leonidas; Papalamprou, Konstantinos; Appa, Gautam; Kotnyek, Balázs

doi:10.1016/j.disc.2009.03.010

Cited by 4 publications

(3 citation statements)

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“…Seymour has shown in [13] that network matrices and their transposes are the main building blocks for TU matrices. Moreover, in [11], it has been shown that the building blocks of TU matrices are matrices associated with bidirected graphs. In this paper we focus on SU matrices which stand between the classes of network and TU matrices and show the network structure of that class.…”

Section: Network and Unimodular Matricesmentioning

confidence: 99%

“…where, in the ⊕ Network matrices and TU matrices are known to be closed under these operations (see [11] and [12], respectively). These operations have been originally defined in the more general framework of regular matroids in [13] and here we present the special form of these operations as they applied to the compact representation matrices in R, i.e.…”

Section: A K-sum Decomposition Of Strongly Unimodular Matricesmentioning

confidence: 99%

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k-Sum decomposition of strongly unimodular matrices

Papalamprou

Pitsoulis

2015

Optim Lett

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Networks are frequently studied algebraically through matrices. In this work, we show that networks may be studied in a more abstract level using results from the theory of matroids by establishing connections to networks by decomposition results of matroids. First, we present the implications of the decomposition of regular matroids to networks and related classes of matrices, and secondly we show that strongly unimodular matrices are closed under k-sums for k = 1, 2 implying a decomposition into highly connected network-representing blocks, which are also shown to have a special structure. * work of this author was conducted at National Research University Higher School of Economics and supported by RSF grant 14-41-00039 network matrix, the transpose of a network matrix, the matrix B 1 or B 2 of (1),or may be constructed recursively by these matrices using matrix 1-, 2-and 3-sums.According to Theorem 3.7, the building blocks of totally unimodular matrices are network matrices and their transposes as well as the matrices B 1 and B 2 in (1).Lemma 3.8. B 1 and B 2 are not SU.Proof: If we make the value of the (4, 3) th -element of B 1 from −1 to 0 then in the matrix so-obtained the 3 × 3 submatrix defined by rows 3, 4 and 5 and columns 2, 3 and 4 has determinant equal to +2. Therefore, B 1 is not SU. Similarly, if we make the value of the (4, 1) th -element of B 2 from +1 to 0 then in the matrix so-obtained, the 3 × 3 submatrix defined by rows 3, 4 and 5 and columns 1, 4 and 5 has determinant equal to −2 and thus, B 2 is not SU.By Theorem 3.7 and Lemma 3.8 we obtain the following result.Theorem 3.9. Any SU matrix is up to row and column permutations and scaling by ±1 factors a network matrix, the transpose of a network matrix, or may be constructed recursively by these matrices using matrix 1-, 2-and 3-sums.The following theorem, known as the splitter theorem for regular matroids, is one of the most important steps which led to the regular matroid decomposition theorem [13].Theorem 3.10. Every regular matroid can be obtained from copies of R 10 and from 3-connected minors without R 10 minors by a sequence of 1-sums and 2-sums.Combining the above we can now state the main result of this section.Theorem 3.11. A matrix is SU if and only if it is decomposable via 1-and 2-sums into strongly unimodular matrices representing 3-connected regular matroids without R 10 minors.

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Section: Network and Unimodular Matricesmentioning

confidence: 99%

Section: A K-sum Decomposition Of Strongly Unimodular Matricesmentioning

confidence: 99%

k-Sum decomposition of strongly unimodular matrices

Papalamprou

Pitsoulis

2015

Optim Lett

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“…Various problems, such as capacitated nonsimple b-matchings, in which 1-matchings and simple b-matchings are included, capacitated b-edge covers, and minimum cost flows, are given a single unified formulation as an optimization problem over bidirected graphs. Bidirected graphs have also gained attention in the studies of nowhere-zero integral flows [1,4,22] and totally unimodular matrices [2,20]. Needless to say, numerous studies exist regarding digraphs and signed graphs; see, e.g., Schrijver [21] or Zaslavsky [23].…”

mentioning

confidence: 99%

Signed analogue of general Kotzig–Lovász decomposition

Kita

2020

Discrete Applied Mathematics

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Optimal Partitioning of Non-Convex Environments for Minimum Turn Coverage Planning

Ramesh

Imeson²,

Fi̇dan

et al. 2022

IEEE Robot. Autom. Lett.

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In this paper, we investigate the problem of decomposing 2D environments for robot coverage planning. Coverage path planning (CPP) involves computing a costminimizing path for a robot equipped with a coverage or sensing tool so that the tool visits all points in the environment. CPP is an NP-Hard problem, so existing approaches simplify the problem by decomposing the environment into the minimum number of sectors. Sectors are sub-regions of the environment that can each be covered using a lawnmower path (i.e., along parallel straight-line paths) oriented at an angle. However, traditional methods either limit the coverage orientations to be axis-parallel (horizontal/vertical) or provide no guarantees on the number of sectors in the decomposition. We introduce an approach to decompose the environment into possibly overlapping rectangular sectors. We provide an approximation guarantee on the number of sectors computed using our approach for a given environment. We do this by leveraging the submodular property of the sector coverage function, which enables us to formulate the decomposition problem as a submodular set cover (SSC) problem with wellknown approximation guarantees for the greedy algorithm. Our approach improves upon existing coverage planning methods, as demonstrated through an evaluation using maps of complex real-world environments.

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On the representability of totally unimodular matrices on bidirected graphs

Cited by 4 publications

References 9 publications

k-Sum decomposition of strongly unimodular matrices

k-Sum decomposition of strongly unimodular matrices

Signed analogue of general Kotzig–Lovász decomposition

Optimal Partitioning of Non-Convex Environments for Minimum Turn Coverage Planning

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