All about Sherali--Adams

Notes: With twin variables \(\overline{x}=1-x\)

Proof Systems

• Sherali--Adams stronger than Resolution
• Source: SA → circRes → Res
• Source: SA → NS_Q → ofPHP → Res
• Sherali--Adams stronger than Truth table
• Source: SA → NS_Q → tlRes → ttp
• Source: SA → PHP → tlResLin_F2 → tlRes → ttp
• Sherali--Adams stronger than Tree-like resolution
• Source: SA → NS_Q → tlRes
• Source: SA → PHP → tlResLin_F2 → tlRes
• Sherali--Adams stronger than Regular resolution
• Source: SA → circRes → Res → regRes
• Source: SA → NS_Q → ofPHP → Res → regRes
• Sherali--Adams stronger than Ordered resolution
• Source: SA → circRes → Res → regRes → ordRes
• Source: SA → NS_Q → ofPHP → Res → regRes → ordRes
• Sherali--Adams stronger than Pool resolution
• Source: SA → circRes → Res → poolRes
• Source: SA → NS_Q → ofPHP → Res → poolRes
• Sherali--Adams stronger than Linear resolution
• Source: SA → circRes → Res → linRes
• Source: SA → NS_Q → ofPHP → Res → linRes
• Sherali--Adams stronger than Reversible resolution
• Source: SA → uSA → revRes
• Source: SA → NS_Q → ofPHP → Res → revRes
• Sherali--Adams incomparable wrt Cutting Planes
• Source: SA → NS_Q → tsQ+ind → CP
• Source: CP → tseitin → SoS → SA
• Sherali--Adams not simulated by Tree-like Cutting Planes
• Source: SA → circRes → Res → regRes → ordRes → peb+ind → tlCP
• Sherali--Adams not simulated by Cutting Planes with Unary Coefficients
• Source: SA → NS_Q → tsQ+ind → CP → uCP
• Sherali--Adams does not simulate Semantic Cutting Planes
• Source: semanticCP → CP → tseitin → SoS → SA
• Sherali--Adams stronger than Cutting Planes with Saturation
• Source: SA → circRes → Res → saturationCP
• Source: SA → NS_Q → ofPHP → Res → saturationCP
• Sherali--Adams does not simulate Stabbing Planes
• Source: SP → tseitin → SoS → SA
• Sherali--Adams not simulated by Stabbing Planes with Unary Coefficients
• Source: SA → NS_Q → tsQ+ind → CP → uSP
• Sherali--Adams does not simulate Res(CP)
• Source: Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS → SA
• Sherali--Adams does not simulate Res(LP)
• Source: Res(LP) → uRes(LP) → tseitin → SoS → SA
• Sherali--Adams does not simulate Res(CP) with unary coefficients
• Source: uRes(CP) → uRes(LP) → tseitin → SoS → SA
• Sherali--Adams does not simulate Res(LP) with unary coefficients
• Source: uRes(LP) → tseitin → SoS → SA
• Sherali--Adams does not simulate Res(L\(\&\)P)
• Source: Res(L&P) → Res(LP) → uRes(LP) → tseitin → SoS → SA
• Sherali--Adams does not simulate Res(L\(\&\)P) with unary coefficients
• Source: uRes(L&P) → uRes(LP) → tseitin → SoS → SA
• Sherali--Adams [missing?] Semantic degree-k threshold system, treelike version
• Sherali--Adams incomparable wrt Polynomial Calculus over \(\mathbb{F}_2\)
• Source: SA → PHP → PC_F2
• Source: PC_F2 → NS_F2 → tseitin → SoS → SA
• Sherali--Adams incomparable wrt Nullstellensatz over \(\mathbb{F}_2\)
• Source: SA → PHP → PC_F2 → NS_F2
• Source: NS_F2 → tseitin → SoS → SA
• Sherali--Adams does not simulate ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: ResLin_Z → uResLin_Z → ResLin_F2 → tseitin → SoS → SA
• Sherali--Adams does not simulate unary ResLin over \(\mathbb{Q}\), ResLin, Resolution over linear equations over rationals
• Source: uResLin_Z → ResLin_F2 → tseitin → SoS → SA
• Sherali--Adams does not simulate ResLin over \(\mathbb{F}_2\), Res(\(\oplus\))
• Source: ResLin_F2 → tseitin → SoS → SA
• Sherali--Adams not simulated by Tree-like ResLin over \(\mathbb{F}_2\), treelike Res(\(\oplus\))
• Source: SA → PHP → tlResLin_F2
