An Integer Programming Approach to Compute Lower Bounds for Ramsey Numbers Using Circulant Graphs

Abstract: The Ramsey number $R(m,n)$ is the smallest order at which every red-blue edge coloring of a complete graph must contain a blue clique (a complete subgraph) of size $m$ or a red clique of size $n$. Determining these numbers exactly is extremely hard, and even certifying a lower bound requires exhibiting an explicit coloring that avoids both cliques. We develop an integer programming framework for certifying such lower bounds, restricting the search to circulant graphs, whose rotational symmetry lets us reformulate the problem in a projected distance space, reducing the number of binary variables from quadratic to linear in the graph order. We strengthen this projected model through coefficient reduction and solve it with a branch-and-cut algorithm whose separation routine exploits the common neighborhood structure of circulant graphs, combining heuristic and exact maximum-clique algorithms. In an extensive computational campaign on circulant graphs with up to 410 vertices, we improve the best lower bounds previously obtained by other methods by up to 11 points for 25 values of $R(3,n)$ with $24\le n\le49$ and $n\neq27$, each backed by an explicit graph certificate that can be independently verified with a stand-alone exact clique solver. To the best of our knowledge, our method also provides the first reproducible optimization-based procedure for certifying circulant Ramsey numbers $R_C(m,n)$, which we use to establish eight new values of $R_C(3,n)$ with $13\le n\le20$. Our framework, graph certificates, and stand-alone checker are provided as supplementary material to support independent verification and reuse.

• The paper developed an integer programming approach for computing lower bounds on Ramsey numbers by constructing circulant $(m, n)$-colorings.
• This framework improved 25 best-known lower bounds for $R(3, n)$ and established eight new exact circulant Ramsey numbers.
• This framework aids researchers investigating graph coloring and combinatorial optimization, particularly within the realm of Ramsey theory.

This paper develops an integer programming framework for certifying lower bounds on Ramsey numbers by constructing circulant $(m,n)$-colorings. A circulant $(m,n)$-coloring of $\mathcal K_t$ is a 2-coloring in which edge colors depend only on circular distance; its existence certifies $R(m,n) \ge t+1$. The authors reformulate the search in a projected "distance space" with $\lfloor t/2 \rfloor$ binary variables instead of $\binom{t}{2}$ edge variables, strengthen the resulting formulation with Turán-derived and coefficient-reduced inequalities, and embed everything in a branch-and-cut algorithm whose separation exploits vertex-transitivity of circulant graphs. The framework improves 25 best known lower bounds for $R(3,n)$ with $24 \le n \le 49$ ($n \neq 27$), each backed by an explicit graph certificate verifiable with an exact clique solver, and establishes eight new exact circulant Ramsey numbers $R_C(3,n)$ for $(m,n)$0.

From edge space to distance space

The paper begins from an edge-space feasibility model with binary variables $(m,n)$1 for each edge of $(m,n)$2, forbidding fully blue $(m,n)$3-cliques and fully red $(m,n)$4-cliques. The feasible integer points lie inside the Cartesian product of two Turán polytopes intersected with the color-partition subspace $(m,n)$5, which allows every valid inequality for the Turán polytope to be transferred. In particular, Turán inequalities of the form $(m,n)$6 are valid for all $(m,n)$7, where $(m,n)$8 is the classical Turán number.

Two structural results refine this picture. First, the authors complete Raymond's characterization of facet-definingness of Turán inequalities: they prove that when $(m,n)$9, the inequality on $\mathcal K_t$0 is implied by those on its $\mathcal K_t$1-subsets, via an explicit primal–dual pair showing that the maximum of $\mathcal K_t$2 under the subset constraints equals exactly $\mathcal K_t$3. Second, projecting out the red variables halves the variable count.

