Strongest-Stable-Set Operator

• Strongest-stable-set operator is a procedure that tightens or re-equilibrates stable set formulations across graph theory, matching, and choice disciplines.
• It bridges local maximal sets with global stability by leveraging decomposition frameworks and lift-and-project hierarchies like LS+ and Handelman's method.
• The operator facilitates computational insights through rank bounds, edge-induced inequalities, and compact extended formulations for specialized graphs.

Searching arXiv for the cited papers to ground the response in current arXiv metadata. arXiv search query: "(Conforti et al., 2019) Extended Formulations for Stable Set Polytopes of Graphs Without Two Disjoint Odd Cycles" The expression strongest-stable-set operator appears in several non-equivalent formal roles in the arXiv literature. In graph theory and polyhedral combinatorics, it is associated with operators that generate or tighten descriptions of stable sets and stable set polytopes; in matching theory, it refers to Tarski-type re-equilibration maps on quasi-stable matchings; and in general choice theory it is tied to contraction-based constructions of stable-set variants over binary relations (0809.1806, Laurent et al., 2013, Conforti et al., 2019, Bonifacio et al., 2024, Andrikopoulos et al., 13 Aug 2025). A common substrate is the stable set problem itself: for a graph $G=(V,E)$,

$$STAB(G)=\operatorname{conv}\{x\in\{0,1\}^{V}\mid x_v+x_w\le 1\ \forall vw\in E\},$$

and many of the operators discussed below act by approximating, decomposing, or reconstructing this object and its analogues (Conforti et al., 2019).

1. Terminological scope and formal settings

In the cited works, the phrase is attached to several operator families rather than to a single canonical map. The main settings are summarized below.

Setting	Operator/formalism	Role
Graph stable sets	$\Psi(G)$	Family of local maximum stable sets
Stable set polytopes	Handelman, $N_+$, $\mathrm{LS}_+$	Hierarchical tightening of relaxations toward $STAB(G)$
Decomposition frameworks	Master/servant, records, extended formulations	Composition of subproblems and compact polyhedral descriptions
Matching markets	Tarski operators $\mathcal{T}^F,\mathcal{T}^W$	Re-equilibration from quasi-stable to stable matchings
Choice theory	Contraction-based stable-set operators	Construction of stable, extended stable, socially stable, $m$-stable, and $w$-stable sets

This plurality of meanings suggests that the term is best understood operationally: it denotes a procedure that extracts, tightens, or re-equilibrates a stable object under the structural rules of the ambient model. In graph optimization, the relevant operators are usually either combinatorial set-valued maps, such as $\Psi$, or lift-and-project maps that move from a fractional relaxation toward the integer hull (0809.1806, Laurent et al., 2013, Au et al., 2023).

2. The Nemhauser–Trotter operator $$STAB(G)=\operatorname{conv}\{x\in\{0,1\}^{V}\mid x_v+x_w\le 1\ \forall vw\in E\},$$0 and local maximum stable sets

A central graph-theoretic meaning is the local maximum stable set operator. A set $$STAB(G)=\operatorname{conv}\{x\in\{0,1\}^{V}\mid x_v+x_w\le 1\ \forall vw\in E\},$$1 is a local maximum stable set of $$STAB(G)=\operatorname{conv}\{x\in\{0,1\}^{V}\mid x_v+x_w\le 1\ \forall vw\in E\},$$2 if $$STAB(G)=\operatorname{conv}\{x\in\{0,1\}^{V}\mid x_v+x_w\le 1\ \forall vw\in E\},$$3 is a maximum stable set in the subgraph induced by its closed neighborhood: $$STAB(G)=\operatorname{conv}\{x\in\{0,1\}^{V}\mid x_v+x_w\le 1\ \forall vw\in E\},$$4 Here $$STAB(G)=\operatorname{conv}\{x\in\{0,1\}^{V}\mid x_v+x_w\le 1\ \forall vw\in E\},$$5, and $$STAB(G)=\operatorname{conv}\{x\in\{0,1\}^{V}\mid x_v+x_w\le 1\ \forall vw\in E\},$$6 denotes the set of all maximum stable sets of $$STAB(G)=\operatorname{conv}\{x\in\{0,1\}^{V}\mid x_v+x_w\le 1\ \forall vw\in E\},$$7 (0809.1806).

The basic structural fact is the Nemhauser–Trotter theorem: every local maximum stable set of a graph is a subset of some maximum stable set. Thus $$STAB(G)=\operatorname{conv}\{x\in\{0,1\}^{V}\mid x_v+x_w\le 1\ \forall vw\in E\},$$8 captures a locally maximal, yet globally extensible, set system (0809.1806). This is the sense in which $$STAB(G)=\operatorname{conv}\{x\in\{0,1\}^{V}\mid x_v+x_w\le 1\ \forall vw\in E\},$$9 functions as an operator on stable sets: it maps a graph to a family of stable sets with strong extension properties.

