De Klerk–Pasechnik Conjecture Overview

• The De Klerk–Pasechnik Conjecture is a principle stating that sum-of-squares based semidefinite programming hierarchies finitely detect exact combinatorial optima in NP-hard optimization problems.
• Copositive formulations and SOS certificates are employed to derive progressively tighter bounds for the stability number of graphs, with convergence guaranteed for acritical cases.
• Extensions to symmetric cones and applications in TSP and crossing numbers demonstrate the conjecture’s potential to improve algorithmic frameworks and establish strong lower bounds in graph theory.

The De Klerk–Pasechnik Conjecture encompasses a family of deep problems in the theory and hierarchy of semidefinite and copositive programming relaxations for NP-hard combinatorial optimization problems, chiefly the stability number of a graph and related structures. Originating from the copositive formulation for $\alpha(G)$, the conjecture postulates that certain sum-of-squares–based semidefinite hierarchies “detect” exact combinatorial optima after a bounded (and often tight) number of steps. This principle connects the geometry of nonconvex cones, algebraic certificates, and classical graph invariants, and extends to approximation hierarchies over symmetric cones, relaxations for the TSP, and lower bounds on crossing numbers. The conjecture serves as a benchmark for hierarchy tightness, convergence, and computational practicality.

1. Copositive Programming Formulation and Hierarchies

The copositive formulation expresses the stability number $\alpha(G)$ as

$$\alpha(G) = \min\{\, t \mid t(I + A_G) - J \text{ is copositive} \,\}$$

where $A_G$ is the adjacency matrix of $G$ and $J$ is the all-ones matrix. Since membership in the copositive cone is NP-hard, inner approximation schemes are employed. De Klerk and Pasechnik (2002) introduced bounds $\vartheta^{(r)}(G)$ based on conic approximations $\mathcal{K}_n^{(r)}$ derived from structured sum-of-squares representations. These hierarchies embed the combinatorial structure into the semidefinite programming (SDP) framework, permitting progressively tighter bounds via increasing the parameter $r$.

Typical relaxations take the form:

• Linearized certificate: $(\sum x_i)^r\, p_M(x) \in \mathbb{R}_+[x]$
• SOS certificate: $(\sum x_i^2)^r\, P_M(x) \in \Sigma$

These preserve combinatorial invariants, and, under the conjecture, converge finitely: $\vartheta^{(\alpha(G)-1)}(G) = \alpha(G)$ (Laurent et al., 2021). Explicit connection with the Motzkin–Straus formulation and SOS/Lasserre hierarchies reveals that the combinatorial optimality can be “witnessed” by finitely many minimizers, especially for acritical graphs (no edge deletion increases $\alpha(G)$).

2. Finite Convergence, Criticality, and Tightness

Finite convergence of the SOS/certificates hierarchy occurs for acritical graphs, as shown via analysis of the Motzkin–Straus minimizers and Nie’s theorem using the Boundary Hessian Condition (Laurent et al., 2021). If $G$ lacks critical edges, the SOS hierarchy exactly recovers $1/\alpha(G)$ at some finite level. For graphs with critical edges, infinite minimizer sets impede direct application, and the conjecture remains open in the general case.

Recent constructions demonstrate tightness: there exist families of graphs for which $r = \alpha(G) - 1$ is necessary for convergence (Vargas et al., 5 Sep 2025). These show that, although lower bounds can sometimes be attained sooner, the upper bound in the conjecture is sharp for specific graph classes.

Related work characterizes the behavior of approximation cones under graph operations, e.g., adding isolated nodes can break exactness ($\vartheta^{(1)}$) (Laurent et al., 2021), indicating strict inclusion $$\bigcup_{r \geq 0} \mathcal{K}_n^{(r)} \subset \text{COP}_n$$ for $n \geq 6$. This disproves earlier conjectures linking cone structure under extension (Laurent et al., 2021).

3. Certificate Translation: Lovász Theta and Algorithmic Reductions

The hierarchy’s certificate language seamlessly translates to well-known SDP graph invariants. For instance, exactness at level zero ($\vartheta^{(0)}(G) = \alpha(G)$) is equivalent to the existence of a Lovász-exactness certificate—a PSD matrix $P$ with precise row and off-diagonal constraints:

• $P \leq M_G = \alpha(G)(A+I)-J$
• $P_{ii} = \alpha(G) - 1$
• $P_{ij} = -1$ for nonedges

Thus, deciding if $\vartheta^{(0)}(G) = \alpha(G)$ can be algorithmically reduced (in polynomial time for fixed $\alpha$) to acritical graphs, with identical reductions and structural findings for the Lovász theta number [c and Laurent (2007), (Laurent et al., 2021)].

