All about Cutting Planes

Proof Systems

• Cutting Planes stronger than Resolution
• Source: CP → uCP → Res
• Source: CP → uCP → PHP → PC_Q → Res
• Cutting Planes stronger than Truth table
• Source: CP → tlCP → tlRes → ttp
• Source: CP → uCP → PHP → tlResLin_F2 → tlRes → ttp
• Cutting Planes stronger than Tree-like resolution
• Source: CP → tlCP → tlRes
• Source: CP → uCP → PHP → tlResLin_F2 → tlRes
• Cutting Planes stronger than Regular resolution
• Source: CP → uCP → Res → regRes
• Source: CP → uCP → PHP → PC_Q → Res → regRes
• Cutting Planes stronger than Ordered resolution
• Source: CP → uCP → Res → regRes → ordRes
• Source: CP → uCP → Res → regRes → pearl → ordRes
• Cutting Planes stronger than Pool resolution
• Source: CP → uCP → Res → poolRes
• Source: CP → uCP → PHP → PC_Q → Res → poolRes
• Cutting Planes stronger than Linear resolution
• Source: CP → uCP → Res → linRes
• Source: CP → uCP → PHP → PC_Q → Res → linRes
• Cutting Planes stronger than Reversible resolution
• Source: CP → uCP → Res → revRes
• Source: CP → uCP → PHP → PC_Q → Res → revRes
• Cutting Planes stronger than Tree-like Cutting Planes
• Source: [subsystem]
• Source: CP → uCP → Res → regRes → ordRes → peb+ind → tlCP
• Cutting Planes simulates Cutting Planes with Unary Coefficients
• Source: [subsystem]
• Cutting Planes weaker than Semantic Cutting Planes
• Source: [subsystem]
• Source: semanticCP → CliqueColouringeq → CP
• Cutting Planes stronger than Cutting Planes with Saturation
• Source: CP → uCP → Res → saturationCP
• Source: CP → uCP → PHP → PC_Q → Res → saturationCP
• Cutting Planes simulated by Stabbing Planes
• Source: FGIPRTW21 On the Power and Limitations of Branch and Cut.
• Cutting Planes simulates Stabbing Planes with Unary Coefficients
• Source: FGIPRTW21 On the Power and Limitations of Branch and Cut.
• Cutting Planes weaker than Res(CP)
• Source: [subsystem]
• Source: Res(CP) → Res(LP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP
• Cutting Planes does not simulate Res(LP)
• Source: Res(LP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP
• Cutting Planes does not simulate Res(CP) with unary coefficients
• Source: uRes(CP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP
• Cutting Planes does not simulate Res(LP) with unary coefficients
• Source: uRes(LP) → Res-k → CliqueColouringmlogm → CP
• Cutting Planes does not simulate Res(L\(\&\)P)
• Source: Res(L&P) → Res(LP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP
• Cutting Planes does not simulate Res(L\(\&\)P) with unary coefficients
• Source: uRes(L&P) → uRes(LP) → Res-k → CliqueColouringmlogm → CP
• Cutting Planes [missing?] Semantic degree-k threshold system, treelike version
• Cutting Planes incomparable wrt Polynomial Calculus over \(\mathbb{F}_2\)
• Source: CP → uCP → PHP → PC_F2
• Source: PC_F2 → NS_F2 → ts+ind → CP
• Cutting Planes incomparable wrt Nullstellensatz over \(\mathbb{F}_2\)
• Source: CP → uCP → PHP → PC_F2 → NS_F2
• Source: NS_F2 → ts+ind → CP
• Cutting Planes does not simulate ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: ResLin_Z → uResLin_Z → uRes(CP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP
• Cutting Planes does not simulate unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: uResLin_Z → uRes(CP) → uRes(LP) → Res-k → CliqueColouringmlogm → CP
• Cutting Planes [missing?] ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Cutting Planes not simulated by Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Source: CP → uCP → PHP → tlResLin_F2
• Cutting Planes incomparable wrt Polynomial Calculus over \(\mathbb{Q}\)
• Source: CP → uCP → PHP → PC_Q
• Source: PC_Q → NS_Q → tsQ+ind → CP
• Cutting Planes incomparable wrt Nullstellensatz over \(\mathbb{Q}\)
• Source: CP → uCP → PHP → PC_Q → NS_Q
• Source: NS_Q → tsQ+ind → CP
• Cutting Planes stronger than Hitting
• Source: CP → tlCP → tlRes → hit
