All about Sum of Squares (Lasserre)

Proof Systems

• Sum of Squares (Lasserre) stronger than Resolution
• Source: SoS → PC_Q → Res
• Source: SoS → sLSn+ → CliqueColouring → Res
• Sum of Squares (Lasserre) stronger than Truth table
• Source: SoS → SA → NS_Q → tlRes → ttp
• Source: SoS → SA → PHP → tlResLin_F2 → tlRes → ttp
• Sum of Squares (Lasserre) stronger than Tree-like resolution
• Source: SoS → SA → NS_Q → tlRes
• Source: SoS → SA → PHP → tlResLin_F2 → tlRes
• Sum of Squares (Lasserre) stronger than Regular resolution
• Source: SoS → PC_Q → Res → regRes
• Source: SoS → sLSn+ → CliqueColouring → Res → regRes
• Sum of Squares (Lasserre) stronger than Ordered resolution
• Source: SoS → PC_Q → Res → regRes → ordRes
• Source: SoS → sLSn+ → CliqueColouring → Res → regRes → ordRes
• Sum of Squares (Lasserre) stronger than Pool resolution
• Source: SoS → PC_Q → Res → poolRes
• Source: SoS → sLSn+ → CliqueColouring → Res → poolRes
• Sum of Squares (Lasserre) stronger than Linear resolution
• Source: SoS → PC_Q → Res → linRes
• Source: SoS → sLSn+ → CliqueColouring → Res → linRes
• Sum of Squares (Lasserre) stronger than Reversible resolution
• Source: SoS → SA → uSA → revRes
• Source: SoS → sLSn+ → CliqueColouring → Res → revRes
• Sum of Squares (Lasserre) incomparable wrt Cutting Planes
• Source: SoS → sLSn+ → CliqueColouring → CP
• Source: CP → tseitin → SoS
• Sum of Squares (Lasserre) not simulated by Tree-like Cutting Planes
• Source: SoS → sLSn+ → CliqueColouring → CP → tlCP
• Sum of Squares (Lasserre) not simulated by Cutting Planes with Unary Coefficients
• Source: SoS → sLSn+ → CliqueColouring → CP → uCP
• Sum of Squares (Lasserre) incomparable wrt Semantic Cutting Planes
• Source: SoS → sLSn+ → CliqueColouring → semanticCP
• Source: semanticCP → CP → tseitin → SoS
• Sum of Squares (Lasserre) stronger than Cutting Planes with Saturation
• Source: SoS → PC_Q → Res → saturationCP
• Source: SoS → sLSn+ → CliqueColouring → Res → saturationCP
• Sum of Squares (Lasserre) incomparable wrt Stabbing Planes
• Source: SoS → sLSn+ → CliqueColouring → SP
• Source: SP → tseitin → SoS
• Sum of Squares (Lasserre) not simulated by Stabbing Planes with Unary Coefficients
• Source: SoS → sLSn+ → CliqueColouring → SP → uSP
• Sum of Squares (Lasserre) does not simulate Res(CP)
• Source: Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate Res(LP)
• Source: Res(LP) → uRes(LP) → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate Res(CP) with unary coefficients
• Source: uRes(CP) → uRes(LP) → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate Res(LP) with unary coefficients
• Source: uRes(LP) → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate Res(L\(\&\)P)
• Source: Res(L&P) → Res(LP) → uRes(LP) → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate Res(L\(\&\)P) with unary coefficients
• Source: uRes(L&P) → uRes(LP) → tseitin → SoS
• Sum of Squares (Lasserre) [missing?] Semantic degree-k threshold system, treelike version
• Sum of Squares (Lasserre) incomparable wrt Polynomial Calculus over \(\mathbb{F}_2\)
• Source: SoS → SA → PHP → PC_F2
• Source: PC_F2 → NS_F2 → tseitin → SoS
• Sum of Squares (Lasserre) incomparable wrt Nullstellensatz over \(\mathbb{F}_2\)
• Source: SoS → SA → PHP → PC_F2 → NS_F2
• Source: NS_F2 → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: ResLin_Z → uResLin_Z → ResLin_F2 → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: uResLin_Z → ResLin_F2 → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Source: ResLin_F2 → tseitin → SoS
• Sum of Squares (Lasserre) not simulated by Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Source: SoS → SA → PHP → tlResLin_F2
• Sum of Squares (Lasserre) stronger than Polynomial Calculus over \(\mathbb{Q}\)
• Source: Berkholz18 The Relation between Polynomial Calculus, Sherali-Adams, and Sum-of-Squares Proofs.
• Source: SoS → SA → PHP → PC_Q
• Sum of Squares (Lasserre) stronger than Nullstellensatz over \(\mathbb{Q}\)
• Source: SoS → SA → NS_Q
• Source: SoS → SA → PHP → PC_Q → NS_Q
• Sum of Squares (Lasserre) stronger than Hitting
• Source: SoS → SA → NS_Q → tlRes → hit
• Source: SoS → SA → PHP → tlResLin_F2 → tlRes → hit
• Sum of Squares (Lasserre) not simulated by Lift and Project
• Source: SoS → sLSn+ → CliqueColouring → L&P
• Sum of Squares (Lasserre) not simulated by Lift and Project with unary coefficients
• Source: SoS → sLSn+ → CliqueColouring → L&P → uL&P
• Sum of Squares (Lasserre) simulated by Positivstellensatz Calculus
• Source: PSC → PS → SoS
• Sum of Squares (Lasserre) simulated by Positivstellensatz
• Source: [subsystem]
• Sum of Squares (Lasserre) [missing?] Lovász--Schrijver (LS)
• Sum of Squares (Lasserre) [missing?] Lovász--Schrijver with squares (LS\(_+\))
• Sum of Squares (Lasserre) does not simulate Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• Source: LSd+ → tseitin → SoS
• Sum of Squares (Lasserre) weaker than Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• Source: LSn+ → PSC → PS → SoS
• Source: LSn+ → LSd+ → tseitin → SoS
• Sum of Squares (Lasserre) weaker than Cone Proof System
• Source: CPS → LSn+ → PSC → PS → SoS
• Source: CPS → LSn+ → LSd+ → tseitin → SoS
• Sum of Squares (Lasserre) simulates tl Lovász--Schrijver (LS)
• Source: SoS → sLSn+ → sLS+ → tlLS+ → tlLS
• Sum of Squares (Lasserre) simulates Lovász--Schrijver with squares (LS\(_+\))
• Source: SoS → sLSn+ → sLS+ → tlLS+
• Sum of Squares (Lasserre) [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• Sum of Squares (Lasserre) simulates static Lovász--Schrijver (static LS)
• Source: SoS → sLSn+ → sLS+ → sLS
• Sum of Squares (Lasserre) simulates static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Source: SoS → sLSn+ → sLS+
• Sum of Squares (Lasserre) simulates static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Source: AtseriasH19 Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs.
• Sum of Squares (Lasserre) simulates Sherali--Adams
• Source: [citation needed]
• Sum of Squares (Lasserre) simulates Circular resolution
• Source: SoS → SA → circRes
• Sum of Squares (Lasserre) stronger than Unary Sherali--Adams
• Source: SoS → SA → uSA
• Source: SoS → PC_Q → Res → sod+xor → uSA
• Sum of Squares (Lasserre) does not simulate Ideal Proof System
• Source: IPS → extFrege → Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate Extended Frege
• Source: extFrege → Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate Extended resolution
• Source: extRes → extFrege → Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate Frege
• Source: Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS
• Sum of Squares (Lasserre) not simulated by \(\mathrm{AC}^0\)-Frege
• Source: SoS → SA → NS_Q → ofPHP → AC0Frege
• Sum of Squares (Lasserre) not simulated by k-DNF Resolution
• Source: SoS → SA → NS_Q → ofPHP → AC0Frege → Res-k
• Sum of Squares (Lasserre) does not simulate \(\mathrm{TC}^0\)-Frege
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Source: AC0Frege+Mod2axioms → NS_F2 → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Source: AC0(+)Frege → ResLin_F2 → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate OBDD(join,weakening)
• Source: OBDDjoinweak → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate LK
• Source: LK → Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS
• Sum of Squares (Lasserre) does not simulate Zermelo-Fraenkl Set Theory with the Axiom of Choice
• Source: ZFC → Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS

