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A Simple LP-Based Approximation Algorithm for the Matching Augmentation Problem

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Integer Programming and Combinatorial Optimization (IPCO 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13265))

Abstract

The Matching Augmentation Problem (MAP) has recently received significant attention as an important step towards better approximation algorithms for finding cheap 2-edge connected subgraphs. This has culminated in a \(\frac{5}{3}\)-approximation algorithm. However, the algorithm and its analysis are fairly involved and do not compare against the problem’s well-known LP relaxation called the cut LP.

In this paper, we propose a simple algorithm that, guided by an optimal solution to the cut LP, first selects a DFS tree and then finds a solution to MAP by computing an optimum augmentation of this tree. Using properties of extreme point solutions, we show that our algorithm always returns (in polynomial time) a better than 2-approximation when compared to the cut LP. We thereby also obtain an improved upper bound on the integrality gap of this natural relaxation.

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Correspondence to Étienne Bamas .

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A Deferred Proofs

A Deferred Proofs

Suppose that \(x^{*}\) is an extreme point solution of LP(GM). We know that \(x^*\) can be defined as the unique solution to the following system of |E| equations, for some \(\mathcal {S} \subseteq 2^V\) and \(E_0 \cup E_1 \subseteq E\).

$$\begin{aligned}&\sum _{e\in \delta (S)} x_e = 2,\quad \text {for all }S \in \mathcal {S} \\&x_e = 0 \quad \quad \quad \forall e\in E_0\\&x_e = 1 \quad \quad \quad \forall e\in E_1\\ \end{aligned}$$

Lemma 3 shows that we can select \(\mathcal {S}\) not too large. The proof of this lemma is the same as Theorem 4.9 from [10].

Lemma 3

(Theorem 4.9 in [10]). Let \(x^{*}\) be an extreme point of the MAP cut LP then the family of equations \(\mathcal {S}\) can be chosen to be a laminar family.

It is well known that any laminar family has size at most \(2n-1\). Therefore the number of fractional edges is at most \(|E|-|E_0|-|E_1|=|\mathcal {S}|\le 2n-1\).

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Bamas, É., Drygala, M., Svensson, O. (2022). A Simple LP-Based Approximation Algorithm for the Matching Augmentation Problem. In: Aardal, K., Sanità, L. (eds) Integer Programming and Combinatorial Optimization. IPCO 2022. Lecture Notes in Computer Science, vol 13265. Springer, Cham. https://doi.org/10.1007/978-3-031-06901-7_5

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  • DOI: https://doi.org/10.1007/978-3-031-06901-7_5

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