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Shorter tours and longer detours: uniform covers and a bit beyond

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Abstract

Motivated by the well known “four-thirds conjecture” for the traveling salesman problem (TSP), we study the problem of uniform covers. A graph \(G=(V,E)\) has an \(\alpha \)-uniform cover for TSP (2EC, respectively) if the everywhere \(\alpha \) vector (i.e., \(\{\alpha \}^{E}\)) dominates a convex combination of incidence vectors of tours (2-edge-connected spanning multigraphs, respectively). The polyhedral analysis of Christofides’ algorithm directly implies that a 3-edge-connected, cubic graph has a 1-uniform cover for TSP. Sebő asked if such graphs have \((1-\epsilon )\)-uniform covers for TSP for some \(\epsilon > 0\). Indeed, the four-thirds conjecture implies that such graphs have \(\frac{8}{9}\)-uniform covers. We show that these graphs have \(\frac{18}{19}\)-uniform covers for TSP. We also study uniform covers for 2EC and show that the everywhere \(\frac{15}{17}\) vector can be efficiently written as a convex combination of 2-edge-connected spanning multigraphs. For a weighted, 3-edge-connected, cubic graph, our results show that if the everywhere \(\frac{2}{3}\) vector is an optimal solution for the subtour elimination linear programming relaxation for TSP, then a tour with weight at most \(\frac{27}{19}\) times that of an optimal tour can be found efficiently. Node-weighted, 3-edge-connected, cubic graphs fall into this category. In this special case, we can apply our tools to obtain an even better approximation guarantee. An essential ingredient in our proofs is decompositions of graphs (e.g., cycle covers) that cover small-cardinality cuts an even (nonzero) number of times. Another essential tool we use is half-integral tree augmentation, which is known to have a small integrality gap. To extend our approach to input graphs that are 2-edge-connected, we present a procedure to decompose a point in the subtour elimination polytope into spanning, connected subgraphs that cover each 2-edge cut an even number of times. Using this decomposition, we obtain a \(\frac{17}{12}\)-approximation algorithm for minimum weight 2-edge-connected spanning subgraphs on subcubic, node-weighted graphs.

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Notes

  1. Applying Theorem 2.3 from [4], we note that this theorem applies to all 3-edge-connected (i.e., possibly noncubic) graphs.

  2. We remark that characterizing instances by their optimal LP solutions is how classes of fundamental points are defined. Incidentally, many fundamental classes of TSP and 2EC extreme points are either cubic or subcubic [1, 10, 11].

  3. [10] do not double half edges, so in fact here we obtain a convex combination of subgraphs.

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Acknowledgements

We would like to thank Jennifer Iglesias for discussions on the tree augmentation problem, Gérard Cornuéjols for his comments on a preliminary draft of this paper, and Sylvia Boyd for clarifications regarding recent work on the 2EC problem. We thank an anonymous referee for pointing out an error in a previous version: we had claimed a smaller approximation ratio in the statement of Theorem 4. The work of A. Haddadan and R. Ravi is supported in part by the U. S. Office of Naval Research under Award Numbers N00014-12-1-1001 and N00014-18-1-2099, and the U. S. National Science Foundation under Award Number CCF-1527032. The work of A. Newman is supported in part by LabEx PERSYVAL-Lab (ANR 11-LABX-0025) and IDEX-IRS SACRE. Our joint work was also supported by a research grant from the Carnegie Bosch Institute.

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  1. Tepper School of Business, Carnegie Mellon University, Pittsburgh, USA

    Arash Haddadan & R. Ravi

  2. CNRS and Université Grenoble-Alpes, Grenoble, France

    Alantha Newman

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  1. Arash Haddadan
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Correspondence to Arash Haddadan.

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Haddadan, A., Newman, A. & Ravi, R. Shorter tours and longer detours: uniform covers and a bit beyond. Math. Program. 185, 245–273 (2021). https://doi.org/10.1007/s10107-019-01426-8

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