Sherali–Adams Hierarchy: LP Relaxations

• Sherali–Adams Hierarchy is a lift-and-project method that tightens 0/1 optimization formulations by introducing local distribution constraints for monomials.
• It offers equivalent formulations using local marginals, tensorisation, and minion frameworks, effectively bridging CSPs, graph isomorphism, and polynomial optimization.
• Its applications span approximation algorithms and integrality gap analyses, yielding practical insights into efficiency and solution accuracy in LP-based methods.

The Sherali–Adams hierarchy is a lift-and-project hierarchy of linear programming relaxations for $0/1$ optimization, graph and relational homomorphism, and constraint satisfaction formulations. At level $t$, it augments a base LP by introducing variables for monomials or, equivalently, by specifying consistent local distributions on subsets of variables of size at most $t$; the resulting relaxations form a nested sequence that reaches the convex hull of integral solutions at full level (Cheriyan et al., 2014, Kolliopoulos et al., 2013). Across recent work, the hierarchy has been characterized through local consistency, fractional isomorphism, higher-dimensional Weisfeiler–Leman equivalence, and tensorisation (Butti et al., 2021, Ciardo et al., 2022).

1. Formal construction and equivalent viewpoints

For a $0/1$-polytope $P\subseteq[0,1]^d$ given by linear constraints $\pi(x)\equiv a^T x-b\le 0$, the level-$t$ Sherali–Adams tightening $SA^t(P)$ is obtained by multiplying each original constraint by all monomials

$\prod_{i\in U\setminus W}x_i\prod_{i\in W}(1-x_i)$

with $|U|\le t$, then linearizing every monomial $t$0 to a new variable $t$1 and replacing $t$2 by $t$3 (Kolliopoulos et al., 2013). In the standard lift-and-project notation, level $t$4 introduces variables $t$5 for all subsets $t$6 with $t$7, with $t$8, and one obtains a descending chain

$t$9

for a $t$0 relaxation $t$1 (Cheriyan et al., 2014).

A second, equivalent presentation uses local marginals. For MAX $t$2-CSP and related formulations, level $t$3 introduces variables $t$4, interpreted as $t$5, for all $t$6 of size at most $t$7, with normalization, nonnegativity, and marginal-consistency constraints (Ghosh et al., 2016). For Max-Cut and general $t$8-CSPs, this takes the form of local marginals $t$9 on assignments $0/1$0, subject only to normalization, nonnegativity, and consistency; no semidefiniteness is imposed (Hopkins et al., 2019). In CSP language, the same object is a consistent collection of local distributions over all subsets of size at most $0/1$1 (Ciardo et al., 2022).

For relational templates, the level-$0/1$2 relaxation can be written explicitly. Let $0/1$3 be an instance structure and $0/1$4 a template with the same signature. Then level $0/1$5 introduces a variable $0/1$6 for every subset $0/1$7 with $0/1$8 and every partial assignment $0/1$9, with constraints

$P\subseteq[0,1]^d$0

together with relation-consistency and feasibility conditions that force zero mass on locally forbidden tuples (Ciardo et al., 2022). This is the standard “local distributions” view of Sherali–Adams on CSPs and homomorphism problems.

2. Constraint satisfaction, valued CSPs, and exactness criteria

For CSP homomorphism, the integer program uses variables $P\subseteq[0,1]^d$1 encoding whether $P\subseteq[0,1]^d$2 is mapped to $P\subseteq[0,1]^d$3, with row-sum constraints and local forbidden-assignment inequalities; the Sherali–Adams level-$P\subseteq[0,1]^d$4 relaxation introduces variables $P\subseteq[0,1]^d$5 for $P\subseteq[0,1]^d$6, $P\subseteq[0,1]^d$7, and assignments $P\subseteq[0,1]^d$8, together with normalization, marginalization, relation-consistency, and forbidden-tuple constraints (Butti et al., 2021). A feasible solution is equivalently a “fractional $P\subseteq[0,1]^d$9-consistency” witness: a global distribution over partial homomorphisms of domain size at most $\pi(x)\equiv a^T x-b\le 0$0 whose marginals agree on overlaps (Butti et al., 2021).

