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. Finally, we explore SET-DR where the realization is restricted to the families of stars, paths, cycles, or caterpillars. For the weighted case, our positive results are that there exist polynomial time algorithms for the 2-SET-DR problem on stars, paths and cycles, and for the 1-SET-DR problem on caterpillars. On the hardness side, we prove that 6-SET-DR is NP-hard for stars and 5-SET-DR is NP-hard for paths, cycles and caterpillars. 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.