All about Clique-Colouring

Proof Systems

• The size complexity of Clique-Colouring in Resolution is exponential
  • Source: Krajicek97 Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic.
• The size complexity of Clique-Colouring in Truth table is exponential
  • Source: CliqueColouring → Res → regRes → tlRes → ttp
• The size complexity of Clique-Colouring in Tree-like resolution is exponential
  • Source: CliqueColouring → Res → regRes → tlRes
• The size complexity of Clique-Colouring in Regular resolution is exponential
  • Source: CliqueColouring → Res → regRes
• The size complexity of Clique-Colouring in Ordered resolution is exponential
  • Source: CliqueColouring → Res → regRes → ordRes
• The size complexity of Clique-Colouring in Pool resolution is exponential
  • Source: CliqueColouring → Res → poolRes
• The size complexity of Clique-Colouring in Linear resolution is exponential
  • Source: CliqueColouring → Res → linRes
• The size complexity of Clique-Colouring in Reversible resolution is exponential
  • Source: CliqueColouring → Res → revRes
• The size complexity of Clique-Colouring in Cutting Planes is exponential
  • Source: Pudlak97 Lower Bounds for Resolution and Cutting Plane Proofs and Monotone Computations.
• The size complexity of Clique-Colouring in Tree-like Cutting Planes is exponential
  • Source: CliqueColouring → CP → tlCP
• The size complexity of Clique-Colouring in Cutting Planes with Unary Coefficients is exponential
  • Source: CliqueColouring → CP → uCP
• The size complexity of Clique-Colouring in Semantic Cutting Planes is exponential
  • Source: FHL16 Semantic Versus Syntactic Cutting Planes.
• The size complexity of Clique-Colouring in Cutting Planes with Saturation is exponential
  • Source: CliqueColouring → Res → saturationCP
• The size complexity of Clique-Colouring in Stabbing Planes is exponential
  • Source: GP23 Sub-Exponential Lower Bounds for Branch-and-Bound with General Disjunctions via Interpolation
• The size complexity of Clique-Colouring in Stabbing Planes with Unary Coefficients is exponential
  • Source: CliqueColouring → SP → uSP
• The size complexity of Clique-Colouring in Res(CP) is [missing?]
• The size complexity of Clique-Colouring in Res(LP) is [missing?]
• The size complexity of Clique-Colouring in Res(CP) with unary coefficients is [missing?]
• The size complexity of Clique-Colouring in Res(LP) with unary coefficients is [missing?]
• The size complexity of Clique-Colouring in Res(L\(\&\)P) is [missing?]
• The size complexity of Clique-Colouring in Res(L\(\&\)P) with unary coefficients is [missing?]
• The size complexity of Clique-Colouring in Semantic degree-k threshold system, treelike version is [missing?]
• The size complexity of Clique-Colouring in Polynomial Calculus over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Clique-Colouring in Nullstellensatz over \(\mathbb{F}_2\) is [missing?]
• The size complexity of Clique-Colouring in ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals is [missing?]
• The size complexity of Clique-Colouring in unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals is [missing?]
• The size complexity of Clique-Colouring in ResLin over \(\mathbb{F}_2\), Res(\(\oplus\)) is [missing?]
• The size complexity of Clique-Colouring in Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\)) is [missing?]
• The size complexity of Clique-Colouring in Polynomial Calculus over \(\mathbb{Q}\) is [missing?]
• The size complexity of Clique-Colouring in Nullstellensatz over \(\mathbb{Q}\) is [missing?]
• The size complexity of Clique-Colouring in Hitting is exponential
  • Source: CliqueColouring → Res → regRes → tlRes → hit
• The size complexity of Clique-Colouring in Lift and Project is exponential
  • Source: Dash02 An Exponential Lower Bound on the Length of Some Classes of Branch-and-Cut Proofs.
• The size complexity of Clique-Colouring in Lift and Project with unary coefficients is exponential
  • Source: CliqueColouring → L&P → uL&P
• The size complexity of Clique-Colouring in Positivstellensatz Calculus is polynomial
  • Source: PSC → PS → SoS → sLSn+ → CliqueColouring
• The size complexity of Clique-Colouring in Positivstellensatz is polynomial
  • Source: PS → SoS → sLSn+ → CliqueColouring
• The size complexity of Clique-Colouring in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Clique-Colouring in Lovász--Schrijver with squares (LS\(_+\)) is [missing?]
• The size complexity of Clique-Colouring in Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree is polynomial
  • Source: GHP02 Complexity of semialgebraic proofs
• The size complexity of Clique-Colouring in Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree is polynomial
  • Source: LSn+ → LSd+ → CliqueColouring
• The size complexity of Clique-Colouring in Cone Proof System is polynomial
  • Source: CPS → LSn+ → LSd+ → CliqueColouring
• The size complexity of Clique-Colouring in tl Lovász--Schrijver (LS) is [missing?]
• The size complexity of Clique-Colouring in Lovász--Schrijver with squares (LS\(_+\)) is [missing?]
• The size complexity of Clique-Colouring in Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike is [missing?]
• The size complexity of Clique-Colouring in static Lovász--Schrijver (static LS) is [missing?]
• The size complexity of Clique-Colouring in static Lovász--Schrijver, with squares of linear functions (static LS\(_+\)) is [missing?]
• The size complexity of Clique-Colouring in static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\)) is polynomial
  • Source: [citation needed]
• The size complexity of Clique-Colouring in Sum of Squares (Lasserre) is polynomial
  • Source: SoS → sLSn+ → CliqueColouring
• The size complexity of Clique-Colouring in Sherali--Adams is [missing?]
• The size complexity of Clique-Colouring in Circular resolution is [missing?]
• The size complexity of Clique-Colouring in Unary Sherali--Adams is [missing?]
• The size complexity of Clique-Colouring in Ideal Proof System is [missing?]
• The size complexity of Clique-Colouring in Extended Frege is [missing?]
• The size complexity of Clique-Colouring in Extended resolution is [missing?]
• The size complexity of Clique-Colouring in Frege is [missing?]
• The size complexity of Clique-Colouring in \(\mathrm{AC}^0\)-Frege is [missing?]
• The size complexity of Clique-Colouring in k-DNF Resolution is [missing?]
• The size complexity of Clique-Colouring in \(\mathrm{TC}^0\)-Frege is [missing?]
• The size complexity of Clique-Colouring in \(\mathrm{AC}^0\)-Frege with mod 2 axioms is [missing?]
• The size complexity of Clique-Colouring in \(\mathrm{AC}^0\)-Frege with mod 2 gates is [missing?]
• The size complexity of Clique-Colouring in OBDD(join,weakening) is polynomial
  • Source: BIKS18 Reordering Rule Makes OBDD Proof Systems Stronger.
• The size complexity of Clique-Colouring in LK is [missing?]
• The size complexity of Clique-Colouring in Zermelo-Fraenkl Set Theory with the Axiom of Choice is [missing?]

This database is still incomplete; missing data may indicate either the information was not yet recorded or an open problem. Users are encouraged to contribute missing proof systems and/or relations at https://gitlab.com/proofcomplexityzoo/zoo.

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