The paper "Graph Operations that are Good for Greedoids" (0809.1806) characterizes when $\Psi(G)$0 forms a greedoid under several graph operations. For a disjoint union $\Psi(G)$1, one has $\Psi(G)$2 iff $\Psi(G)$3 for each $\Psi(G)$4, and $\Psi(G)$5 is a greedoid iff each $\Psi(G)$6 is a greedoid. For a Zykov sum $\Psi(G)$7 with $\Psi(G)$8, $\Psi(G)$9 is a greedoid iff all $N_+$0 are greedoids, exactly one $N_+$1 is not complete, and $N_+$2. For a corona $N_+$3, $N_+$4 is a greedoid iff each $N_+$5 is a greedoid (0809.1806).

These results place $N_+$6 at the interface of graph structure and feasible-set geometry. In this usage, the strongest-stable-set operator is not a relaxation hierarchy but a graph-to-set-system map whose output can inherit accessibility and exchange.

3. Hierarchy operators for stable set optimization

A second meaning arises in relaxation hierarchies for the maximum stable set problem. The paper "Handelman's hierarchy for the maximum stable set problem" (Laurent et al., 2013) defines, for order $N_+$7,

$N_+$8

and the corresponding Handelman bound

$N_+$9

For the stable set problem, the hierarchy is monotone, exact at order $\mathrm{LS}_+$0, and its exactness order is the Handelman rank (Laurent et al., 2013).

This hierarchy is explicitly compared with other stable-set operators. The Sherali–Adams bound satisfies

$\mathrm{LS}_+$1

while Lasserre dominates both: $\mathrm{LS}_+$2 The same paper states that the Lovász–Schrijver hierarchy often achieves exactness $\mathrm{LS}_+$3 steps faster than Handelman, and gives the representative relation $\mathrm{LS}_+$4 (Laurent et al., 2013). In this literature, “strongest” refers to comparative tightening power inside a family of lift-and-project or moment-type operators.

Concrete graph classes make the operator viewpoint explicit. The order-$\mathrm{LS}_+$5 Handelman bound equals the fractional stability number, $\mathrm{LS}_+$6. For weighted bipartite graphs, $\mathrm{LS}_+$7; for odd cycles $\mathrm{LS}_+$8, $\mathrm{LS}_+$9; for odd wheels $STAB(G)$0, the rank is at most $STAB(G)$1; and for vertex-transitive perfect graphs, $STAB(G)$2 (Laurent et al., 2013).

A related SDP-based line of work applies the Lovász–Schrijver $STAB(G)$3 operator to compact stable set formulations. The paper "Application of the Lovász-Schrijver Lift-and-Project Operator to Compact Stable Set Integer Programs" (Battista et al., 2024) studies clique-based and nodal formulations and shows that the resulting SDP bounds are at least as strong as $STAB(G)$4. Its computational findings indicate that stronger upper bounds than $STAB(G)$5 can be accessed by a reasonable additional effort using the clique-based formulation on sparse graphs and the nodal-based one on dense graphs (Battista et al., 2024).

4. $STAB(G)$6-rank and the “strongest” classical lift-and-project operator

The most explicit use of “strongest” in the stable-set-operator literature concerns the Lovász–Schrijver SDP operator $STAB(G)$7. For a graph $STAB(G)$8, the fractional stable set polytope is

$STAB(G)$9

and the $\mathcal{T}^F,\mathcal{T}^W$0-rank $\mathcal{T}^F,\mathcal{T}^W$1 is the least $\mathcal{T}^F,\mathcal{T}^W$2 such that

$\mathcal{T}^F,\mathcal{T}^W$3

The operator is defined through a lifted positive-semidefinite matrix system and repeated projection, with $\mathcal{T}^F,\mathcal{T}^W$4 for sets $\mathcal{T}^F,\mathcal{T}^W$5 (Au et al., 2023).

The paper "Stable Set Polytopes with High Lift-and-Project Ranks for the Lovász-Schrijver SDP Operator" (Au et al., 2023) established families of graphs with asymptotically linear $\mathcal{T}^F,\mathcal{T}^W$6-rank. For the graph family $\mathcal{T}^F,\mathcal{T}^W$7 on $\mathcal{T}^F,\mathcal{T}^W$8 vertices, it proves

$\mathcal{T}^F,\mathcal{T}^W$9

improving on the older $m$0 lower bounds from line graphs of odd cliques and matching the known upper bound $m$1 up to constants (Au et al., 2023).

The later paper "Stable Set Polytopes with Rank $m$2 for the Lovász--Schrijver SDP Operator" (Au et al., 13 Jan 2025) sharpens this picture decisively. It proves that for every positive integer $m$3, the smallest possible graph with $m$4-rank $m$5 contains $m$6 vertices, so $m$7, and this bound is sharp. It also states that for every positive integer $m$8 there exists a vertex-transitive graph on $m$9 vertices with $w$0-rank at least $w$1 (Au et al., 13 Jan 2025).

