Degeneracy Cutting (DC) Techniques

• Degeneracy Cutting (DC) is a set of methods that systematically remove structural redundancies across disciplines like graph theory, optimization, and quantum error correction.
• DC techniques transform complex discrete problems into tractable formulations via approaches such as DC programming and multirow cuts, yielding rapid convergence and stronger bounds.
• In quantum LDPC decoding, local removal of degenerate nodes simplifies belief propagation and enhances decoding efficiency, paving the way for scalable fault-tolerant architectures.

Degeneracy Cutting (DC) refers to a family of techniques across combinatorics, optimization, integer programming, computational geometry, and quantum error correction, where degeneracy—either in the form of structural redundancy, solution multiplicity, or unwanted symmetries—is systematically identified and removed, cut, or exploited to improve theoretical bounds, algorithmic performance, or solution interpretability. The concept is strongly linked to the classical graph-theoretic degeneracy parameter, but recent research has extended and generalized DC to dense graph invariants, polyhedral methods for optimization, and local algorithmic strategies in quantum decoding.

1. Classical Degeneracy and Cutwidth: Quantitative Bounds in Graph Theory

The foundational aspect of DC arises in the context of graph theory, where the degeneracy δ(G) of a graph G is the maximum k such that the k-core is nonempty—i.e., iteratively removing vertices of degree less than k leaves a subgraph with minimum degree at least k. Degeneracy is a measure of graph "thickness" and immediately provides upper bounds on the chromatic and list chromatic numbers ($\chi(G) \leq \chi_{\ell}(G) \leq \delta(G) + 1$).

In "^^^^1^^^^" (0907.5138), a rigorous quantitative connection is proved:

• For any (ρ, λ)-uniformly sparse graph,

$$\mathrm{cw}(G) \geq \lceil \rho n \rceil \cdot \left( \delta(G) - \frac{\lceil \rho n \rceil - 1}{\lambda} \right)$$

• For general graphs $(0,1)$-uniform sparsity yields

$$\mathrm{cw}(G) \geq \frac{1}{4} \delta(G)^2 + \frac{1}{2} \delta(G)$$

• For triangle-free graphs (via Turán's theorem),

$\mathrm{cw}(G) \geq \frac{1}{2} \delta(G)^2$

These results show that "degeneracy cutting"—interpreted as partitioning or ordering the graph to minimize the cutwidth—is fundamentally lower-bounded by quadratic functions of the graph's degeneracy, reflecting intrinsic topological complexity and establishing hard limitations for graph layout and crossing minimization algorithms.

2. Degeneracy Cutting in Optimization: DC Programming and Cutting Planes

DC techniques are key to modern optimization via the Difference-of-Convex (DC) programming paradigm, which reformulates discrete (often NP-hard) problems as minimization of DC functions.

• In the constrained non-guillotine cutting (NGC) problem, binary indicator variables for rectangular cuts are relaxed using a concave penalty $\alpha(x) = \sum_{i,p,q} x_{i,p,q}(1 - x_{i,p,q})$, which is zero if and only if $x_{i,p,q}$ is binary, and large otherwise. The reformulated objective is DC: $F(x) = -\sum v_{i}x_{i,p,q} + t\alpha(x)$ with $x$ subject to box and geometric constraints. The DC Algorithm (DCA) solves linearized convex subproblems at each iteration, achieving rapid convergence and strong performance for large problem instances compared to CPLEX (Moeini et al., 2014).
• For mixed-binary linear programs (MBLPs), the exact penalization function $p(x) = \sum \min\{x_i, 1 - x_i\}$ allows formulating global DC cuts that exclude regions corresponding to fractional or infeasible solutions. These cuts (type-I and type-II) directly tighten the relaxation polytope. When combined with classical global cuts (e.g., Lift-and-Project), the DCCUT algorithm achieves fast gap closure and competitive global optimality; parallelization furthers computational efficiency (Niu et al., 2021).

These developments demonstrate that DC in optimization can be systematically applied to convert combinatorial constraints into analytically tractable forms, exploiting problem-specific degeneracy for targeted cutting and relaxation strengthening.

3. Degeneracy Cutting in Integer Programming: Multirow Tableau Cuts

In mixed-integer programming (MIP), degeneracy cutting methods construct deep, multirow cutting planes—especially from degenerate tableaux where LP relaxations have multiple basic integer variables.

• "Experiments with two-row cuts from degenerate tableaux" (Basu et al., 2017) studies intersection cuts generated from two rows, exploiting "lattice-free" convex sets (splits, triangles) in the tableau coordinates. Type 2 triangle cuts, which cover cases with fractional vertices and extended integer points, can theoretically dominate the split closure.
• Despite theoretical strength, empirical evaluation indicates that adding two-row cuts provides only marginal improvement over highly tuned Gomory Mixed Integer (GMI) generators, highlighting sensitivity to cut generator parameters (aggressiveness vs reliability). The effectiveness of DC-style multirow cuts is thus context-dependent, requiring nuanced numerical treatment.

