arXiv subjects

Levent Tunçel

Publications and source records attributed to Levent Tunçel.

At least 19 recordsLinked to original sources

arXiv2026

On some k-fold generalizations of Lovász theta and their sandwich theorems

We study several $k$-fold generalizations of the Lovász theta function associated with the maximum $k$-colorable induced subgraph problem. The first is the Narasimhan--Manber parameter $\vartheta_k$. We prove that, for graphs whose adjacency matrix belongs to a homogeneous partially coherent algebra, this parameter is recovered by the theta number of the Cartesian product with the complete graph on $k$ vertices. This class includes distance-regular and $1$-walk-regular graphs, and thus our result generalizes a theorem by Sinjorgo and Sotirov (2022) for graphs that are vertex- and edge-transitive. We introduce a new parameter $φ_k$ obtained from orthonormal representations of graphs and show the inequality $φ_k \leq \vartheta_k$. For both parameters, we study the smallest $k$ for which the parameter is equal to the number of vertices; these saturation parameters yield lower bounds on the chromatic number. We determine which vertex-weighted versions of these parameters are gauges, and discuss a natural definition for the $k$-fold theta body of a graph. We conclude with open questions comparing $\vartheta_k$, $φ_k$, $\vartheta(G\square K_k)$, and related convexifications.

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arXiv2026

New complexity bounds for primal--dual interior-point algorithms in conic optimization

We provide improved complexity results for symmetric primal--dual interior-point algorithms in conic optimization. The results follow from new uniform bounds on a key complexity measure for primal--dual metrics at pairs of primal and dual points. The complexity measure is defined as the largest eigenvalue of the product of the Hessians of the primal and dual barrier functions, normalized by the proximity of the points to the central path. For algorithms based on self-scaled barriers for symmetric cones, we determine the exact value of the complexity measure. In the significantly more general case of self-concordant barriers with negative curvature, we provide the asymptotically tight upper bound of 4/3. This result implies $O(\vartheta^{1/2}\ln(1/ε))$ iteration complexity for a variety of symmetric (and some nonsymmetric) primal--dual interior-point algorithms. Finally, in the case of general self-concordant barriers, we give improved bounds for some variants of the complexity measure.

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arXiv2026

Stable Set Polytopes with Rank $|V(G)|/3$ for the Lovász--Schrijver SDP Operator

We study the lift-and-project rank of the stable set polytope of graphs with respect to the Lovász--Schrijver SDP operator $\text{LS}_+$ applied to the fractional stable set polytope. In particular, we show that for every positive integer $\ell$, the smallest possible graph with $\text{LS}_+$-rank $\ell$ contains $3\ell$ vertices. This result is sharp and settles a conjecture posed by Lipták and the second author in 2003, as well as answers a generalization of a problem posed by Knuth in 1994. We also show that for every positive integer $\ell$ there exists a vertex-transitive graph on at most $4\ell+12$ vertices with $\text{LS}_+$-rank at least $\ell$.

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arXiv2026

Maximum Cuts and Fractional Cut Covers: A Computational Study of a Randomized Semidefinite Programming Approach

We present experimental work on a primal-dual framework simultaneously approximating maximum cut and weighted fractional cut-covering instances. In this primal-dual framework, we solve a semidefinite programming (SDP) relaxation to either the maximum cut problem or to the weighted fractional cut-covering problem, and then independently sample a collection of cuts via the random-hyperplane technique. We then simultaneously certify the approximate optimality of a cut and a fractional cut cover. We present several implementations which reliably achieve the celebrated Goemans and Williamson approximation ratio of $α_{\mathrm{GW}} \approx 0.878$ for both optimization problems simultaneously, after $\lceil 128 \ln m \rceil$ samples, a number significantly smaller than the best theoretical bounds. This is the first experimental work approximating the weighted fractional cut-covering problem, and we deliver robust and repeatable results despite the use of randomized algorithms and floating-point arithmetic. Careful pre-processing of instances and post-processing of numeric results allow for good empirical outcomes with both first-order and second-order SDP solvers. Nearly optimal SDP solutions are suitably perturbed to ensure better probabilistic and numerical behavior. Our experiments deviate from theory by using a linear programming (LP) solver to compute fractional cut covers. For most instances studied, LP solving produces certifiably better results than the theoretical algorithm after $\lceil 128 \ln m \rceil$ samples. All our experiments strictly follow a unified pipeline which explicitly documents all parameters used in each run.

