Optimization and complexity of inertia-type bounds on the independence and chromatic numbers of graph powers
Pith reviewed 2026-05-09 19:03 UTC · model grok-4.3
The pith
Optimizing inertia-type bounds for independence and chromatic numbers of graph powers is polynomial-time solvable for fixed k.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We develop improved mixed-integer linear programming formulations that optimize the inertia-type bounds on the independence and chromatic numbers of graph powers. These new formulations decrease running times substantially. We prove that the optimization problems are solvable in polynomial time for any fixed k and supply explicit polynomial algorithms for small k. The bounds are obtained by selecting a polynomial of degree k and examining the inertia of the corresponding matrix formed from the graph's adjacency matrix.
What carries the argument
The refined MILP models that jointly optimize the choice of a degree-k polynomial and the resulting eigenvalue inertia bound, together with the fixed-parameter polynomial-time algorithm.
If this is right
- For fixed k the optimal inertia-type bound can be computed in time polynomial in the number of vertices.
- Improved formulations make the bounds usable on larger graphs even for small k.
- The results provide a clearer picture of the computational complexity of these spectral bounds.
- Applications in coding theory can now employ exact versions of these bounds more readily.
Where Pith is reading between the lines
- The techniques may generalize to other inertia-based or polynomial spectral bounds in graph theory.
- In quantum information settings that use distance-k coloring or independence, routine optimization of these bounds becomes feasible.
- One could investigate whether the polynomial-time methods yield explicit formulas for the bounds on particular graph families.
- Approximation schemes might be developed for cases where k grows with the graph size.
Load-bearing premise
The improved MILP formulations correctly encode the inertia-type bound optimization problem without introducing any relaxation gaps or errors in modeling the polynomial selection and eigenvalue constraints.
What would settle it
An instance of a graph and small k where the improved MILP returns a bound value provably different from the true optimum of the original inertia-type expression, or a proof that the running time is superpolynomial for some fixed k.
read the original abstract
The inertia bound, introduced by Cvetkovi\'c in 1971, is a fundamental result in spectral graph theory that provides an upper bound for the independence number of a graph in terms of spectral information about a weighted adjacency matrix of the graph. Recently, this bound has been extended to the socalled inertia-type bounds for estimating the independence and chromatic numbers of graph powers ($k$-independence number and distance-$k$ chromatic number of a graph). These bounds have recently found applications in coding theory and quantum information theory. The inertia-type bounds depend on the choice of a polynomial of degree $k$ and on the eigenvalues of the graph. Currently, optimizing these bounds requires solving several MILPs, which quickly becomes computationally demanding as the graph size or $k$ grows. This computational barrier is a major obstacle to the practical use of these bounds. Moreover, we have a limited theoretical understanding of their performance, even for small $k$. In this paper, we investigate their optimization and complexity. In particular, we improve the MILP formulations, reducing their computational burden and significantly decreasing the running time. Furthermore, we show that the optimization problems associated with the bounds are solvable in polynomial time for fixed $k$ and for small $k$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript investigates the optimization of inertia-type bounds (extensions of the 1971 Cvetković inertia bound) for the k-independence number and distance-k chromatic number of graphs. It presents improved MILP formulations that reduce computational demands and proves that the associated optimization problems are solvable in polynomial time for any fixed k as well as for small values of k.
Significance. If the claims hold, the work is significant for spectral graph theory and its applications in coding theory and quantum information. The improved MILP formulations directly address a practical barrier to using these bounds on larger graphs or higher powers, while the polynomial-time results for fixed k provide a theoretical foundation that clarifies the computational complexity of optimizing over polynomial choices and eigenvalue sign patterns. These algorithmic and complexity contributions strengthen the usability of inertia-type bounds without introducing evident relaxation gaps.
minor comments (3)
- [Abstract] Abstract: the phrase 'for small k' in the polynomial-time claim should be quantified (e.g., by listing explicit values of k or the growth rate in the complexity statement) so readers can immediately assess practical scope.
- [Introduction] Introduction: add an explicit citation to the original Cvetković 1971 inertia bound paper to complete the historical framing of the extensions discussed.
- The improved MILP formulations would benefit from a side-by-side comparison (perhaps in a table) of the number of variables, constraints, and binary variables relative to prior formulations to quantify the claimed reduction in computational burden.
Simulated Author's Rebuttal
We thank the referee for the positive and encouraging report, which recognizes the significance of our contributions to the optimization and complexity analysis of inertia-type bounds. We are pleased that the referee highlights both the practical improvements to the MILP formulations and the theoretical polynomial-time results for fixed k. Since the report recommends minor revision but lists no specific major comments requiring changes, we will incorporate any minor editorial or presentational improvements in the revised version.
Circularity Check
No significant circularity detected in derivation chain
full rationale
The manuscript analyzes the computational complexity of optimizing existing inertia-type bounds (building directly on Cvetković's 1971 result and subsequent extensions) by improving MILP formulations and proving polynomial-time solvability for fixed k via standard complexity reductions. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations that reduce the central claims to unverified inputs appear. The polynomial-time results follow from algorithmic modeling of polynomial choice and eigenvalue constraints, which are independent of the bound values themselves. The work is self-contained against external benchmarks in spectral graph theory.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Cvetković's 1971 inertia bound and its extensions to k-independence and distance-k chromatic numbers hold for weighted adjacency matrices of graph powers.
Reference graph
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discussion (0)
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