The red-blue-yellow matching problem

Abstract

We consider the red-blue-yellow matching problem: given two natural numbers $k_R$, $k_B$ and a graph $G$ whose edges are colored red, blue or yellow, the goal is to find a matching of $G$ that contains exactly $k_R$ red edges and exactly $k_B$ blue edges, and is of maximum cardinality subject to these constraints. This is a natural generalization of the well known red-blue matching problem, whose complexity status is unknown: although a randomized polynomial-time algorithm exists, a deterministic algorithm has remained elusive for nearly four decades. The best known deterministic approach to the red-blue matching problem, due to Yuster (2012), gives an additive approximation. In this paper, we show a similar result for the red-blue-yellow matching problem, giving a polynomial-time deterministic algorithm that, under natural assumptions, finds a matching satisfying the color requirements almost exactly and has cardinality within 3 of the optimal solution. Our algorithm is a mix of classic linear programming techniques and ad hoc existence results on restricted classes of graphs such as paths and cycles. As a key ingredient, we prove a curious topological property of plane curves, which is a strengthened version of a result by Grandoni and Zenklusen (2010) in the related context of budgeted matchings.

Figures (8)

• Figure 1: In the cycle above, a matching with two red edges cannot have any blue edges.
• Figure 2: Examples of crossing pairs, with $f$ shown in black, its extension $f^\infty$ in gray, and $f+q-f(0)$ in red. Squares indicate crossings.
• Figure 3: The imbalance curve $d: [0,\ell]\rightarrow \mathbb R^2$, with $\ell=9$, corresponding to the 18-edge cycle with the following coloring (where the edges are numbered from 0 to 17): Y B Y B Y R Y R Y B R B Y R B R B R. The point $q$ (in red) belongs to the segment between $d(0),d(\ell)$. While $d$ does not contain $q$, it does contain several pairs of points whose difference is exactly $q$, in particular $d(u)$ and $d(v)$. As the reader can easily verify, this translates into the existence of intersecting pairs for $(d^\infty,d+q)$ (some of which are crossing pairs).
• Figure 4: Examples of an overlapping crossing of type (a) (left) and of type (b) (right), where we only show $d$ (black) and $d+q$ (red). In both cases, the crossing pair is $(u,v)=(6,3)$ and the ' ' length" of the overlapping part is $i=2$. At the side of each plot we draw (translates of) the vectors $d_\mathrm{in}$, $\bar{d}_\mathrm{out}$, $d_\mathrm{out}$, $\bar{d}_\mathrm{in}$, in clockwise order, with $d_\mathrm{in}= (1,0)$ in both cases.
• Figure 5: An example of the contradictions obtained in case \ref{['case:a)']} (left) and in case \ref{['case:b)']} (right). The possible moves for vectors $d_\mathrm{in}$, $d_\mathrm{out}$, $\bar{d}_\mathrm{in}$, $\bar{d}_\mathrm{out}$ conflict with $(u,v)$ being a crossing pair for $(d^\infty,d+q)$.

Theorems & Definitions (43)

• Theorem 1
• Proposition 2
• Theorem 3
• Lemma 4
• Lemma 5: Crossing Lemma
• Theorem 6
• Theorem 7
• proof : Proof of Theorem \ref{['thm:cycle_paths_choice']}
• Remark 8
• Lemma 9