• Source: CP → uCP → PHP → tlResLin_F2 → tlRes → hit
• Cutting Planes [missing?] Lift and Project
• Cutting Planes simulates Lift and Project with unary coefficients
• Source: CP → uCP → uL&P
• Cutting Planes does not simulate Positivstellensatz Calculus
• Source: PSC → PS → SoS → sLSn+ → CliqueColouring → CP
• Cutting Planes does not simulate Positivstellensatz
• Source: PS → SoS → sLSn+ → CliqueColouring → CP
• Cutting Planes [missing?] Lovász--Schrijver (LS)
• Cutting Planes [missing?] Lovász--Schrijver with squares (LS\(_+\))
• Cutting Planes does not simulate Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• Source: LSd+ → CliqueColouring → CP
• Cutting Planes does not simulate Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• Source: LSn+ → LSd+ → CliqueColouring → CP
• Cutting Planes weaker than Cone Proof System
• Source: CPS → IPS → extFrege → Frege → CP
• Source: CPS → LSn+ → LSd+ → CliqueColouring → CP
• Cutting Planes not simulated by tl Lovász--Schrijver (LS)
• Source: CP → tseitin → sLS+ → tlLS+ → tlLS
• Cutting Planes not simulated by Lovász--Schrijver with squares (LS\(_+\))
• Source: CP → tseitin → sLS+ → tlLS+
• Cutting Planes [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• Cutting Planes not simulated by static Lovász--Schrijver (static LS)
• Source: CP → tseitin → sLS+ → sLS
• Cutting Planes not simulated by static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Source: CP → tseitin → sLS+
• Cutting Planes incomparable wrt static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Source: CP → tseitin → SoS → sLSn+
• Source: sLSn+ → CliqueColouring → CP
• Cutting Planes incomparable wrt Sum of Squares (Lasserre)
• Source: CP → tseitin → SoS
• Source: SoS → sLSn+ → CliqueColouring → CP
• Cutting Planes incomparable wrt Sherali--Adams
• Source: CP → tseitin → SoS → SA
• Source: SA → NS_Q → tsQ+ind → CP
• Cutting Planes incomparable wrt Circular resolution
• Source: CP → tseitin → SoS → SA → circRes
• Source: circRes → SA → NS_Q → tsQ+ind → CP
• Cutting Planes not simulated by Unary Sherali--Adams
• Source: CP → uCP → Res → sod+xor → uSA
• Cutting Planes weaker than Ideal Proof System
• Source: IPS → extFrege → Frege → CP
• Source: IPS → extFrege → Frege → AC0Frege → CliqueColouring2mm → CP
• Cutting Planes weaker than Extended Frege
• Source: extFrege → Frege → CP
• Source: extFrege → Frege → AC0Frege → CliqueColouring2mm → CP
• Cutting Planes weaker than Extended resolution
• Source: extRes → extFrege → Frege → CP
• Source: extRes → extFrege → Frege → AC0Frege → CliqueColouring2mm → CP
• Cutting Planes weaker than Frege
• Source: Goerdt92 Cutting plane versus Frege proof systems
• Source: Frege → AC0Frege → CliqueColouring2mm → CP
• Cutting Planes incomparable wrt \(\mathrm{AC}^0\)-Frege
• Source: CP → uCP → PHP → fPHP → ofPHP → AC0Frege
• Source: AC0Frege → CliqueColouring2mm → CP
• Cutting Planes incomparable wrt k-DNF Resolution
• Source: CP → uCP → PHP → fPHP → ofPHP → AC0Frege → Res-k
• Source: Res-k → CliqueColouringmlogm → CP
• Cutting Planes weaker than \(\mathrm{TC}^0\)-Frege
• Source: TC0Frege → Res(CP) → CP
• Source: TC0Frege → AC0Frege → CliqueColouring2mm → CP
• Cutting Planes does not simulate \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Source: AC0Frege+Mod2axioms → AC0Frege → CliqueColouring2mm → CP
• Cutting Planes does not simulate \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Source: AC0(+)Frege → AC0Frege → CliqueColouring2mm → CP
• Cutting Planes does not simulate OBDD(join,weakening)
• Source: OBDDjoinweak → CliqueColouring → CP
• Cutting Planes weaker than LK
• Source: LK → Frege → CP
• Source: LK → Frege → AC0Frege → CliqueColouring2mm → CP
• Cutting Planes weaker than Zermelo-Fraenkl Set Theory with the Axiom of Choice
• Source: ZFC → Res(CP) → CP
• Source: ZFC → extFrege → Frege → AC0Frege → CliqueColouring2mm → CP

