Improved SDP-Based Algorithm for Coloring 3-Colorable Graphs

We present a polynomial-time algorithm that colors any 3-colorable $n$-vertex graph using $O(n^{0.19539})$ colors, improving upon the previous best bound of $\widetilde{O}(n^{0.19747})$ by Kawarabayashi, Thorup, and Yoneda [STOC 2024]. Our result constitutes the first progress in nearly two decades on SDP-based approaches to this problem. The earlier SDP-based algorithms of Arora, Chlamtáč, and Charikar [STOC 2006] and Chlamtáč [FOCS 2007] rely on extracting a large independent set from a suitably “random-looking” second-level neighborhood, under the assumption that the KMS algorithm [Karger, Motwani, and Sudan, JACM 1998] fails to find one globally. We extend their analysis to third-level neighborhoods. We then come up with a new vector $5/2$-coloring, which allows us to extract a large independent set from some third-level neighborhood. The new vector coloring construction may be of independent interest.

💡 Research Summary

The paper presents a polynomial‑time algorithm that colors any 3‑colorable graph on n vertices with O(n^0.19539) colors, improving the previous best bound of Õ(n^0.19747) achieved by Kawarabayashi, Thorup, and Yoneda (STOC 2024). This is the first progress on SDP‑based methods for this problem in almost two decades.

The authors start by reviewing the classic Karger‑Motwani‑Sudan (KMS) algorithm, which solves a simple SDP relaxation (vector 3‑coloring) and then samples a Gaussian vector r, selecting vertices whose projection exceeds a threshold t. The expected size of the sampled set S is about n·Δ^{‑1/3}, where Δ is the maximum degree. By keeping only isolated vertices in S, KMS guarantees an independent set of size Ω(n·Δ^{‑1/3}).

Arora‑Chlamtác‑Charikar (STOC 2006) and Chlamtác (FOCS 2007) observed that if KMS returns a set much smaller than S, the vectors associated with the neighbors of a typical vertex must be “c‑inefficient” covers: they are spread out and roughly orthogonal. This orthogonality allows one to argue that the second‑level neighborhood N^{(2)}(i) of a typical vertex i is large and that a vector 2‑coloring can be extracted on N^{(2)}(i), yielding an independent set proportional to |N^{(2)}(i)|. Combining this with a dense‑graph combinatorial algorithm (KTY24) gives the Õ(n^0.19747) bound.

The new contribution is to push the analysis one level deeper, to third‑level neighborhoods N^{(3)}(i). The authors decompose the vector of a vertex ℓ at distance three from i as

v_ℓ = –1/8 v_i + √(3/8) v_{ij} – √(3/4) v_{jk} + √(3/2) v_{kℓ}.

The main technical challenges are: (1) proving that the four vectors v_i, v_{ij}, v_{jk}, v_{kℓ} are pairwise almost orthogonal, and (2) composing the “cover” obtained from the second‑level analysis with an additional step without incurring a prohibitive loss.

To address (2) the authors develop a “win‑win” argument based on a refined Cover‑Composition Lemma (Lemma 4.14). If the two‑step cover is highly inefficient (large c′), then N^{(2)}(i) is already much larger than the second‑level analysis predicts, and the existing vector 2‑coloring argument yields a bigger independent set. If the cover is efficient (small c′), the loss term O(√{c′ t}) in the composition lemma is controllable, and one can prove that N^{(3)}(i) is significantly larger than N^{(2)}(i).

A crucial obstacle is that the vector 2‑coloring technique does not extend naturally to N^{(3)}(i). The authors therefore introduce a novel vector 5/2‑coloring on the third‑level neighborhood. They employ the Sum‑of‑Squares (Lasserre) hierarchy, which provides, for each vertex ℓ, three SDP vectors v(ℓ,R), v(ℓ,G), v(ℓ,B) corresponding to the three colors. By examining the local distribution over colorings of the triple {i,ℓ₁,ℓ₂} (where ℓ₁,ℓ₂∈N^{(3)}(i) share an edge), they define

u_ℓ = (v(ℓ,G) – v(ℓ,B)) / ‖v(ℓ,G) – v(ℓ,B)‖.

For adjacent ℓ₁,ℓ₂ this yields u_{ℓ₁}·u_{ℓ₂} = –2/3, i.e., the inner product is at most –2/3, which satisfies the definition of a 5/2‑coloring (a vector coloring where adjacent vertices have inner product ≤ –2/3). Theorem 2.3 shows that a vector 5/2‑coloring guarantees an independent set of size Ω(|N^{(3)}(i)|·Δ^{‑1/5}). Because N^{(3)}(i) is provably larger than N^{(2)}(i), the Δ^{‑1/5} loss is outweighed, leading to an overall improvement.

The algorithm proceeds by a standard sparse‑dense split: if the minimum degree of the graph exceeds n^{1/2}, the dense‑graph combinatorial algorithm of Kawarabayashi‑Thorup‑Yoneda (Theorem 1.2) is applied, achieving a k‑coloring with k = n^{o(1)}·n/Δ. If the maximum degree is at most n^{3/5}, the new SDP‑based routine extracts a large independent set from a third‑level neighborhood as described above, removes it, and recurses. By carefully balancing the choice of Δ, the recursion yields a coloring using O(n^{0.19539}) colors in polynomial time.

The paper concludes that the introduction of third‑level neighborhood analysis and the vector 5/2‑coloring technique revives the power of SDP‑based methods for approximate graph coloring. Moreover, the construction of a 5/2‑vector coloring via high‑level Lasserre relaxations may find applications beyond coloring, such as in Promise CSPs, graph partitioning, and other problems where high‑quality vector colorings are useful.