Graph Realization of Distance Sets
The Distance Realization problem is defined as follows. Given an n × n matrix D of nonnegative integers, interpreted as inter-vertex distances, find an n-vertex weighted or unweighted graph G realizing D, i.e., whose inter-vertex distances satisfy dist_G(i,j) = D_{i,j} for every 1 ≤ i < j ≤ n, or decide that no such realizing graph exists. The problem was studied for general weighted and unweighted graphs, as well as for cases where the realizing graph is restricted to a specific family of graphs (e.g., trees or bipartite graphs). An extension of Distance Realization that was studied in the past is where each entry in the matrix D may contain a range of consecutive permissible values. We refer to this extension as Range Distance Realization (or Range-DR). Restricting each range to at most k values yields the problem k-Range Distance Realization (or k-Range-DR). The current paper introduces a new extension of Distance Realization, in which each entry D_{i,j} of the matrix may contain an arbitrary set of acceptable values for the distance between i and j, for every 1 ≤ i < j ≤ n. We refer to this extension as Set Distance Realization (Set-DR), and to the restricted problem where each entry may contain at most k values as k-Set Distance Realization (or k-Set-DR).
We first show that 2-Range-DR is NP-hard for unweighted graphs (implying the same for 2-Set-DR). Next we prove that 2-Set-DR is NP-hard for unweighted and weighted trees. We then explore Set-DR where the realization is restricted to the families of stars, paths, or cycles. For the weighted case, our positive results are that for each of these families there exists a polynomial time algorithm for 2-Set-DR. On the hardness side, we prove that 6-Set-DR is NP-hard for stars and 5-Set-DR is NP-hard for paths and cycles. For the unweighted case, our results are the same, except for the case of unweighted stars, for which k-Set-DR is polynomially solvable for any k.
Graph Realization
distance realization
network design
Mathematics of computing~Graph algorithms
13:1-13:14
Regular Paper
Supported in part by a US-Israel BSF grant (2018043).
Amotz
Bar-Noy
Amotz Bar-Noy
City University of New York (CUNY), NY, USA
David
Peleg
David Peleg
Weizmann Institute of Science, Rehovot, Israel
Mor
Perry
Mor Perry
The Academic College of Tel-Aviv-Yaffo, Israel
Dror
Rawitz
Dror Rawitz
Bar Ilan University, Ramat-Gan, Israel
10.4230/LIPIcs.MFCS.2022.13
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Amotz Bar-Noy, David Peleg, Mor Perry, and Dror Rawitz
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