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arXiv2026

A Computational Search for Minimal Obstruction Graphs for the Lovász--Schrijver SDP Hierarchy

We study the lift-and-project relaxations of the stable set polytope of graphs generated by $\text{LS}_+$, the SDP lift-and-project operator devised by Lovász and Schrijver. Our focus is on $\ell$-minimal graphs: graphs on $3\ell$ vertices with $\text{LS}_+$-rank $\ell$, i.e., the smallest graphs realizing rank $\ell$. This manuscript makes two complementary contributions. First, we introduce $\text{LS}_+$ certificate packages, a modular framework for certifying membership in $\text{LS}_+$-relaxations using only integer arithmetic and simple, concise calculations, thereby making numerical lower-bound proofs more transparent, reliable, and easier to verify. Second, we apply this framework to a computational search for extremal graphs. We prove that there are at least 49 non-isomorphic 3-minimal graphs and at least 4,107 non-isomorphic 4-minimal graphs, improving the previously known counts of 14 and 588, respectively. Beyond the increase in counts, the new examples sharpen the emerging structural picture: stretched cliques remain central but are not exhaustive, clique number is informative but not decisive, and some extremal graphs exhibit previously unseen graph minor and edge density behaviour. We also determine the smallest vertex-transitive graphs of $\text{LS}_+$-rank $\ell$ for every $\ell \leq 4$.

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arXiv2025

On Connections Between Association Schemes and Analyses of Polyhedral and Positive Semidefinite Lift-and-Project Relaxations

We explore some connections between association schemes and the analyses of the semidefinite programming (SDP) based convex relaxations of combinatorial optimization problems in the Lovász--Schrijver lift-and-project hierarchy. Our analysis of the relaxations of the stable set polytope leads to bounds on the clique and stability numbers of some regular graphs reminiscent of classical bounds by Delsarte and Hoffman, as well as the notion of deeply vertex-transitive graphs -- highly symmetric graphs that we show arise naturally from some association schemes. We also study relaxations of the hypergraph matching problem, and determine exactly or provide bounds on the lift-and-project ranks of these relaxations. Our proofs for these results also inspire the study of a homogeneous coherent configuration based on hypermatchings, which is an association scheme except it is generally non-commutative. We then illustrate the usefulness of obtaining commutative subschemes from non-commutative homogeneous coherent configurations via contraction in this context.

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arXiv2025

Generalizations of Total Dual Integrality

We design new tools to study variants of Total Dual Integrality. As an application, we obtain a geometric characterization of Total Dual Integrality for the case where the associated polyhedron is non-degenerate. We also give sufficient conditions for a system to be Totally Dual Dyadic, and prove new special cases of Seymour's Dyadic conjecture on ideal clutters.

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arXiv2025

A Primal-Dual Extension of the Goemans--Williamson Algorithm for the Weighted Fractional Cut-Covering Problem

We study a weighted generalization of the fractional cut-covering problem, which we relate to the maximum cut problem via antiblocker and gauge duality. This relationship allows us to introduce a semidefinite programming (SDP) relaxation whose solutions may be rounded into fractional cut covers by sampling via the random hyperplane technique. We then provide a $1/α_{\scriptscriptstyle \mathrm{GW}}$-approximation algorithm for the weighted fractional cut-covering problem, where $α_{\scriptscriptstyle \mathrm{GW}} \approx 0.878$ is the approximation factor of the celebrated Goemans--Williamson algorithm for the maximum cut problem. Nearly optimal solutions of the SDPs in our duality framework allow one to consider instances of the maximum cut and the fractional cut-covering problems as primal-dual pairs, where cuts and fractional cut covers simultaneously certify each other's approximation quality. We exploit this relationship to introduce new combinatorial certificates for both problems, as well as a randomized polynomial-time algorithm for producing such certificates. In~particular, we~show how the Goemans--Williamson algorithm implicitly approximates a weighted instance of the fractional cut-covering problem, and how our new algorithm explicitly approximates a weighted instance of the maximum cut problem. We conclude by discussing the role played by geometric representations of graphs in our results, and by proving our algorithms and analyses to be optimal in several aspects.

