Online Algorithm for Fractional Matchings with Edge Arrivals in Graphs of Maximum Degree Three

TL;DR

This paper tackles online fractional matching under adversarial edge arrivals in graphs with maximum degree three, establishing that the best possible competitive ratio is $c=\frac{4}{9-\\sqrt{5}}\approx0.5914$. It provides an online primal–dual algorithm achieving this ratio and proves its optimality by matching the known upper bound, while also showing a gap for integral matchings at $0.5807$ and confirming the same bound for vertex- and edge-arrival models at degree three. The approach builds on Buchbinder's consistent-instances framework, introducing a nuanced edge-type partition (path, spoke, bridge) and a careful bridge-update mechanism to preserve feasibility and tightness. The work also shows additional bounds: for maximum degree four, the fractional guarantee cannot reach $c$, being bounded by $0.58884$, and it leaves open questions about the precise behavior of vertex vs. edge arrivals beyond degree three.

Abstract

We study online algorithms for maximum cardinality matchings with edge arrivals in graphs of low degree. Buchbinder, Segev, and Tkach showed that no online algorithm for maximum cardinality fractional matchings can achieve a competitive ratio larger than $4/(9-\sqrt 5)\approx 0.5914$ even for graphs of maximum degree three. The negative result of Buchbinder et al. holds even when the graph is bipartite and edges are revealed according to vertex arrivals, i.e. once a vertex arrives, all edges are revealed that include the newly arrived vertex and one of the previously arrived vertices. In this work, we complement the negative result of Buchbinder et al. by providing an online algorithm for maximum cardinality fractional matchings with a competitive ratio at least $4/(9-\sqrt 5)\approx 0.5914$ for graphs of maximum degree three. We also demonstrate that no online algorithm for maximum cardinality integral matchings can have the competitive guarantee $0.5807$, establishing a gap between integral and fractional matchings for graphs of maximum degree three. Note that the work of Buchbinder et al. shows that for graphs of maximum degree two, there is no such gap between fractional and integral matchings, because for both of them the best achievable competitive ratio is $2/3$. Also, our results demonstrate that for graphs of maximum degree three best possible competitive ratios for fractional matchings are the same in the vertex arrival and in the edge arrival models.

Figures (13)

• Figure 1: Example of a consistent instance with $n = 4$ rounds.
• Figure 2: The construction of the fractional matching and increases in the fractional vertex cover, when Algorithm \ref{['alg: one']} has a consistent instance as an input.
• Figure 3: An example of a potential partition indirectly maintained by Algorithm \ref{['alg: one']} (subject to arrival order) into subgraphs of the consistent instances. Here, the straight green edges represent path edges, the wavy red edges represent spokes, and the dashed blue edges represent bridges.
• Figure 4: Edge assignment and cover updates upon the arrival of a bridge $e$ with $\{\text{type}(\delta(u) \setminus \{e\}), \text{type}(\delta(v) \setminus \{e\})\} = \{(1,0,0)\}$. The dashed cover assignments represent the values of $x^{old}$ under the assumption that prior to the arrival of $e$, assignments and cover construction are as per Figure \ref{['fig: cover construction']}. Whereas, the solid cover assignments represent the contribution of $e$ to the updated cover $x^{new}$. In particular, the value of $x^{new}_{u}$ is the sum of the dashed and solid assignments into $u$, and the same holds for $x^{new}_v$.
• Figure 5: Edge assignment and cover updates upon the arrival of a bridge $e$ with $\{\text{type}(\delta(u) \setminus \{e\}), \text{type}(\delta(v) \setminus \{e\})\} = \{(1,0,0), (0,1,0)\}$. The dashed cover assignments represent the values of $x^{old}$ under the assumption that prior to the arrival of $e$, assignments and cover construction are as per Figure \ref{['fig: cover construction']}. Whereas, the solid cover assignments represent the contribution of $e$ to the updated cover $x^{new}$. In particular, the value of $x^{new}_{u}$ is the sum of the dashed and solid assignments into $u$, and the same holds for $x^{new}_v$.

Theorems & Definitions (19)

• Lemma 3.1: Main Properties
• Theorem 3.2
• Lemma 3.3: Additional Properties
• Lemma 3.4
• proof : Proof of Lemma \ref{['lma x non-neg']}
• Lemma 3.5
• proof : Proof of Lemma \ref{['lma y non-neg']}
• Claim 1
• proof : Proof of Claim \ref{['claim 1']}
• Theorem 4.1