Formulas

• The size complexity of Pigeonhole principle in Cutting Planes is polynomial
• Source: CP → uCP → PHP
• The size complexity of Pigeonhole principle with functionality axioms in Cutting Planes is polynomial
• Source: CP → uCP → PHP → fPHP
• The size complexity of Pigeonhole principle with functionality and onto axioms in Cutting Planes is polynomial
• Source: CP → uCP → PHP → fPHP → ofPHP
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Cutting Planes is polynomial
• Source: CP → uCP → PHP → PHP2nn
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Cutting Planes is polynomial
• Source: CP → uCP → PHP → PHP2nn → PHPn2n
• The size complexity of Bitwise pigeonhole principle in Cutting Planes is exponential
• Source: HP17 Random Formulas, Monotone Circuits, and Interpolation.
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Cutting Planes is [missing?]
• The size complexity of Ordering principle in Cutting Planes is polynomial
• Source: CP → uCP → Res → regRes → ordering
• The size complexity of Guarded ordering principle in Cutting Planes is polynomial
• Source: CP → uCP → Res → poolRes → ordering+guard
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Cutting Planes is at most quasipolynomial
• Source: DT20 On the Complexity of Branching Proofs.
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Cutting Planes is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Cutting Planes is [missing?]
• The size complexity of Flow in Cutting Planes is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Cutting Planes is exponential
• Source: GGKS20 Monotone Circuit Lower Bounds from Resolution.
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Cutting Planes is exponential
• Source: GGKS20 Monotone Circuit Lower Bounds from Resolution.
• The size complexity of Random CNF in Cutting Planes is [missing?]
• 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 encoded as equalities in Cutting Planes is exponential
• Source: FHL16 Semantic Versus Syntactic Cutting Planes.
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Cutting Planes is exponential
• Source: Pudlak97 Lower Bounds for Resolution and Cutting Plane Proofs and Monotone Computations.
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Cutting Planes is superpolynomial
• Source: [citation needed]
• The size complexity of Clique-Colouring composed with a permutation in Cutting Planes is [missing?]
• The size complexity of Pebbling \(\circ\) IND in Cutting Planes is polynomial
• Source: CP → uCP → Res → regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Cutting Planes is polynomial
• Source: CP → uCP → Res → poolRes → peb+xor+guard
• The size complexity of Stone formula in Cutting Planes is polynomial
• Source: CP → uCP → Res → poolRes → stone
• The size complexity of String of pearls in Cutting Planes is polynomial
• Source: CP → uCP → Res → regRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Cutting Planes is polynomial
• Source: CP → uCP → Res → sod+xor

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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