This area exemplifies how degeneracy in integer program relaxations motivates the search for cutting planes tailored to redundant or ambiguous structural features in the problem formulation.

4. Degeneracy Cutting in Computational Geometry: Predicate Detection and Canonical Representation

DC also plays a central role in computational geometry, particularly in efficient degeneracy detection among algebraic predicates arising in free space construction for polyhedra motion.

• In motion planning, rigidity constraints are encoded via the signs and roots of angle polynomials. Degenerate predicates arise when two polynomials share a common factor—i.e., when geometric configurations coincide in such a way that standard algebraic evaluation would fail.
• "Table Based Detection of Degenerate Predicates in Free Space Construction" (Milenkovic et al., 2018) develops a table-based approach: every possible angle polynomial is factored and mapped to a unique canonical representative (a-poly), allowing ultra-efficient degeneracy detection via precomputation. This method is over three orders of magnitude faster than GCD-based checking and dramatically improves free space algorithm runtimes.

Canonical representation and systematic factorization are thus powerful DC techniques in exact geometry computations, eliminating need for ad hoc "degeneracy logic" and supporting robust, scalable implementations.

5. Degeneracy Cutting in Dense Graph Invariants: sd-degeneracy and Signed Tree Models

Recent advances have extended DC concepts from sparse graphs to dense analogues via the symmetric-difference degeneracy (sd-degeneracy) parameter.

• A graph is sd-degenerate at most $d$ if an elimination order allows removal of vertices with a "d-twin": another vertex differing on fewer than $d$ neighbors (formally $$sd_G(u,v) = |(N(u) \setminus N[v]) \cup (N(v) \setminus N[u])| < d$$). This relaxation generalizes both classical degeneracy and flip-width, allowing for broader implicit representation schemes.
• Unlike classical degeneracy, sd-degeneracy is not hereditary; a supergraph of a given graph may have lower sd-degeneracy. Computing sd-degeneracy and symmetric-difference is computationally hard (NP-complete and co-NP-complete, respectively) (Bonnet et al., 15 May 2024).
• Signed tree models extend twin-decompositions, using tree representations with transversal edges and anti-edges, capturing broader classes of graphs efficiently for adjacency labeling and first-order model-checking.

This dense DC framework hints at new cutting methodologies for handling graphs with near-twin neighborhood structures, going beyond traditional sparsity.

6. Degeneracy Cutting in Quantum LDPC Decoding: Local Loop Removal

DC has practical impact in quantum information processing, specifically in error correction code decoding.

• qLDPC codes suffer from syndrome degeneracy, where distinct errors produce the same syndrome due to stabilizer symmetries. After belief propagation (BP) decoding, many variables involved in the same stabilizer generator have similar soft error probabilities, reflecting local ambiguity.
• "Degeneracy Cutting: A Local and Efficient Post-Processing for Belief Propagation Decoding of Quantum Low-Density Parity-Check Codes" (Tsubouchi et al., 9 Oct 2025) introduces DC as a strictly local post-processing: for each stabilizer (row in the parity-check matrix), remove the variable node (qubit) with lowest error probability from consideration. This step is repeated for all stabilizers (or detector degeneracy matrix generalizations for realistic noise), then BP is rerun on the modified graph. The algorithm achieves performance comparable to ordered statistics decoding (which is cubic-time), but with only linear-time complexity and amenability to parallel implementation.
• The DC step crucially resolves local degeneracies in the Tanner graph, facilitating fast, scalable quantum decoding.

This illustrates how DC is used as an efficient, structurally motivated correction for inherent ambiguity in quantum error models, with immediate benefits for fault-tolerant quantum computing architectures.

7. Implications and Future Directions

Degeneracy Cutting transcends discipline boundaries, providing a unifying theme where structural redundancy is algorithmically or analytically addressed—by quadratic bounds (in graphs), penalized relaxations (in optimization), deep intersection cuts (in MIP), canonical representations (in geometry), dense decompositions (for signed tree models), or local ambiguity elimination (in quantum decoding).

Further research directions include:

• Tightening extremal bounds for DC operations in both sparse and dense graph classes.
• Developing polynomial-time algorithms exploiting DC invariants for nonhereditary graph parameters.
• Extending table-based DC approaches to higher-dimensional geometric and algebraic problems.
• Integrating DC techniques into real-time optimization, coding, and model-checking systems for large-scale applications.

Overall, DC remains a vibrant area for advancing both theoretical understanding and computational practice, particularly in problems where degeneracy is a central structural or operational bottleneck.