All about Lovász--Schrijver (LS)

Definition from: On the complexity of the propositional calculus

Notes: degree two, dynamic, no squares

Proof Systems

• Lovász--Schrijver (LS) stronger than Resolution
• Source: LS → L&P → uL&P → Res
• Source: LS → PHP → PC_F2 → Res
• Lovász--Schrijver (LS) stronger than Truth table
• Source: LS → tlLS → tlRes → ttp
• Source: LS → PHP → tlResLin_F2 → tlRes → ttp
• Lovász--Schrijver (LS) stronger than Tree-like resolution
• Source: LS → tlLS → tlRes
• Source: LS → PHP → tlResLin_F2 → tlRes
• Lovász--Schrijver (LS) stronger than Regular resolution
• Source: LS → L&P → uL&P → Res → regRes
• Source: LS → PHP → PC_F2 → Res → regRes
• Lovász--Schrijver (LS) stronger than Ordered resolution
• Source: LS → L&P → uL&P → Res → regRes → ordRes
• Source: LS → PHP → PC_F2 → Res → regRes → ordRes
• Lovász--Schrijver (LS) stronger than Pool resolution
• Source: LS → L&P → uL&P → Res → poolRes
• Source: LS → PHP → PC_F2 → Res → poolRes
• Lovász--Schrijver (LS) stronger than Linear resolution
• Source: LS → L&P → uL&P → Res → linRes
• Source: LS → PHP → PC_F2 → Res → linRes
• Lovász--Schrijver (LS) stronger than Reversible resolution
• Source: LS → L&P → uL&P → Res → revRes
• Source: LS → PHP → PC_F2 → Res → revRes
• Lovász--Schrijver (LS) [missing?] Cutting Planes
• Lovász--Schrijver (LS) not simulated by Tree-like Cutting Planes
• Source: LS → L&P → uL&P → Res → regRes → ordRes → peb+ind → tlCP
• Lovász--Schrijver (LS) [missing?] Cutting Planes with Unary Coefficients
• Lovász--Schrijver (LS) [missing?] Semantic Cutting Planes
• Lovász--Schrijver (LS) stronger than Cutting Planes with Saturation
• Source: LS → L&P → uL&P → Res → saturationCP
• Source: LS → PHP → PC_F2 → Res → saturationCP
• Lovász--Schrijver (LS) [missing?] Stabbing Planes
• Lovász--Schrijver (LS) [missing?] Stabbing Planes with Unary Coefficients
• Lovász--Schrijver (LS) [missing?] Res(CP)
• Lovász--Schrijver (LS) [missing?] Res(LP)
• Lovász--Schrijver (LS) [missing?] Res(CP) with unary coefficients
• Lovász--Schrijver (LS) [missing?] Res(LP) with unary coefficients
• Lovász--Schrijver (LS) [missing?] Res(L\(\&\)P)
• Lovász--Schrijver (LS) [missing?] Res(L\(\&\)P) with unary coefficients
• Lovász--Schrijver (LS) [missing?] Semantic degree-k threshold system, treelike version
• Lovász--Schrijver (LS) not simulated by Polynomial Calculus over \(\mathbb{F}_2\)
• Source: LS → PHP → PC_F2
• Lovász--Schrijver (LS) not simulated by Nullstellensatz over \(\mathbb{F}_2\)
• Source: LS → PHP → PC_F2 → NS_F2
• Lovász--Schrijver (LS) [missing?] ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Lovász--Schrijver (LS) [missing?] unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Lovász--Schrijver (LS) [missing?] ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Lovász--Schrijver (LS) not simulated by Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Source: LS → PHP → tlResLin_F2
• Lovász--Schrijver (LS) not simulated by Polynomial Calculus over \(\mathbb{Q}\)
• Source: LS → PHP → PC_Q
• Lovász--Schrijver (LS) not simulated by Nullstellensatz over \(\mathbb{Q}\)
• Source: LS → PHP → PC_Q → NS_Q
• Lovász--Schrijver (LS) stronger than Hitting
• Source: LS → tlLS → tlRes → hit
• Source: LS → PHP → tlResLin_F2 → tlRes → hit
• Lovász--Schrijver (LS) simulates Lift and Project
• Source: [subsystem]
• Lovász--Schrijver (LS) simulates Lift and Project with unary coefficients
• Source: LS → L&P → uL&P
• Lovász--Schrijver (LS) [missing?] Positivstellensatz Calculus
• Lovász--Schrijver (LS) [missing?] Positivstellensatz
• Lovász--Schrijver (LS) simulated by Lovász--Schrijver with squares (LS\(_+\))
• Source: [subsystem]
• Lovász--Schrijver (LS) simulated by Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• Source: LSd+ → LS+ → LS
• Lovász--Schrijver (LS) simulated by Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• Source: LSn+ → LSd+ → LS+ → LS
• Lovász--Schrijver (LS) simulated by Cone Proof System
• Source: CPS → LSn+ → LSd+ → LS+ → LS
• Lovász--Schrijver (LS) simulates tl Lovász--Schrijver (LS)
• Source: [subsystem]
• Lovász--Schrijver (LS) [missing?] Lovász--Schrijver with squares (LS\(_+\))
• Lovász--Schrijver (LS) [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• Lovász--Schrijver (LS) [missing?] static Lovász--Schrijver (static LS)
• Lovász--Schrijver (LS) [missing?] static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Lovász--Schrijver (LS) [missing?] static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Lovász--Schrijver (LS) [missing?] Sum of Squares (Lasserre)
• Lovász--Schrijver (LS) [missing?] Sherali--Adams
• Lovász--Schrijver (LS) [missing?] Circular resolution
• Lovász--Schrijver (LS) not simulated by Unary Sherali--Adams
• Source: LS → L&P → uL&P → Res → sod+xor → uSA
• Lovász--Schrijver (LS) [missing?] Ideal Proof System
• Lovász--Schrijver (LS) [missing?] Extended Frege
• Lovász--Schrijver (LS) [missing?] Extended resolution
• Lovász--Schrijver (LS) [missing?] Frege
• Lovász--Schrijver (LS) not simulated by \(\mathrm{AC}^0\)-Frege
• Source: LS → PHP → fPHP → ofPHP → AC0Frege
• Lovász--Schrijver (LS) not simulated by k-DNF Resolution
• Source: LS → PHP → fPHP → ofPHP → AC0Frege → Res-k
• Lovász--Schrijver (LS) [missing?] \(\mathrm{TC}^0\)-Frege
• Lovász--Schrijver (LS) [missing?] \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Lovász--Schrijver (LS) [missing?] \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Lovász--Schrijver (LS) [missing?] OBDD(join,weakening)
• Lovász--Schrijver (LS) [missing?] LK
• Lovász--Schrijver (LS) [missing?] Zermelo-Fraenkl Set Theory with the Axiom of Choice

