Beyond Vizing Chains: Improved Recourse in Dynamic Edge Coloring

TL;DR

The paper tackles dynamic edge coloring with a fixed palette of size $\Delta+C$, aiming to minimize recourse, the number of edge recolorings per update. It introduces the shift-tree, a tree-encoded collection of shift-based recolorings that extends a partial coloring to a new uncolored edge without using Vizing fans or bicolored-path fans, and shows how to exploit multiple shift paths to guarantee a color extension. For large palettes ($C$ near $\Delta$ with $C/\Delta$ bounded away from $1$) and $\Delta-C = O(n^{1-\delta})$, it proves a tight worst-case recourse bound of $O\left( \frac{\log n}{\log \frac{\Delta+C}{\Delta-C}} \cdot \frac{1}{(C/\Delta) - 1/\phi} \right)$ per update, where $\phi$ is the golden ratio; it also provides a $\Delta$-adaptive result for graphs with low arboricity $\alpha$ when $C \ge (2+\varepsilon)\alpha-1$, achieving $O\left( \frac{1}{\varepsilon} \log n \right)$ recourse. The paper further demonstrates a separation between shift-based and general recoloring approaches and discusses extensions and open questions toward smaller palettes and improved update times.

Abstract

We study the maintenance of a $(Δ+C)$-edge-coloring ($C\ge 1$) in a fully dynamic graph $G$ with maximum degree $Δ$. We focus on minimizing \emph{recourse} which equals the number of recolored edges per edge updates. We present a new technique based on an object which we call a \emph{shift-tree}. This object tracks multiple possible recolorings of $G$ and enables us to maintain a proper coloring with small recourse in polynomial time. We shift colors over a path of edges, but unlike many other algorithms, we do not use \emph{fans} and \emph{alternating bicolored paths}. We combine the shift-tree with additional techniques to obtain an algorithm with a \emph{tight} recourse of $O\big( \frac{\log n}{\log \frac{Δ+C}{Δ-C}}\big)$ for all $C \ge 0.62Δ$ where $Δ-C = O(n^{1-δ})$. Our algorithm is the first deterministic algorithm to establish tight bounds for large palettes, and the first to do so when $Δ-C=o(Δ)$. This result settles the theoretical complexity of the recourse for large palettes. Furthermore, we believe that viewing the possible shifts as a tree can lead to similar tree-based techniques that extend to lower values of $C$, and to improved update times. A second application is to graphs with low arboricity $α$. Previous works [BCPS24, CRV24] achieve $O(ε^{-1}\log n)$ recourse per update with $C\ge (4+ε)α$, and we improve by achieving the same recourse while only requiring $C \ge (2+ε)α- 1$. This result is $Δ$-adaptive, i.e., it uses $Δ_t+C$ colors where $Δ_t$ is the current maximum degree. Trying to understand the limitations of our technique, and shift-based algorithms in general, we show a separation between the recourse achievable by algorithms that only shift colors along a path, and more general algorithms such as ones using the Nibbling Method [BGW21, BCPS24].

Figures (9)