The key modeling step restricts attention to circulant colorings. Rotational and reflectional symmetry (the dihedral group $\mathcal K_t$4) lets the model be rewritten over distance variables $\mathcal K_t$5, where $\mathcal K_t$6 iff distance $\mathcal K_t$7 belongs to the blue distance list. Constraints are indexed not by subsets but by distinct coefficient vectors in $\mathcal K_t$8 — the set of distance-multiplicity profiles induced by $\mathcal K_t$9-subsets. An orbit-counting argument via Burnside's lemma shows that, for fixed $R(m,n) \ge t+1$0, the number of distinct inequalities shrinks by a factor asymptotic to $R(m,n) \ge t+1$1 relative to all $R(m,n) \ge t+1$2 subset-indexed inequalities. This reduction is what makes instances with hundreds of vertices tractable at all; the paper shows explicitly that removing the circulant restriction makes even $R(m,n) \ge t+1$3 certification prohibitive within a 30-minute limit, while $R(m,n) \ge t+1$4 and $R(m,n) \ge t+1$5 remain solvable without it.

Strengthened inequalities

The basic distance-space model is strengthened by coefficient reduction: each coefficient $R(m,n) \ge t+1$6 is capped at $R(m,n) \ge t+1$7, preserving all binary solutions while strictly shrinking the LP relaxation (a small fractional counterexample demonstrates strictness). At the exact cardinalities $R(m,n) \ge t+1$8 and $R(m,n) \ge t+1$9, coefficients collapse to binary values, yielding covering-type inequalities indexed by distance sets $\lfloor t/2 \rfloor$0; retaining only inclusion-wise minimal distance sets eliminates support-dominated inequalities.

The practical impact is substantial. For $\lfloor t/2 \rfloor$1 and $\lfloor t/2 \rfloor$2, anchored edge-space enumeration requires roughly $\lfloor t/2 \rfloor$3 inequalities, coefficient-reduced distance-set identification about $\lfloor t/2 \rfloor$4, and minimal distance sets only about $\lfloor t/2 \rfloor$5. Additional structured families — arithmetic-progression, triangle, and quadrangle inequalities — are small enough to be enumerated entirely during preprocessing.

Branch-and-cut and separation

Since full enumeration remains impossible, the strengthened model is solved by branch-and-cut with lazy constraints separated only at integer solutions. Separation asks whether the candidate coloring contains a blue $\lfloor t/2 \rfloor$6 or red $\lfloor t/2 \rfloor$7. Because circulant graphs are vertex-transitive, $\lfloor t/2 \rfloor$8 for any neighborhood graph $\lfloor t/2 \rfloor$9, so clique detection reduces to a single, much smaller induced subgraph. The separation routine chains a greedy target-clique heuristic, the adaptive multistart tabu search AMTS, and finally the exact CliSAT branch-and-bound solver; failure of the heuristics cannot cause a missed cut because CliSAT either finds a clique or certifies none exists. A heuristic deletion-filter minimization then reduces cut supports, though the paper notes this procedure does not certify inclusion-wise minimality since it uses only heuristic clique tests.

The separation problem itself remains NP-hard even on circulant graphs, which the authors establish by reducing threshold clique on circulants (Codenotti et al.) to violation detection with $\binom{t}{2}$0. Vertex-transitivity thus reduces problem size in practice but does not remove worst-case difficulty.

Tuning results

On a pilot testbed of 72 provably infeasible instances (all orders between $\binom{t}{2}$1 and $\binom{t}{2}$2 for $\binom{t}{2}$3), an ablation study isolates four features of the Baseline configuration:

Configuration removed	Node ratio	CPU time ratio
Triangle inequalities	$\binom{t}{2}$4	$\binom{t}{2}$5
AP inequalities	$\binom{t}{2}$6	$\binom{t}{2}$7
Coefficient reduction	$\binom{t}{2}$8	$\binom{t}{2}$9
Cut minimization	$R(3,n)$0	$R(3,n)$1
All four	$R(3,n)$2	$R(3,n)$3

