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On Minimum- and Maximum-Weight Minimum Spanning Trees with Neighborhoods

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Abstract

We study optimization problems for the Euclidean Minimum Spanning Tree (MST) problem on imprecise data. To model imprecision, we accept a set of disjoint disks in the plane as input. From each member of the set, one point must be selected, and the MST is computed over the set of selected points. We consider both minimizing and maximizing the weight of the MST over the input. The minimum weight version of the problem is known as the Minimum Spanning Tree with Neighborhoods (MSTN) problem, and the maximum weight version (max-MSTN) has not been studied previously to our knowledge. We provide deterministic and parameterized approximation algorithms for the max-MSTN problem, and a parameterized algorithm for the MSTN problem. Additionally, we present hardness of approximation proofs for both settings.

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Notes

  1. For an informal discussion of sum-of-square-roots-hard problems, see [14].

  2. Separability is similar in spirit to the notion of a well-separated pair; see [8]

  3. Note that the number of edges in the construction is a polynomial in the size of the input. Since all we need to determine is whether the weight of the solution is at least 0.34 units less than w tot, we can round the measure of the total weight to a number of bits that is logarithmic in the size of the input.

  4. If the Z D configurations were maintained, then the e-wire could be joined sub-optimally to the variable gadget with weight 5.5, since the assignment is not satisfying. However, a larger MST can be achieved by deviating from the Z D configuration in the variable gadget.

  5. Using this construction, pairs of disks of the gadget trivially intersect at a single point, which simplifies our analysis. To achieve strict disjointedness, the disks of the gadget may be contracted to have radius 1−γ so that the tangent point is now distance γ from the nearest point in the adjacent disks. Any path which uses the tangent point in our analysis will have less than 2γ units of additional weight on these shrunken disks, and there are fewer than n(8m + 6) disks, where n and m are the number of variables and clauses respectively. Choosing an appropriate value of γ so that 2γ n(8m + 6) ≪ 0.0735 achieves the same result as our simplified analysis.

  6. D i+2 may be more generally indexed as D i+2+4c , where there is a block of 4c disks in the variable gadget joined to the interior of the gadget in a truth configuration opposite of that of the neighboring disks in the variable gadget. This does not affect the analysis, it simply relocates the singleton disk. Recall that by Lemma 9, such singletons would exist rather than having three disks connected by a path to a single edge of the interior of the gadget.

  7. Note that the number of edges in the construction is a polynomial in the size of the input. Since all we need to determine is whether the weight of the solution is at least 0.0735 units greater than w tot, we can round the measure of the total weight to a number of bits logarithmic in the size of the input.

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Authors and Affiliations

  1. Dalhousie University, Halifax, Canada

    Reza Dorrigiv & Meng He

  2. University of Manitoba, Winnipeg, Canada

    Robert Fraser

  3. University of Waterloo, Waterloo, Canada

    Shahin Kamali & Alejandro López-Ortiz

  4. University of Tokyo, Tokyo, Japan

    Akitoshi Kawamura

  5. University of Concepción, Concepción, Chile

    Diego Seco

Authors
  1. Reza Dorrigiv
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  2. Robert Fraser
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  3. Meng He
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  4. Shahin Kamali
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  5. Akitoshi Kawamura
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  6. Alejandro López-Ortiz
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  7. Diego Seco
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Correspondence to Robert Fraser.

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Dorrigiv, R., Fraser, R., He, M. et al. On Minimum- and Maximum-Weight Minimum Spanning Trees with Neighborhoods. Theory Comput Syst 56, 220–250 (2015). https://doi.org/10.1007/s00224-014-9591-3

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