• Sherali--Adams not simulated by Polynomial Calculus over \(\mathbb{Q}\)
• Source: SA → PHP → PC_Q
• Sherali--Adams stronger than Nullstellensatz over \(\mathbb{Q}\)
• Source: [subsystem]
• Source: SA → PHP → PC_Q → NS_Q
• Sherali--Adams stronger than Hitting
• Source: SA → NS_Q → tlRes → hit
• Source: SA → PHP → tlResLin_F2 → tlRes → hit
• Sherali--Adams [missing?] Lift and Project
• Sherali--Adams not simulated by Lift and Project with unary coefficients
• Source: SA → NS_Q → tsQ+ind → CP → uCP → uL&P
• Sherali--Adams simulated by Positivstellensatz Calculus
• Source: PSC → PS → SoS → SA
• Sherali--Adams simulated by Positivstellensatz
• Source: PS → SoS → SA
• Sherali--Adams [missing?] Lovász--Schrijver (LS)
• Sherali--Adams [missing?] Lovász--Schrijver with squares (LS\(_+\))
• Sherali--Adams does not simulate Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree
• Source: LSd+ → tseitin → SoS → SA
• Sherali--Adams weaker than Lovász--Schrijver with squares (LS\(_+^\infty\)), unbounded degree
• Source: LSn+ → PSC → PS → SoS → SA
• Source: LSn+ → LSd+ → tseitin → SoS → SA
• Sherali--Adams weaker than Cone Proof System
• Source: CPS → LSn+ → PSC → PS → SoS → SA
• Source: CPS → LSn+ → LSd+ → tseitin → SoS → SA
• Sherali--Adams [missing?] tl Lovász--Schrijver (LS)
• Sherali--Adams [missing?] Lovász--Schrijver with squares (LS\(_+\))
• Sherali--Adams [missing?] Lovász--Schrijver with squares (LS\(_+^d\)), bounded degree, treelike
• Sherali--Adams [missing?] static Lovász--Schrijver (static LS)
• Sherali--Adams [missing?] static Lovász--Schrijver, with squares of linear functions (static LS\(_+\))
• Sherali--Adams [missing?] static Lovász--Schrijver, with squares of linear functions (static LS\(_+^n\))
• Sherali--Adams simulated by Sum of Squares (Lasserre)
• Source: [citation needed]
• Sherali--Adams equivalent Circular resolution
• Source: AL19 Circular (Yet Sound) Proofs.
• Sherali--Adams stronger than Unary Sherali--Adams
• Source: [subsystem]
• Source: SA → circRes → Res → sod+xor → uSA
• Sherali--Adams does not simulate Ideal Proof System
• Source: IPS → extFrege → Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS → SA
• Sherali--Adams does not simulate Extended Frege
• Source: extFrege → Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS → SA
• Sherali--Adams does not simulate Extended resolution
• Source: extRes → extFrege → Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS → SA
• Sherali--Adams does not simulate Frege
• Source: Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS → SA
• Sherali--Adams not simulated by \(\mathrm{AC}^0\)-Frege
• Source: SA → NS_Q → ofPHP → AC0Frege
• Sherali--Adams not simulated by k-DNF Resolution
• Source: SA → NS_Q → ofPHP → AC0Frege → Res-k
• Sherali--Adams does not simulate \(\mathrm{TC}^0\)-Frege
• Source: TC0Frege → Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS → SA
• Sherali--Adams does not simulate \(\mathrm{AC}^0\)-Frege with mod 2 axioms
• Source: AC0Frege+Mod2axioms → NS_F2 → tseitin → SoS → SA
• Sherali--Adams does not simulate \(\mathrm{AC}^0\)-Frege with mod 2 gates
• Source: AC0(+)Frege → ResLin_F2 → tseitin → SoS → SA
• Sherali--Adams does not simulate OBDD(join,weakening)
• Source: OBDDjoinweak → tseitin → SoS → SA
• Sherali--Adams does not simulate LK
• Source: LK → Frege → AC0(+)Frege → ResLin_F2 → tseitin → SoS → SA
• Sherali--Adams does not simulate Zermelo-Fraenkl Set Theory with the Axiom of Choice
• Source: ZFC → Res(CP) → Res(LP) → uRes(LP) → tseitin → SoS → SA