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arXiv2025

Normalizations of factorizations over convex cones and their effects on extension complexity

Factorizations over cones and their duals play central roles for many areas of mathematics and computer science. One of the reasons behind this is the ability to find a representation for various objects using a well-structured family of cones, where the representation is captured by the factorizations over these cones. Several major questions about factorizations over cones remain open even for such well-structured families of cones as non-negative orthants and positive semidefinite cones. Having said that, we possess a far better understanding of factorizations over non-negative orthants and positive semidefinite cones than over other families of cones. One of the key properties that led to this better understanding is the ability to normalize factorizations, i.e., to guarantee that the norms of the vectors involved in the factorizations are bounded in terms of an input and in terms of a constant dependent on the given cone. Our work aims at understanding which cones guarantee that factorizations over them can be normalized, and how this effects extension complexity of polytopes over such cones.

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arXiv2024

On Rank-Monotone Graph Operations and Minimal Obstruction Graphs for the Lovász--Schrijver SDP Hierarchy

We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lovász--Schrijver SDP operator $\text{LS}_+$, with a particular focus on finding and characterizing the smallest graphs with a given $\text{LS}_+$-rank (the needed number of iterations of the $\text{LS}_+$ operator on the fractional stable set polytope to compute the stable set polytope). We introduce a generalized vertex-stretching operation that appears to be promising in generating $\text{LS}_+$-minimal graphs and study its properties. We also provide several new $\text{LS}_+$-minimal graphs, most notably the first known instances of $12$-vertex graphs with $\text{LS}_+$-rank $4$, which provides the first advance in this direction since Escalante, Montelar, and Nasini's discovery of a $9$-vertex graph with $\text{LS}_+$-rank $3$ in 2006.

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arXiv2024

Everything is possible: constructing spectrahedra with prescribed facial dimensions

Given any finite set of nonnegative integers, there exists a closed convex set whose facial dimension signature coincides with this set of integers, that is, the dimensions of its nonempty faces comprise exactly this set of integers. In this work, we show that such sets can be realised as solution sets of systems of finitely many convex quadratic inequalities, and hence are representable via second-order cone programming problems, and are, in particular, spectrahedral. It also follows that these sets are facially exposed, in contrast to earlier constructions. We obtain a lower bound on the minimum number of convex quadratic inequalities needed to represent a closed convex set with prescribed facial dimension signature, and show that our bound is tight for some special cases. Finally, we relate the question of finding efficient representations with indecomposability of integer sequences and other topics, and discuss a substantial number of open questions.

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arXiv2024

Generalized Cuts and Grothendieck Covers: a Primal-Dual Approximation Framework Extending the Goemans--Williamson Algorithm

We provide a primal-dual framework for randomized approximation algorithms utilizing semidefinite programming (SDP) relaxations. Our framework pairs a continuum of APX-complete problems including MaxCut, Max2Sat, MaxDicut, and more generally, Max-Boolean Constraint Satisfaction and MaxQ (maximization of a positive semidefinite quadratic form over the hypercube) with new APX-complete problems which are stated as convex optimization problems with exponentially many variables. These new dual counterparts, based on what we call Grothendieck covers, range from fractional cut covering problems (for MaxCut) to tensor sign covering problems (for MaxQ). For each of these problem pairs, our framework transforms the randomized approximation algorithms with the best known approximation factors for the primal problems to randomized approximation algorithms for their dual counterparts with reciprocal approximation factors which are tight with respect to the Unique Games Conjecture. For each APX-complete pair, our algorithms solve a single SDP relaxation and generate feasible solutions for both problems which also provide approximate optimality certificates for each other. Our work utilizes techniques from areas of randomized approximation algorithms, convex optimization, spectral sparsification, as well as Chernoff-type concentration results for random matrices.

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arXiv2024

Stable Set Polytopes with High Lift-and-Project Ranks for the Lovász-Schrijver SDP Operator

We study the lift-and-project rank of the stable set polytopes of graphs with respect to the Lovász-Schrijver SDP operator $\text{LS}_+$. In particular, we focus on a search for relatively small graphs with high $\text{LS}_+$-rank (i.e., the least number of iterations of the $\text{LS}_+$ operator on the fractional stable set polytope to compute the stable set polytope). We provide families of graphs whose $\text{LS}_+$-rank is asymptotically a linear function of its number of vertices, which is the best possible up to improvements in the constant factor. This improves upon the previous best result in this direction from 1999, which yielded graphs whose $\text{LS}_+$-rank only grew with the square root of the number of vertices.

