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arxiv: 2504.07204 · v2 · submitted 2025-04-09 · 🧮 math.OC

Rounding the Lov\'asz Theta Function with a Value Function Approximation

Rui Gong , Diego Cifuentes , Alejandro Toriello This is my paper

Pith reviewed 2026-05-22 19:50 UTC · model grok-4.3

classification 🧮 math.OC
keywords Lovász theta functionmaximum weighted stable setperfect graphsrounding schemesemidefinite programmingdynamic programmingvalue function approximationgeneralized split graphs
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The pith

A value function approximation from the Lovász theta SDP, optimized by dynamic programming, recovers a maximum weighted stable set exactly on generalized split graphs and other large subclasses of perfect graphs.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The Lovász theta function gives a tight SDP relaxation for the maximum weighted stable set problem on perfect graphs, yet no prior rounding method was known to extract the exact optimum from the SDP solution. This paper constructs a value function approximation directly from that SDP solution and applies dynamic programming to produce a stable set. The resulting procedure is proven to return a maximum weighted stable set on generalized split graphs, which asymptotically include almost all perfect graphs. The approach solves only a single SDP and performs well even when tested on imperfect graphs.

Core claim

The paper claims that converting the Lovász theta SDP solution into a value function approximation and then taking its optimal policy under dynamic programming yields a maximum weighted stable set for several subclasses of perfect graphs, including generalized split graphs.

What carries the argument

A value function approximation constructed from the theta SDP solution, whose optimal dynamic-programming policy produces the stable set.

If this is right

  • The method is the first known rounding strategy for the theta function that is guaranteed to recover maximum weighted stable sets on large classes of perfect graphs.
  • Only one SDP needs to be solved, in contrast to earlier theta-based methods that require multiple SDPs.
  • The same procedure produces good solutions on imperfect graphs in computational tests.
  • Because generalized split graphs asymptotically cover almost all perfect graphs, the exact-recovery guarantee applies to a broad measure of the perfect-graph family.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The dynamic-programming step may generalize to other SDP relaxations whose solutions admit a natural ordering or recursive structure.
  • The approach could be tested on random perfect graphs generated outside the generalized-split class to check whether the recovery property extends further.
  • Efficiency gains from solving only one SDP may make the method attractive for large-scale stable-set instances where multiple-SDP baselines become prohibitive.

Load-bearing premise

The SDP solution for the Lovász theta function can be converted into a value function approximation whose optimal policy under dynamic programming yields a maximum weighted stable set for generalized split graphs and other stated perfect-graph subclasses.

What would settle it

A generalized split graph on which the procedure returns a stable set whose weight is strictly less than the maximum would falsify the exact-recovery claim.

Figures

Figures reproduced from arXiv: 2504.07204 by Alejandro Toriello, Diego Cifuentes, Rui Gong.

Figure 1
Figure 1. Figure 1: Paths from i, j to v ∗ . Proposition 3.6 implies that Algorithm 1 with VLP returns an optimal stable set on house-free perfect graphs without applying the look-ahead. And Proposition 3.4 provides a necessary condition for the algorithm to possibly fail: in some iteration, V (I) > V (I ′ ) +wi . As we indicate above, VLP(I) > VLP(I ′ ) +wi is equivalent to discarding all vertices of an essential clique. A h… view at source ↗
Figure 3
Figure 3. Figure 3: Look-ahead counter-example. complementary solution of (LP-D). A maximum stable set of this graph has size 9. As in Example 3.8, suppose 2,7,12 are selected; we then obtain the graph in [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Look-ahead counter-example after selecting nodes 2 [PITH_FULL_IMAGE:figures/full_fig_p016_4.png] view at source ↗
read the original abstract

The Lov\'asz theta function is a semidefinite programming (SDP) relaxation for the maximum weighted stable set problem, which is tight for perfect graphs. However, even for perfect graphs, there is no known rounding method guaranteed to extract a maximum weighted stable set from the SDP solution. In this paper, we develop a novel rounding scheme for the theta function that constructs a value function approximation from the SDP solution and then generates a stable set using dynamic programming. Our method provably recovers a maximum weighted stable set in several sub-classes of perfect graphs, including generalized split graphs, which asymptotically cover almost all perfect graphs. To the best of our knowledge, this is the only known rounding strategy for the theta function that recovers a maximum weighted stable set for large classes of perfect graphs. Our rounding scheme relies on simple linear algebra computations; we only solve one SDP. In contrast, existing methods that leverage the theta function require solving multiple SDPs. Computational experiments show that our method produces good solutions even on imperfect graphs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper develops a rounding procedure for the Lovász theta SDP relaxation of the maximum weighted stable set problem. A value function approximation is constructed directly from the SDP solution via a linear-algebra step; this approximation is then used inside a dynamic program whose optimal policy is shown to return a maximum-weight stable set. The authors prove exact recovery on generalized split graphs and several other subclasses of perfect graphs, note that the procedure requires only a single SDP solve, and report competitive numerical performance on imperfect graphs.

Significance. If the stated guarantees hold, the contribution is notable: it supplies an explicit, single-SDP rounding scheme that provably recovers the exact optimum on large families of perfect graphs (generalized split graphs asymptotically cover almost all perfect graphs). The constructive extraction of the value function from the SDP vectors together with the polynomial-state DP that exploits the clique–independent-set partition of generalized split graphs is a clear technical strength. The work also contrasts favorably with prior theta-based methods that require multiple SDPs.

minor comments (3)
  1. [§3.2] §3.2: the linear-algebra step that converts the SDP vectors into the value-function approximation would benefit from an explicit formula or small worked example on a 4-vertex generalized split graph.
  2. [§5] §5, Table 2: the running-time column should separate SDP solve time from the subsequent DP time so that the single-SDP claim can be verified numerically.
  3. The statement that generalized split graphs 'asymptotically cover almost all perfect graphs' should be accompanied by a precise reference to the relevant asymptotic result in the literature.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. We are pleased that the single-SDP rounding procedure, the value-function extraction, and the exact-recovery guarantees for generalized split graphs (and other perfect-graph subclasses) are viewed as notable contributions.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central contribution is an explicit constructive rounding procedure: the value-function approximation is obtained directly from the single Lovász theta SDP solution via a linear-algebra step on the SDP vectors, after which dynamic programming is run on a polynomially sized state space that exploits the explicit clique-plus-independent-set partition defining generalized split graphs. The optimality proof for these (and a few other perfect-graph subclasses) proceeds by verifying that the DP policy recovers a maximum-weight stable set using only the SDP tightness property for perfect graphs and the graph partition structure; no parameter is fitted to data, no result is renamed as a prediction, and no load-bearing step reduces to a self-citation or prior ansatz. The manuscript is therefore internally consistent and self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard fact that the Lovász theta SDP is tight for perfect graphs and on the structural properties of the mentioned subclasses (generalized split graphs). No free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption The Lovász theta function is tight for perfect graphs.
    Stated as background in the abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Copositive Matrices with Ordered Off-Diagonal Entries

    math.OC 2026-05 unverdicted novelty 7.0

    Copositive matrices with nondecreasing off-diagonal entries admit a PSD plus nonnegative decomposition, which implies exactness of a natural relaxation for separable quadratic optimization over the simplex.

Reference graph

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