Formulas

• The size complexity of Pigeonhole principle in Lovász--Schrijver (LS) is polynomial
• Source: Pud99 On the complexity of the propositional calculus
• The size complexity of Pigeonhole principle with functionality axioms in Lovász--Schrijver (LS) is polynomial
• Source: LS → PHP → fPHP
• The size complexity of Pigeonhole principle with functionality and onto axioms in Lovász--Schrijver (LS) is polynomial
• Source: LS → PHP → fPHP → ofPHP
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Lovász--Schrijver (LS) is polynomial
• Source: LS → PHP → PHP2nn
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Lovász--Schrijver (LS) is polynomial
• Source: LS → PHP → PHP2nn → PHPn2n
• The size complexity of Bitwise pigeonhole principle in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Ordering principle in Lovász--Schrijver (LS) is polynomial
• Source: LS → L&P → uL&P → Res → regRes → ordering
• The size complexity of Guarded ordering principle in Lovász--Schrijver (LS) is polynomial
• Source: LS → L&P → uL&P → Res → poolRes → ordering+guard
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Flow in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Random CNF in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Clique-Colouring in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Clique-Colouring encoded as equalities in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Clique-Colouring composed with a permutation in Lovász--Schrijver (LS) is [missing?]
• The size complexity of Pebbling \(\circ\) IND in Lovász--Schrijver (LS) is polynomial
• Source: LS → L&P → uL&P → Res → regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Lovász--Schrijver (LS) is polynomial
• Source: LS → L&P → uL&P → Res → poolRes → peb+xor+guard
• The size complexity of Stone formula in Lovász--Schrijver (LS) is polynomial
• Source: LS → L&P → uL&P → Res → poolRes → stone
• The size complexity of String of pearls in Lovász--Schrijver (LS) is polynomial
• Source: LS → L&P → uL&P → Res → regRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Lovász--Schrijver (LS) is polynomial
• Source: LS → L&P → uL&P → 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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