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arXiv2023

Computational complexity of decomposing a symmetric matrix as a sum of positive semidefinite and diagonal matrices

We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades. On the one hand, we prove that when the rank of the positive semidefinite matrix in the decomposition is bounded above by an absolute constant, the problem can be solved in polynomial time. On the other hand, we prove that, in general, these problems as well as their certain approximation versions are all NP-hard. Finally, we prove that many of these low-rank decomposition problems are complete in the first-order theory of the reals; i.e., given any system of polynomial equations, we can write down a low-rank decomposition problem in polynomial time so that the original system has a solution iff our corresponding decomposition problem has a feasible solution of certain (lowest) rank.

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arXiv2023

Dyadic linear programming and extensions

A rational number is dyadic if it has a finite binary representation $p/2^k$, where $p$ is an integer and $k$ is a nonnegative integer. Dyadic rationals are important for numerical computations because they have an exact representation in floating-point arithmetic on a computer. A vector is dyadic if all its entries are dyadic rationals. We study the problem of finding a dyadic optimal solution to a linear program, if one exists. We show how to solve dyadic linear programs in polynomial time. We give bounds on the size of the support of a solution as well as on the size of the denominators. We identify properties that make the solution of dyadic linear programs possible: closure under addition and negation, and density, and we extend the algorithmic framework beyond the dyadic case.

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arXiv2022

Graphs with large girth and chromatic number are hard for Nullstellensatz

We study the computational efficiency of approaches, based on Hilbert's Nullstellensatz, which use systems of linear equations for detecting non-colorability of graphs having large girth and chromatic number. We show that for every non-$k$-colorable graph with $n$ vertices and girth $g>4k$, the algorithm is required to solve systems of size at least $n^{Ω(g)}$ in order to detect its non-$k$-colorability.

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arXiv2022

Linear optimization over homogeneous matrix cones

A convex cone is homogeneous if its automorphism group acts transitively on the interior of the cone, i.e., for every pair of points in the interior of the cone, there exists a cone automorphism that maps one point to the other. Cones that are homogeneous and self-dual are called symmetric. The symmetric cones include the positive semidefinite matrix cone and the second order cone as important practical examples. In this paper, we consider the less well-studied conic optimization problems over cones that are homogeneous but not necessarily self-dual. We start with cones of positive semidefinite symmetric matrices with a given sparsity pattern. Homogeneous cones in this class are characterized by nested block-arrow sparsity patterns, a subset of the chordal sparsity patterns. We describe transitive subsets of the automorphism groups of the cones and their duals, and important properties of the composition of log-det barrier functions with the automorphisms in this set. Next, we consider extensions to linear slices of the positive semidefinite cone, i.e., intersection of the positive semidefinite cone with a linear subspace, and review conditions that make the cone homogeneous. In the third part of the paper we give a high-level overview of the classical algebraic theory of homogeneous cones due to Vinberg and Rothaus. A fundamental consequence of this theory is that every homogeneous cone admits a spectrahedral (linear matrix inequality) representation. We conclude by discussing the role of homogeneous cone structure in primal-dual symmetric interior-point methods.

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arXiv2022

Total dual dyadicness and dyadic generating sets

A vector is \emph{dyadic} if each of its entries is a dyadic rational number, i.e. of the form $\frac{a}{2^k}$ for some integers $a,k$ with $k\geq 0$. A linear system $Ax\leq b$ with integral data is \emph{totally dual dyadic} if whenever $\min\{b^\top y:A^\top y=w,y\geq {\bf 0}\}$ for $w$ integral, has an optimal solution, it has a dyadic optimal solution. In this paper, we study total dual dyadicness, and give a co-NP characterization of it in terms of \emph{dyadic generating sets for cones and subspaces}, the former being the dyadic analogue of \emph{Hilbert bases}, and the latter a polynomial-time recognizable relaxation of the former. Along the way, we see some surprising turn of events when compared to total dual integrality, primarily led by the \emph{density} of the dyadic rationals. Our study ultimately leads to a better understanding of total dual integrality and polyhedral integrality. We see examples from dyadic matrices, $T$-joins, cycles, and perfect matchings of a graph.

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