Formulas

• The size complexity of Pigeonhole principle in Sherali--Adams is polynomial
• Source: Rod07 Rank Lower Bounds for the Sherali-Adams Operator.
• The size complexity of Pigeonhole principle with functionality axioms in Sherali--Adams is polynomial
• Source: SA → PHP → fPHP
• The size complexity of Pigeonhole principle with functionality and onto axioms in Sherali--Adams is polynomial
• Source: SA → NS_Q → ofPHP
• The size complexity of Weak pigeonhole principle (2n pigeons, n holes) in Sherali--Adams is polynomial
• Source: SA → PHP → PHP2nn
• The size complexity of Weak pigeonhole principle (\(n^2\) pigeons, n holes) in Sherali--Adams is polynomial
• Source: SA → PHP → PHP2nn → PHPn2n
• The size complexity of Bitwise pigeonhole principle in Sherali--Adams is [missing?]
• The size complexity of Relativized pigeonhole principle (n,\(n^2\),n-1) in Sherali--Adams is exponential
• Source: ALN16 Narrow Proofs May Be Maximally Long.
• The size complexity of Ordering principle in Sherali--Adams is polynomial
• Source: SA → circRes → Res → regRes → ordering
• The size complexity of Guarded ordering principle in Sherali--Adams is polynomial
• Source: SA → circRes → Res → poolRes → ordering+guard
• The size complexity of Tseitin over \(\mathbb{F}_2\) in Sherali--Adams is exponential
• Source: tseitin → SoS → SA
• The size complexity of Tseitin over \(\mathbb{F}_2\) with k-ary AND gadget in Sherali--Adams is [missing?]
• The size complexity of Tseitin over \(\mathbb{Q}\) in Sherali--Adams is polynomial
• Source: SA → NS_Q → tseitinQ
• The size complexity of Flow in Sherali--Adams is [missing?]
• The size complexity of Tseitin \(\mathbb{F}_2\) \(\circ\) IND in Sherali--Adams is [missing?]
• The size complexity of Tseitin \(\mathbb{Q}\) \(\circ\) IND in Sherali--Adams is at most quasipolynomial
• Source: SA → NS_Q → tsQ+ind
• The size complexity of Random CNF in Sherali--Adams is [missing?]
• The size complexity of Clique-Colouring in Sherali--Adams is [missing?]
• The size complexity of Clique-Colouring encoded as equalities in Sherali--Adams is [missing?]
• The size complexity of Weak Clique-Colouring (2m clique, m colours) in Sherali--Adams is [missing?]
• The size complexity of Weak Clique-Colouring (m^1/2 clique, log^2 m colours) in Sherali--Adams is [missing?]
• The size complexity of Clique-Colouring composed with a permutation in Sherali--Adams is [missing?]
• The size complexity of Pebbling \(\circ\) IND in Sherali--Adams is polynomial
• Source: SA → circRes → Res → regRes → ordRes → peb+ind
• The size complexity of Pebbling \(\circ\) XOR, then guarded in Sherali--Adams is polynomial
• Source: SA → circRes → Res → poolRes → peb+xor+guard
• The size complexity of Stone formula in Sherali--Adams is polynomial
• Source: SA → circRes → Res → poolRes → stone
• The size complexity of String of pearls in Sherali--Adams is polynomial
• Source: SA → circRes → Res → regRes → pearl
• The size complexity of Sink of DAG \(\circ\) XOR in Sherali--Adams is polynomial
• Source: SA → circRes → 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.

Licensed under CC BY 4.0