Levent Tunçel’s research while affiliated with University of Waterloo and other places

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 Lov\'{a}sz and Schrijver. In particular, we focus on searching for $\ell$-minimal graphs, which are graphs on $3\ell$ vertices whose stable set polytope has rank $\ell$ with respect to $\text{LS}_+$. These are the graphs which are the most challenging for the $\text{LS}_+$ operator according to one of the main complexity measures (smallest graphs with largest $\text{LS}_+$-rank). We introduce the notion of $\text{LS}_+$ certificate packages, which is a framework that allows for efficient and reliable verification of membership of points in $\text{LS}_+$-relaxations. Using this framework, we present numerical certificates which (combined with other results) show that there are at least 49 3-minimal graphs, as well as over 4000 4-minimal graphs. This marks a significant leap from the 14 3-minimal and 588 4-minimal graphs known before this work, with many of the newly-discovered graphs containing novel structures which helps enrich and recalibrate our understanding of $\ell$-minimal graphs. Some of this computational work leads to interesting conjectures. We also find all of the smallest vertex-transitive graphs with $\text{LS}_+$-rank $\ell$ for every $\ell \leq 4$.

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 $1/\alpha _{\scriptscriptstyle \textrm{GW}}$-approximation algorithm for the weighted fractional cut-covering problem, where $\alpha _{\scriptscriptstyle \textrm{GW}}\approx 0.878$ 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.

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.

We study the lift-and-project rank of the stable set polytope of graphs with respect to the Lov{\'a}sz--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 Lipt{\'a}k 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 $4\ell+12$ vertices with $\text{LS}_+$-rank at least $\ell$.

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.

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

Quantum relative entropy (QRE) programming is a recently popular and challenging class of convex optimization problems with significant applications in quantum computing and quantum information theory. We are interested in modern interior-point (IP) methods based on optimal self-concordant barriers for the QRE cone. A range of theoretical and numerical challenges associated with such barrier functions and the QRE cones have hindered the scalability of IP methods. To address these challenges, we propose a series of numerical and linear algebraic techniques and heuristics aimed at enhancing the efficiency of gradient and Hessian computations for the self-concordant barrier function, solving linear systems, and performing matrix-vector products. We also introduce and deliberate about some interesting concepts related to QRE such as symmetric quantum relative entropy. We design a two-phase method for performing facial reduction that can significantly improve the performance of QRE programming. Our new techniques have been implemented in the latest version (DDS 2.2) of the software package Domain-Driven Solver (DDS). In addition to handling QRE constraints, DDS accepts any combination of several other conic and nonconic convex constraints. Our comprehensive numerical experiments encompass several parts, including (1) a comparison of DDS 2.2 with Hypatia for the nearest correlation matrix problem, (2) using DDS 2.2 for combining QRE constraints with various other constraint types, and (3) calculating the key rate for quantum key distribution (QKD) channels and presenting results for several QKD protocols. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing and Operations Research. Accepted for Special Issue. Funding: This work was supported by the National Science Foundation [Grant CMMI-2347120] and Discovery Grants from the Natural Sciences and Engineering Research Council of Canada. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0570 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0570 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Domain-Driven Solver (DDS) is a MATLAB-based software package for convex optimization problems in Domain-Driven form.

A rational number is dyadic if it has a finite binary representation $p/2^k$ 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.