At level $\pi(x)\equiv a^T x-b\le 0$1, Sherali–Adams coincides with the basic LP for CSP. One exactness theorem states that $\pi(x)\equiv a^T x-b\le 0$2 is the basic LP and that $\pi(x)\equiv a^T x-b\le 0$3 decides $\pi(x)\equiv a^T x-b\le 0$4 exactly if and only if $\pi(x)\equiv a^T x-b\le 0$5 admits symmetric polymorphisms of all arities (Butti et al., 2021). In the reformulation through Weisfeiler–Leman invariance, the families of CSPs solvable by this linear program are precisely those closed under the corresponding equivalence relation (Barto et al., 2024).

For valued CSPs, the hierarchy is commonly indexed as $\pi(x)\equiv a^T x-b\le 0$6, with local distributions $\pi(x)\equiv a^T x-b\le 0$7 on scopes $\pi(x)\equiv a^T x-b\le 0$8, consistency on all projections of size at most $\pi(x)\equiv a^T x-b\le 0$9, and an objective minimizing the expected weighted relation cost (Thapper et al., 2015). The principal exactness result is that if $t$0 satisfies the bounded width condition, then $t$1 has valued relational width $t$2; equivalently, the level-$t$3 Sherali–Adams relaxation is exact on $t$4 (Thapper et al., 2015). The proof strategy passes through support closure under fractional polymorphisms, a derived ordinary CSP on support relations, and the $t$5-minimality characterization of bounded relational width (Thapper et al., 2015).

The 2024 extension to Promise Valued CSPs preserves the same pattern. In that setting, $t$6 exactly captures the class of Weisfeiler–Leman invariant templates, and the decomposition $t$7 is used to describe feasibility and objective preservation (Barto et al., 2024).

3. Graph isomorphism, Weisfeiler–Leman, and pebble-game characterizations

Applied to graph isomorphism, Sherali–Adams acts on the graph-isomorphism polytope

$t$8

or on related semi-algebraic formulations whose integer points encode isomorphisms (Malkin, 2011). The lifted variables correspond to partial matchings $t$9, and the resulting constraints generalize the Birkhoff equations and the adjacency-compatibility equations (Malkin, 2011).

A central conclusion is that Sherali–Adams characterizes higher-dimensional color-refinement procedures. On graph isomorphism polytopes, the hierarchy characterizes a generalized vertex classification algorithm, and $SA^t(P)$0 exactly captures the $SA^t(P)$1-dimensional Weisfeiler–Lehman partition of $SA^t(P)$2 (Malkin, 2011). The same work proves an $SA^t(P)$3 lower bound on the number of rounds needed in the worst case, via Cai–Fürer–Immerman constructions (Malkin, 2011).

The correspondence sharpens further in the pebble-game formulation. For graphs $SA^t(P)$4, the intermediate systems $SA^t(P)$5 and $SA^t(P)$6 match, respectively, bijective $SA^t(P)$7-pebble counting equivalence and the weak $SA^t(P)$8-pebble game; the two hierarchies interleave strictly, and Boolean-semiring analogues recover the non-counting $SA^t(P)$9 pebble games (Grohe et al., 2012). This identifies Sherali–Adams not merely as a hierarchy of LPs, but as a hierarchy of logical indistinguishability tests.

For arbitrary relational structures, the same picture persists. One theorem states that

$\prod_{i\in U\setminus W}x_i\prod_{i\in W}(1-x_i)$0

and equivalent characterizations are given in terms of counting homomorphisms from relational trees of treewidth $\prod_{i\in U\setminus W}x_i\prod_{i\in W}(1-x_i)$1, $\prod_{i\in U\setminus W}x_i\prod_{i\in W}(1-x_i)$2 counting logic, and doubly-stochastic witnesses of fractional isomorphism (Butti et al., 2021). A later refinement lifts the level-$\prod_{i\in U\setminus W}x_i\prod_{i\in W}(1-x_i)$3 decomposition to all $\prod_{i\in U\setminus W}x_i\prod_{i\in W}(1-x_i)$4: $\prod_{i\in U\setminus W}x_i\prod_{i\in W}(1-x_i)$5 is feasible if and only if there exists a chain $\prod_{i\in U\setminus W}x_i\prod_{i\in W}(1-x_i)$6 (Barto et al., 2024).