Triangle and arithmetic-progression inequalities dominate; cut minimization reduces nodes everywhere but costs CPU time on small instances (ratio $R(3,n)$4 at $R(3,n)$5, rising to $R(3,n)$6 at $R(3,n)$7), so it is retained only because its benefit concentrates on hard instances. Replacing the specialized separator with a CPLEX-based clique oracle produces identical trees but is $R(3,n)$8 slower overall. Fitted exponential growth across the testbed gives multipliers of $R(3,n)$9 per unit $24 \le n \le 49$0 in branching nodes and $24 \le n \le 49$1 per unit $24 \le n \le 49$2 in total CPU time, both with high fit quality — though the paper cautions these fits cover only five values of $24 \le n \le 49$3 and should not be extrapolated beyond $24 \le n \le 49$4.

Certified lower bounds

The large-scale campaign produced certificates for 25 improved lower bounds, summarized below alongside representative improvements over prior independent bounds:

Parameter	Prior bound	New bound	Improvement
$24 \le n \le 49$5	143	151	8
$24 \le n \le 49$6	252	261	9
$24 \le n \le 49$7	338	348	10
$24 \le n \le 49$8	360	371	11
$24 \le n \le 49$9	402	411	9

Improvements range up to 11 points, with certificates on circulant graphs of up to 410 vertices. Every certificate was verified post hoc by a stand-alone checker based on CliSAT that treats stored edges as red and missing edges as blue and proves absence of both forbidden cliques; the checker depends neither on CPLEX nor on the distance-space representation, and applies to arbitrary (noncirculant) colorings. Notably, several bounds were recovered through post-processing of incumbents whose separation had not completed within the 24-hour limit, meaning the branch-and-cut proof of feasibility was incomplete but the coloring itself was independently certified. The one exception in the range is $n \neq 27$0: the framework reproduces the known lower bound but does not improve it.

For circulant Ramsey numbers $n \neq 27$1 — the smallest order beyond which no circulant $n \neq 27$2-coloring exists at any larger order below $n \neq 27$3 — the campaign established eight new exact values: $n \neq 27$4 for $n \neq 27$5. Each required proving infeasibility at every order between the last feasible order and $n \neq 27$6, exploiting the fact that circulant colorability is non-monotone in order. The same procedure also provides the first reproducible optimization-based certification of previously reported circulant Ramsey values that had been described only as "determined with the help of a computer." For $n \neq 27$7 and $n \neq 27$8, at least one order could not be resolved within the time limit, so no further exact circulant values were obtained.

Limitations and open questions

The framework is specialized to the $n \neq 27$9 case at scale: all blue triangles can be pre-enumerated, eliminating blue separation entirely. For $R_C(3,n)$0 the quadrangle index set grows as $R_C(3,n)$1, making preprocessing increasingly costly, and for $R_C(3,n)$2 both colors rely on dynamic separation; the paper claims applicability beyond triangles for circulant Ramsey numbers but not comparable large-scale performance. Campaign runs use parallel CPLEX search and hence nondeterministic execution, so single-run wall-clock times are not stable performance measures. The growth fits are confined to $R_C(3,n)$3. Open questions left explicit include whether other structured graph families (block-circulant, multicolor, book, cycle Ramsey problems) admit analogous compact symmetry-class formulations, and how dominance relations among distance-space inequalities can be characterized beyond the exact-cardinality case.

Conclusion

The paper contributes a complete optimization pipeline — distance-space projection, polyhedral strengthening from the Turán polytope, minimal-distance-set reduction, and clique-based lazy separation — that converts the construction of Ramsey lower-bound certificates into a reproducible computational procedure. Its strongest quantitative outcomes are 25 improved lower bounds on $R_C(3,n)$4 up to $R_C(3,n)$5, improvements of up to 11 points over prior methods, eight new exact circulant Ramsey numbers, and independently verifiable certificates released together with source code and a stand-alone checker.