In this line of work, the strongest-stable-set operator is therefore the operator that most aggressively collapses $w$2 to $w$3 among the standard lift-and-project constructions considered. The rank results delimit its power: even the strongest classical operator may require $w$4 iterations on explicit graph families (Au et al., 13 Jan 2025).

5. Decomposition, extended formulations, and polyhedral operator behavior

A different operator perspective comes from polyhedral decomposition and compact extended formulations. The paper "Extended Formulations for Stable Set Polytopes of Graphs Without Two Disjoint Odd Cycles" (Conforti et al., 2019) studies graphs with odd cycle packing number $w$5, equivalently graphs without two disjoint odd cycles. It constructs a size-$w$6 extended formulation for $w$7 and shows that the maximum weight stable set problem can be solved in strongly polynomial time via the Artmann–Weismantel–Zenklusen algorithm for bimodular integer programs (Conforti et al., 2019).

The structural input is that sufficiently connected such graphs either have odd cycle transversal number at most $w$8 or admit an even-face embedding in the projective plane; more generally, they decompose into a projective-planar core and bipartite attachments through a star decomposition. In the even-face embedded case, the construction maps node variables to edge variables and characterizes the stable set polytope by integer circulations in the dual graph with an additional parity constraint; in the general case, the authors glue extended formulations for the core and the bipartite pieces while preserving an overall $w$9 extension complexity (Conforti et al., 2019).

This paper also states an operator-theoretic consequence: facet-defining inequalities of $\Psi$0 for these graphs have a special edge-induced structure, simplifying the behavior of operators like the strongest-stable-set operator. The point is not merely algorithmic solvability. The authors emphasize that strongly polynomial optimization for bimodular programs does not by itself imply a compact extended formulation, whereas the explicit EF supplies a concrete linear description and new geometric information (Conforti et al., 2019).

Related decomposition ideas appear in "Stable Sets and Graphs with no Even Holes" (Conforti et al., 2013). There the master/servant decomposition, rooted graphs, templates, and linearization by records provide a recursive framework for solving maximum weight stable set and for constructing compact systems. For cap-free graphs with no even holes, the paper proves polynomial-time solvability and compact extended formulations for the stable set polytope, and presents the strongest stable set operator as a recursive composition rule realized through clique cutsets, amalgams, and record variables (Conforti et al., 2013).

6. Extensions beyond graph independence

Outside graph independence, the operator language reappears in matching and in general solution theory for binary relations. In "Lattice operations for the stable set in substitutable matching markets via re-equilibration dynamics" (Bonifacio et al., 2024), the stable object is the lattice of stable matchings. The paper defines worker-quasi-stable and firm-quasi-stable matchings and introduces Tarski operators $\Psi$1 and $\Psi$2, interpreted as lay-off and vacancy chain dynamics. Fixed points of these operators are exactly the stable matchings, and joins or meets are computed by constructing a quasi-stable candidate and iterating the relevant Tarski operator until a fixed point is reached (Bonifacio et al., 2024).

A complementary fractional viewpoint appears in "On the many-to-one strongly stable fractional matching set" (Neme et al., 2019). For many-to-one markets with strict and $\Psi$3-responsive preferences, the paper characterizes the set of strongly stable fractional matchings as

$\Psi$4

the union of the convex hulls of all connected sets of stable matchings. It also proves that every strongly stable fractional matching is a convex combination of stable matchings ordered in the common preferences of all firms (Neme et al., 2019). In this setting, the operator is a convex-hull construction over connected components of the stable-matching space.

The most abstract extension is "A Characterization Framework for Stable Sets and Their Variants" (Andrikopoulos et al., 13 Aug 2025), which studies stable sets, extended stable sets, socially stable sets, and $\Psi$5- and $\Psi$6-stable sets for irreflexive binary relations over infinite sets of alternatives. Its framework uses contraction onto strong components and topological conditions such as compactness, $\Psi$7-order separation, and Nachbin closedness. For the extended stable set, the paper states that a set $\Psi$8 is extended stable iff it selects exactly one element from each strong component of the contraction, thereby turning the strongest-stable-set operator into a component-selection rule under a topological existence theory (Andrikopoulos et al., 13 Aug 2025).

Taken together, these usages show that the strongest-stable-set operator is not a single universally accepted formal object. In graph optimization it usually denotes a tightening or decomposition mechanism aimed at $\Psi$9; in matching it is a re-equilibration operator on quasi-stable states; and in abstract choice theory it is a contraction-based selector for stable-set variants. The common invariant is that the operator organizes stability by converting local admissibility, fractional feasibility, or quasi-stability into a structurally maximal stable object.