• Figure 1: An example of a colored graph $G$ and a shift-tree $T$ (Definition \ref{['definition_shift_tree_and_terminology']}). (a) The graph $G$, with $\Delta=4$ and extra colors $C=2$ (total of $\Delta+C = 6$). White vertices have less than $\Delta$ edges, the edge $(u,v)$ is considered part of the graph but uncolored yet. Fat edges (such as $(u,a)$) mark the ones that appear in the shift-tree, thin edges (such as $(j,h)$) do not. (b) The shift-tree $T$, expanded $3$ levels down. Every vertex of $G$ may correspond to multiple nodes of $T$ (for example $x$), but is only expanded with children once. A yellow/green node is an active copy of a vertex, gray nodes are inactive and remain leaves. The green nodes have less than $C+1 = 3$ children, which by Lemma \ref{['lemma_basic_shift_children']} guarantees a useful path (Definition \ref{['definition_chain_shift']}). For example, shifting over $(u,v,b,e)$ lets us color $(b,e)$ by either red or blue. Notice that $(u,v)$ appears as a red edge in $T$ because if we track the color shifting on the path to $u$, at the moment we expand children for $u$ the edge $(u,v)$ is red (stole the color from $(v,b)$). (c) The skeleton of $T$ when reduced only to paths that lead to $x$. (d) In the proof of Theorem \ref{['theorem_large_palette_general_parameterized']} we expand each leaf of $T^x$ one step further regardless of being active or inactive. Then, some neighbour of $x$, say $y$, appears multiple times, so there are multiple possible paths to shift colors and reach the edge $(x,y)$.
• Figure 2: An example for Definition \ref{['definition_dangerous']}. (a) Edge $e$ and its neighbours. In (b) and (c) the numbers correspond to the colors, assume a total of $4$ colors ($\Delta=3$, $C=1$): black, red, orange, blue. (b) The edge $e$ is dangerous with bad color orange whose bad neighbours pair is $e_2,e_3$. The good neighbours of $e$ are $e_1,e_4$. (c) The edge $e$ is not dangerous and all the neighbours are good (blue is always available).
• Figure 3: A visual example for the proof of Theorem \ref{['theorem_gap_shift_recoloring_versus_general_separation_simple']} for $\Delta=3$ and $C=0$. (a) The graph is a union of a red matching (vertical edges, $(v_i,u_i)$) and a subgraph $H$ constructed over the vertices $v_i$ by Theorem \ref{['theorem_lower_bound_generalized']} for $\Delta'=2$, which in this case is a path (horizontal edges $(v_i,v_{i+1})$) colored as two bicolored paths and separated by an uncolored edge $e$ in the middle. (b) Extending the coloring to $e$ requires $O(1)$ recourse if we simply color it red. (c) If we insist on only shifting colors over a chain of edges, we must shift colors horizontally towards one of the ends, resulting in $\Omega(n)$ recourse.
• Figure 4: An example of the expansion game in the proof of Lemma \ref{['lemma_shift_tree_max_depth']}. (a) A standard tree $T$, with expansion of $2$ ($C=1$). Blue marks active nodes and gray marks inactive nodes. (b) A presentation of $T$ as integer segments, same color-coding. To the left of each segment we have two numbers, black for the length of the entire segment and blue for the length of its active subsegment. The length of the whole segment is twice the length of the active subsegment of the previous level. The red funnels illustrate the fractional ancestry hierarchy of these segments, where a segment $[x,y]$ of level $k$ is a child of the segment $[\frac{x}{2},\frac{y}{2}]$ of level $k-1$ and, if active, the parent of segment $[2x,2y]$ of level $k+1$. (c) Illustration of a game with fractional inactivations. The first two fractional inactivations are $\frac{1}{2}$. Observe that the total inactive amount is $4$ in both (b) and (c), yet the total blue area (active) in (c) is smaller. This is natural since an early inactivation reduces the size of future segments.
• Figure 5: Three cases for the modification from strategy $S$ (top) to $S'$ (bottom) as detailed in the proof of Lemma \ref{['lemma_shift_tree_max_depth']}. We focus on the full active (blue) segment at step $t$, and the next inactivation (grey) is at step $j$, if such exists. The growth factor of blue segments in this example is $2$ ($C=1$). The segments are divided for visual purposes to ease comparing sizes, these are not necessarily integer units. Phantom dashed segments in $S'$ mark quantities that are saved (do not exist) by early inactivation. (a) $S$ makes no inactivation after $t$ ($j = \infty$), in this case we just inactivate in step $t$ an extra segment such that $S'$ conforms with Greedy at step $t$. (b) Past step $t$, $S$ makes a little inactivation at step $j$, insufficient for $S'$ to conform to Greedy in step $t$. Then we inactivate an extra $\epsilon$ such that $\epsilon \cdot (C+1)^{t-j}$ exactly voids the inactivation of $S$ at step $j$. (c) Past step $t$, $S$ makes a large inactivation at step $j$ that lets us conform $S'$ to Greedy in step $t$. Some leftover inactivation in $S'$ may remain at step $j$. Observe that in both (b) and (c) $S'$ accumulates less activity in steps $t+1$ to $j-1$, but the current active segment in step $j$ is the same for $S$ and $S'$ by construction.

Theorems & Definitions (43)

• Definition 2.1: Chain, Useful, Shift-based
• Definition 2.2: Shift-tree
• Lemma 2.3: Few Children Imply a Useful Shiftable Path
• proof
• Definition 2.4: Good/Bad Neighbours/Colors, Dangerous Edge
• Theorem 2.5: Tight Recourse
• Theorem 2.6: Lower Bound by LowerBoundRecourse2018
• Theorem 2.7: Low Arboricity Graphs
• Theorem 2.8: $\Delta$-adaptive for Low Arboricity
• Theorem 2.9: Lower Bound with Arboricity