4. Tensorisation, minions, and promise CSPs

A recent multilinear characterization recasts Sherali–Adams for Promise CSPs through tensor powers. For $\prod_{i\in U\setminus W}x_i\prod_{i\in W}(1-x_i)$7 and $\prod_{i\in U\setminus W}x_i\prod_{i\in W}(1-x_i)$8-enhanced structures $\prod_{i\in U\setminus W}x_i\prod_{i\in W}(1-x_i)$9, the main theorem states

$|U|\le t$0

where $|U|\le t$1 and $|U|\le t$2 are tensor powers of the original structures and $|U|\le t$3 is the free structure of stochastic tensors (Ciardo et al., 2022). The universe of $|U|\le t$4 consists of nonnegative real tensors $|U|\le t$5 satisfying $|U|\le t$6, and the theorem is equivalently stated as: the $|U|\le t$7-th Sherali–Adams relaxation on $|U|\le t$8 is exactly the basic LP applied to the tensorised pair $|U|\le t$9 (Ciardo et al., 2022).

This tensor viewpoint yields concrete lower bounds for promise coloring. For any loopless digraph $t$00 and any $t$01, $t$02 accepts, and by combining this with the line-digraph construction one obtains that no constant level of Sherali–Adams solves $t$03 for any $t$04 (Ciardo et al., 2022). The construction is driven by the geometry of $t$05, whose tensors are highly symmetric under coordinate permutations but still admit homomorphisms from arbitrary $t$06 (Ciardo et al., 2022).

The tensor framework was then generalized into a minion-and-tensorisation program. In that formulation, the base minion for Sherali–Adams is the minion of all finite-ary stochastic vectors over $t$07, minor maps are row-summing maps, and the hierarchy arises by repeated tensor-power construction (Ciardo et al., 2022). The same framework captures bounded width, Sherali–Adams LP, affine IP, Sum-of-Squares SDP, and combined “LP + affine IP” hierarchies (Ciardo et al., 2022). In that sense, Sherali–Adams becomes one specific linear minion test inside a broader algebraic taxonomy.

5. Relations to other hierarchies in polynomial optimization

On the Boolean hypercube, Sherali–Adams admits a certificate interpretation through nonnegative $t$08-juntas. A degree-$t$09 SA certificate writes $t$10 as a sum $t$11, where each multiplier $t$12 is a nonnegative $t$13-junta and $t$14 (Kurpisz et al., 2019). The Schmüdgen-type variant $t$15 allows arbitrary products of Boolean constraints, and on $t$16 it has a Positivstellensatz degree bound $t$17 (Kurpisz et al., 2019).

The comparative results are unusually sharp. Over the Boolean hypercube, the hierarchies $t$18, $t$19, and $t$20 are polynomially equivalent, and any Schmüdgen-type hierarchy with an $t$21 degree bound contains $t$22 up to constant-factor degree blow-up (Kurpisz et al., 2019). The same source states that $t$23, $t$24, and $t$25 each admit an $t$26-degree Positivstellensatz on $t$27, which implies containment of $t$28 and, together with the equivalence theorem, gives full polynomial equivalence among $t$29, $t$30, and $t$31 (Kurpisz et al., 2019).

This Boolean-hypercube result is distinct from the tensor-minion comparison. The former is a certificate-complexity equivalence among LP, SOCP, and relative-entropy based systems on $t$32; the latter is a structural framework in which Sherali–Adams, affine IP, and SDP hierarchies appear as different choices of base minion and tensorisation scheme (Ciardo et al., 2022).