Formulas

• The size complexity of Pigeonhole principle in Sum of Squares (Lasserre) is polynomial
• Source: SoS → SA → PHP
• The size complexity of Pigeonhole principle with functionality axioms in Sum of Squares (Lasserre) is polynomial
• Source: SoS → SA → PHP → fPHP
• The size complexity of Pigeonhole principle with functionality and onto axioms in Sum of Squares (Lasserre) is polynomial
• Source: SoS → SA → NS_Q → ofPHP
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Sum of Squares (Lasserre) is polynomial
• Source: SoS → SA → PHP → PHP2nn
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Sum of Squares (Lasserre) is polynomial
• Source: SoS → SA → PHP → PHP2nn → PHPn2n
• The size complexity of Bitwise pigeonhole principle in Sum of Squares (Lasserre) is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Sum of Squares (Lasserre) is [missing?]
• The size complexity of Ordering principle in Sum of Squares (Lasserre) is polynomial
• Source: SoS → PC_Q → Res → regRes → ordering
• The size complexity of Guarded ordering principle in Sum of Squares (Lasserre) is polynomial
• Source: SoS → PC_Q → Res → poolRes → ordering+guard
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Sum of Squares (Lasserre) is exponential
• Source: AtseriasH19 Size-Degree Trade-Offs for Sums-of-Squares and Positivstellensatz Proofs.
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Sum of Squares (Lasserre) is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Sum of Squares (Lasserre) is polynomial
• Source: SoS → SA → NS_Q → tseitinQ
• The size complexity of Flow in Sum of Squares (Lasserre) is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Sum of Squares (Lasserre) is [missing?]
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Sum of Squares (Lasserre) is at most quasipolynomial
• Source: SoS → SA → NS_Q → tsQ+ind
• The size complexity of Random CNF in Sum of Squares (Lasserre) is [missing?]
• The size complexity of Clique-Colouring in Sum of Squares (Lasserre) is polynomial
• Source: SoS → sLSn+ → CliqueColouring
• The size complexity of Clique-Colouring encoded as equalities in Sum of Squares (Lasserre) is [missing?]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Sum of Squares (Lasserre) is polynomial
• Source: SoS → sLSn+ → CliqueColouring → CliqueColouring2mm
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Sum of Squares (Lasserre) is polynomial
• Source: SoS → sLSn+ → CliqueColouring → CliqueColouringmlogm
• The size complexity of Clique-Colouring composed with a permutation in Sum of Squares (Lasserre) is [missing?]
• The size complexity of Pebbling \(\circ\) IND in Sum of Squares (Lasserre) is polynomial
• Source: SoS → PC_Q → Res → regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Sum of Squares (Lasserre) is polynomial
• Source: SoS → PC_Q → Res → poolRes → peb+xor+guard
• The size complexity of Stone formula in Sum of Squares (Lasserre) is polynomial
• Source: SoS → PC_Q → Res → poolRes → stone
• The size complexity of String of pearls in Sum of Squares (Lasserre) is polynomial
• Source: SoS → PC_Q → Res → regRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Sum of Squares (Lasserre) is polynomial
• Source: SoS → PC_Q → 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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