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oai:drops-oai.dagstuhl.de:24275 2025-11-12T13:39:44Z ddc:004 open_access

Streaming Algorithms for Conflict-Free Coloring Mathew, Rogers Panolan, Fahad Seshikanth Streaming algorithm conflict-free coloring vertex coloring randomized algorithms Conflict-free coloring of a hypergraph ℋ = (V,ℰ) using k colors is a function f:V → {1,2, …, k} such that for all E ∈ ℰ, there exists a vertex v ∈ E with a unique color. That is, f(v)≠ f(u) for all u ∈ E ⧵ {v}. The minimum k for which ℋ has a conflict-free coloring using k colors is called the conflict-free chromatic number of ℋ. For a simple graph G, a conflict-free coloring of the hypergraph with vertex set V(G) and edge set being the set of all closed neighborhoods of the vertices in G is called a conflict-free closed neighborhood (CFCN) coloring of G. CFCN chromatic number, denoted by χ_{CN}(G), is the minimum number of colors used in a conflict-free closed neighborhood coloring of G. Analogously, we define conflict-free open neighborhood (CFON) coloring and CFON chromatic number, χ_{ON}(G), of a graph G. There are various works on proving upper and lower bounds of χ_{ON}(G) and χ_{CN}(G). In this work, we develop streaming algorithms for CFCN and CFON coloring of a graph where the number of colors used matches the best-known upper bounds of χ_{ON}(G) and χi_{CN}(G). Our algorithms use as input an edge stream of the graph G in the insertion-only model. Our results and the best-known bounds for χ_{ON}(G) and χ_{CN}(G) are given below. 1. Pach and Tardos [Combinatorics, Probability and Computing, 2009] showed that, for any n vertex graph G, χ_{CN}(G) = O(ln² n). Glebov, Szabó and Tardos [Combinatorics, Probability and Computing, 2014] showed the existence of graphs G with χ_{CN}(G) = Ω(ln² n). We design a randomized single-pass semi-streaming algorithm (i.e., it uses O(n ln n) space that, given an n-vertex graph G, outputs a CFCN coloring of G using O(ln² n) colors with probability at least (1-2/n). 2. Bhyravarapu, Kalyanasundaram, Mathew [Journal of Graph Theory, 2021] showed that for a graph G with maximum degree Δ, χ_{CN}(G) = O(ln² Δ). The methods used by our algorithms give rise to a simpler, alternate proof for this bound. 3. It is known that χ_{ON}(G) ≤ 1/2 + √{2n + 1/4} (See Pach and Tardos [Combinatorics, Probability and Computing, 2009] and Ph.D. thesis of Cheilaris). This bound is asymptotically tight. - We design a deterministic single-pass O(n√n) space streaming algorithm that, given a graph G on n vertices, finds a CFON coloring using 2√n colors. - We design a randomized, single-pass, semi-streaming algorithm to find a CFON coloring of a graph G using O(√n ln² n) colors with success probability at least (1-2/n). 4. It is known that χ_{ON}(G) ≤ Δ+1, where Δ is the maximum degree of a vertex in G. Further, there are graphs G known with χ_{ON}(G) = Δ + 1. We design a randomized two-pass semi-streaming algorithm (uses O(1/(ε²) n ln³ n) space) that outputs a CFON coloring of G using (1+ε)Δ colors, for any ε > 0, with a probability at least (1-1/n). Schloss Dagstuhl – Leibniz-Zentrum für Informatik Rogers Mathew and Fahad Panolan and Seshikanth 2025 Is Part Of LIPIcs, Volume 349, 19th International Symposium on Algorithms and Data Structures (WADS 2025) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.WADS.2025.44 urn:nbn:de:0030-drops-242756 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.WADS.2025.44 eng https://creativecommons.org/licenses/by/4.0/legalcode