6. Algorithmic consequences and positive results

Low and medium levels of Sherali–Adams yield exact algorithms and approximation schemes on several structured problem classes. For Sparsest Cut with general demands in graphs of treewidth $t$33, the level-$t$34 relaxation provides local distributions on all bags needed by the rounding, leading to the first constant-factor approximation on bounded-treewidth graphs (Chlamtac et al., 2010). For Graph Vertex Pricing, if $t$35 has treewidth $t$36, then solving $t$37 exactly gives integrality gap $t$38, while for genus-$t$39 graphs, $t$40 rounds yield a $t$41-approximation (Chalermsook et al., 2012).

The hierarchy also supports quasi-polynomial approximation in scheduling. For unit-size jobs with precedence constraints on $t$42 identical machines, $t$43 rounds of Sherali–Adams applied to the natural time-indexed LP provide a $t$44-approximation algorithm running in quasi-polynomial time for fixed $t$45 and $t$46 (Garg, 2017). The proof uses repeated conditioning in the lifted space and a laminar decomposition of the time horizon (Garg, 2017).

In clustering, Sherali–Adams improves on the standard triangle LP. For Correlation Clustering on complete graphs, $t$47 rounds, combined with correlated rounding and a global charging scheme, yield a $t$48-approximation algorithm; equivalently, the $t$49-round relaxation has integrality gap at most $t$50 (Cohen-Addad et al., 2022).

For Max-Cut, the hierarchy exhibits both certification and approximation phenomena. If an $t$51-vertex graph has random-walk matrix with all nontrivial eigenvalues bounded in magnitude by $t$52, then $t$53 rounds of Sherali–Adams certify that the maximum cut is at most $t$54 (O'Donnell et al., 2018). More generally, for every $t$55 there is $t$56 such that the degree-$t$57 Sherali–Adams LP approximates Max-Cut within a factor $t$58 on every $t$59-vertex graph, and the same paper extends this subexponential-time LP approach to Unique Games (Hopkins et al., 2019).

7. Lower bounds, integrality gaps, and limitations

A broad meta-result for CSP approximation states that any integrality-gap instance for the basic LP lifts to an integrality-gap instance for

$t$60

levels of Sherali–Adams on instances of size $t$61 (Ghosh et al., 2016). This shows that, for every predicate, $t$62 levels may still be no stronger than the basic LP, and it implies that polynomial-size LP extended formulations cannot beat the basic LP for CSPs (Ghosh et al., 2016).

Problem-specific lower bounds are equally strong. For decoding random $t$63-regular LDPC codes, even $t$64 rounds of Sherali–Adams cannot correct more than an $t$65 fraction of errors (Ghazi et al., 2014). For uniform capacitated facility location, there exist instances for which the level-$t$66 relaxation of the natural LP has integrality gap $t$67 for all $t$68, and the same family remains hard even after adding generalized flow-cover inequalities (Kolliopoulos et al., 2013).

For ATSP, level-$t$69 Sherali–Adams applied to the standard DFJ LP has integrality ratio at least

$t$70

while for the balanced LP the ratio is at least

$t$71

on the Charikar–Goemans–Karloff digraph family (Cheriyan et al., 2014). These constructions identify structural properties of digraphs that preserve fractional feasibility through many lift-and-project rounds (Cheriyan et al., 2014).

The hierarchy can also fail sharply at very low levels. In the MaxMinDegree Arborescence problem, a depth-$t$72 instance has an $t$73 integrality gap that survives one round of Sherali–Adams, and the paper lifts this construction to all depths $t$74 (Bamas, 2024). For Bounded Color Matching, the natural LP can retain large gaps after multiple rounds on several instance families, although the second round already implies all bi-chromatic-4-cycle inequalities and yields upper bounds $t$75 in bipartite graphs and $t$76 in general graphs (Kelk et al., 2018).

These results delimit the hierarchy’s power rather than negating it. They show that Sherali–Adams can be exact on bounded-width and bounded-treewidth regimes, can produce nontrivial subexponential-time approximations and certificates, and can nevertheless remain far from the integer hull on canonical hard instances, even after many rounds (Thapper et al., 2015, Hopkins et al., 2019).