4. Extensions to Symmetric Cones and Computational Hierarchies

Recent advances “lift” the copositive cone and its hierarchies to symmetric cone settings, e.g., cones formed by direct products of nonnegative orthants and second-order cones. The de Klerk–Pasechnik-type inner approximation hierarchies

$$I_{dP,r}(E_+) := \{\, A \mid A(c_1,\ldots,c_k) \in I_{dP,r} \,\}$$

converge as $r\uparrow\infty$ to the full copositive cone $\text{COP}(E_+)$ (Nishijima et al., 2022). Yıldırım-type outer approximations similarly converge from above. Computational efficiency is achieved by formulating moderate-sized block-diagonal SDP constraints, permitting substantial numerical progress—particularly when the nonnegative orthant is small. Empirical evidence supports practical exactness for moderate $r$ (Nishijima et al., 2022).

5. SDP Relaxations for the TSP and Cycle Cover Problems

The De Klerk–Pasechnik–Sotirov SDP relaxation for the TSP employs matrix variables and cosine-weighted LMIs reflecting the structure of distance matrices for Hamiltonian cycles:

$$\text{minimize}\quad \tfrac{1}{2}\,\text{trace}(C X^{(1)})$$

subject to:

• $X^{(k)} \succeq 0$
• $\sum_{j=1}^d X^{(j)} = J - I$
• $I + \sum_{j=1}^d \cos(2\pi jk/n) X^{(j)} \succeq 0$

Explicitly constructed “cut semimetric” instances show that for certain cost matrices, the integrality gap grows linearly with $n$—the SDP relaxation can be made arbitrarily weaker than the integer optimum (Gutekunst et al., 2017). This stark failure contrasts with the subtour LP, which maintains a gap at most $1.5$. Extension to $k$-cycle cover problems confirms the unbounded gap for all fixed $k$.

A further SDP constraint developed from the matrix-tree theorem, enforcing that the sum of spanning tree weights is at least $n$, is shown to be implied by standard subtour elimination LP constraints (Gutekunst et al., 2019). This implies that such SDP constraints do not strengthen TSP relaxations beyond the well-known linear programming bounds.

6. Sum-of-Squares Bounds and Turán-type Lemmas

SOS hierarchies derived from the copositive formulation not only help determine stability numbers but, in low degrees, recapitulate classical extremal bounds. For example, the greedy Turán-type lemma is captured by

$\alpha(G) \geq |G|/(d+1)$

with generalizations to Caro–Wei bounds via hierarchical sum-of-squares. Recent work shows that structured SOS relaxations yield tight bounds in classes of graphs where generic hierarchies may not (Vargas et al., 5 Sep 2025). Key novelty includes an SDP feasibility formulation at each hierarchy level and preservation of essential combinatorial structure.

7. Applications to Crossing Numbers and Topological Graph Theory

Semidefinite relaxations leveraging representation theory have advanced lower bounds for the crossing numbers of complete bipartite graphs $K_{m,n}$ (Brosch et al., 2022). By full block-diagonalization of the invariant matrix algebra and relaxing only certain key blocks to be positive semidefinite, new bounds significantly improve upon earlier results: $$\begin{align*} \text{cr}(K_{10,n}) &\geq 4.87057 n^2 - 10n \ \text{cr}(K_{11,n}) &\geq 5.99939 n^2 - 12.5n \ \text{cr}(K_{12,n}) &\geq 7.25579 n^2 - 15n \ \text{cr}(K_{13,n}) &\geq 8.65675 n^2 - 18n \end{align*}$$ The SDP relaxations, particularly those requiring only a “special” block to be PSD, are computationally efficient and nearly as strong as full SDP bounds. These methods approach Zarankiewicz’s conjectured crossing numbers, and could, with further hierarchy refinement, resolve central open problems in topological graph theory.

8. Open Problems, Complexity Barriers, and Future Directions

Despite substantial progress on finite convergence in specialized graph classes, the general validity of the De Klerk–Pasechnik Conjecture—that all combinatorial invariants of the stability number are realized at a bounded hierarchy level—remains open. Recent work presents algorithmic reductions to the acritical case and tightness examples. Complexity results highlight NP-hardness of determining finite minimizer sets in quadratic optimization, underscoring intrinsic computational limits (Laurent et al., 2021). There exists sensitivity under graph augmentation; isolated node addition can invalidate exactness at fixed hierarchy levels (Laurent et al., 2021).

Further investigation into the interaction of nonnegative forms with their zeros, the extension of block-diagonal SDP relaxations to broader combinatorial problems, and the precise asymptotic behavior of approximation ratios remain major research avenues. Integration of algebraic geometry, spectral graph theory, and combinatorial optimization is likely to yield further insights into these hierarchies and their practical utility.

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In summary, the De Klerk–Pasechnik Conjecture is central to understanding finite convergence and tightness of SDP/certificate-based relaxations for the stability number, copositivity, TSP, and related combinatorial optimization domains. It is a touchstone for the performance and design of approximating cones, SOS hierarchies, and semidefinite relaxations, linking fundamental algebraic, geometric, and computational aspects of graph theory.