An overview of graph covering and partitioning

Schwartz, Stephan

doi:10.1016/j.disc.2022.112884

Cited by 8 publications

(3 citation statements)

References 192 publications

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“…Our goal is to penalize the inhomogeneity of a template

T

with respect to

d

, that is, the values

{\left({d}_v\right)}_{v\in T}

should be as similar as possible. A number of measures for (in)homogeneity are listed by Hansen and Jaumard [38], a recent survey can be found in [73]. Our penalty function is a weighted version of the star measure discussed in both articles, that is, in the star objective function each

{w}_v

is replaced by 1.…”

Section: Problem Formulation and A Column Generation Approachmentioning

confidence: 99%

“…While these concepts are general and should carry over to other (graph partitioning) problems involving connectivity, our exposition will be based upon and is motivated by a concrete real-world application of toll enforcement. A recent survey on homogeneous covering and partitioning problems, as well as heuristic and exact solution approaches, can be found in [73].…”

Section: Introductionmentioning

confidence: 99%

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Vertex covering with capacitated trees

2022

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The covering of a graph with (possibly disjoint) connected subgraphs is a fundamental problem in graph theory. In this paper, we study a version to cover a graph's vertices by connected subgraphs subject to lower and upper weight bounds, and propose a column generation approach to dynamically generate feasible and promising subgraphs. Our focus is on the solution of the pricing problem which turns out to be a variant of the NP-hard Maximum Weight Connected Subgraph Problem. We compare different formulations to handle connectivity, and find that a single-commodity flow formulation performs best. This is notable since the respective literature seems to have widely dismissed this formulation. We improve it to a new coarse-to-fine flow formulation that is theoretically and computationally superior, especially for large instances with many vertices of degree 2 like highway networks, where it provides a speed-up factor of 5 over the non-flow-based formulations. We also propose a preprocessing method that exploits a median property of weight-constrained subgraphs, a primal heuristic, and a local search heuristic. In an extensive computational study we evaluate the presented connectivity formulations on different classes of instances, and demonstrate the effectiveness of the proposed enhancements. Their speed-ups essentially multiply to an overall factor of well over 10. Overall, our approach allows the reliable solution of instances with several hundreds of vertices in a few minutes. These findings are further corroborated in a comparison to existing districting models on a set of test instances from the literature.

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“…Our goal is to penalize the inhomogeneity of a template

T

with respect to

d

, that is, the values

{\left({d}_v\right)}_{v\in T}

{w}_v

is replaced by 1.…”

Section: Problem Formulation and A Column Generation Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Vertex covering with capacitated trees

2022

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“…The binary rank of 0, 1 matrices has been intensively studied in combinatorics under various equivalent formulations (see, e.g., [7,9,13,15]), as well as in the area of communication complexity, where it is closely related to the unambiguous non-deterministic communication complexity of functions (see, e.g., [12,Chapter 2]). It can be observed that every 0, 1 matrix M satisfies rank bin (M) ≥ rank R (M), where rank R (M) stands for the standard rank of the matrix M over the reals.…”

Section: Introductionmentioning

confidence: 99%

The Binary Rank of Circulant Block Matrices

Haviv¹,

Parnas²

2022

Preprint

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The binary rank of a 0, 1 matrix is the smallest size of a partition of its ones into monochromatic combinatorial rectangles. A matrix M is called (k 1 , . . . , k m ; n 1 , . . . , n m ) circulant block diagonal if it is a block matrix with m diagonal blocks, such that for each i ∈ [m], the ith diagonal block of M is the circulant matrix whose first row has k i ones followed by n i − k i zeros, and all of whose other entries are zeros. In this work, we study the binary rank of these matrices as well as the binary rank of their complement, obtained by replacing the zeros by ones and the ones by zeros. In particular, we compare the binary rank of these matrices to their rank over the reals, which forms a lower bound on the former.We present a general method for proving upper bounds on the binary rank of block matrices that have diagonal blocks of some specified structure and ones elsewhere. Using this method, we prove that the binary rank of the complement of a (k 1 , . . . , k m ; n 1 , . . . , n m ) circulant block diagonal matrix for integers satisfying n i > k i > 0 for each i ∈ [m] exceeds its real rank by no more than the maximum of gcd(n i , k i ) − 1 over all i ∈ [m]. We further present several sufficient conditions for the binary rank of these matrices to strictly exceed their real rank. By combining the upper and lower bounds, we determine the exact binary rank of various families of matrices and, in addition, significantly generalize a result of Gregory (J. Comb. Math. & Comb. Comp., 1989).Motivated by a question of Pullman (Linear Algebra Appl. ,1988), we study the binary rank of k-regular 0, 1 matrices, those having precisely k ones in every row and column, and the binary rank of their complement. As an application of our results on circulant block diagonal matrices, we show that for every k ≥ 2, there exist k-regular 0, 1 matrices whose binary rank is strictly larger than that of their complement. Furthermore, we exactly determine for every integer r, the smallest possible binary rank of the complement of a 2-regular 0, 1 matrix with binary rank r.

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Political Districting

Buchanan

2023

Encyclopedia of Optimization

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An overview of graph covering and partitioning

Cited by 8 publications

References 192 publications

Vertex covering with capacitated trees

Vertex covering with capacitated trees

The Binary Rank of Circulant Block Matrices

Political Districting

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