arXiv subjects

Renata Sotirov

Publications and source records attributed to Renata Sotirov.

At least 19 recordsLinked to original sources

arXiv2026

Improved semidefinite programming bounds for the maximum $k$-colorable subgraph problem

We study the maximum $k$-colorable subgraph (M$k$CS) problem, which consists in finding a largest $k$-colorable induced subgraph in a given graph. We consider a Semidefinite Programming (SDP) relaxation for the M$k$CS problem and regard its resulting upper bound as a graph parameter. We present several properties of this graph parameter, from which we obtain that the M$k$CS problem is solvable in polynomial time for $k$-perfect graphs. We further derive two novel families of valid inequalities to strengthen the SDP relaxation. The first family reduces to a family of inequalities for the Boolean quadric polytope when $k = 1$, and the second family generalizes the family of rank inequalities for binary linear programming formulations of the stable set problem. We efficiently solve the strengthened SDP relaxation using a cutting-plane algorithm that is based on the Alternating Direction Method of Multipliers (ADMM). Extensive computational experiments show that the obtained upper bounds outperform the best upper bounds from the literature. To complement our SDP-based upper bounds, we propose an integer ADMM variant that uses an exact Binary Semidefinite Programming (BSDP) formulation of the M$k$CS problem to produce high-quality feasible solutions. To the best of our knowledge, this is the first application of the ADMM to compute integer solutions to a BSDP problem.

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arXiv2026

Beyond binarity: Semidefinite programming for ternary quadratic problems

We study the ternary quadratic problem (TQP), a quadratic optimization problem with linear constraints where the variables take values in $\{0, \pm 1\}$. While semidefinite programming (SDP) techniques are well established for $\{0,1\}$- and $\{\pm 1\}$-valued quadratic problems, no dedicated integer semidefinite programming framework exists for the ternary case. In this paper, we introduce a ternary SDP formulation for the TQP that forms the basis of an exact solution approach. We derive new theoretical insights in rank-one ternary positive semidefinite matrices, which lead to a basic SDP relaxation that is further strengthened by valid triangle, RLT, split and $k$-gonal inequalities. These are embedded in a tailored branch-and-bound algorithm that iteratively solves strengthened SDPs, separates violated inequalities, applies a ternary branching strategy and computes high-quality feasible solutions. We test our algorithm on TQP variations motivated by practice, including unconstrained, linearly constrained and quadratic ratio problems. Computational results on these instances demonstrate the effectiveness of the proposed algorithm.

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arXiv2025

Spanning and Splitting: Integer Semidefinite Programming for the Quadratic Minimum Spanning Tree Problem

In the quadratic minimum spanning tree problem (QMSTP) one wants to find the minimizer of a quadratic function over all possible spanning trees of a graph. We present a formulation of the QMSTP as a mixed-integer semidefinite program exploiting the algebraic connectivity of a graph. Based on this formulation, we derive a doubly nonnegative relaxation for the QMSTP and investigate classes of valid inequalities to strengthen the relaxation using the Chvátal-Gomory procedure for mixed-integer conic programming. Solving the resulting relaxations is out of reach for off-the-shelf software. We therefore develop and implement a version of the Peaceman-Rachford splitting method that allows to compute the new bounds for graphs from the literature. The computational results demonstrate that our bounds significantly improve over existing bounds from the literature in both quality and computation time, in particular for graphs with more than 30 vertices. This work is further evidence that semidefinite programming is a valuable tool to obtain high-quality bounds for problems in combinatorial optimization, in particular for those that can be modelled as a quadratic problem.

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arXiv2025

Exploiting Symmetries in Optimal Quantum Circuit Design

A physical limitation in quantum circuit design is the fact that gates in a quantum system can only act on qubits that are physically adjacent in the architecture. To overcome this problem, SWAP gates need to be inserted to make the circuit physically realizable. The nearest neighbour compliance problem (NNCP) asks for an optimal embedding of qubits in a given architecture such that the total number of SWAP gates to be inserted is minimized. In this paper we study the NNCP on general quantum architectures. Building upon an existing linear programming formulation, we show how the model can be reduced by exploiting the symmetries of the graph underlying the formulation. The resulting model is equivalent to a generalized network flow problem and follows from an in-depth analysis of the automorphism group of specific Cayley graphs. As a byproduct of our approach, we show that the NNCP is polynomial time solvable for several classes of symmetric quantum architectures. Numerical tests on various architectures indicate that the reduction in the number of variables and constraints is on average at least 90%. In particular, NNCP instances on the star architecture can be solved for quantum circuits up to 100 qubits and more than 1000 quantum gates within a very short computation time. These results are far beyond the computational capacity when solving the instances without the exploitation of symmetries.

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arXiv2025

Lagrangian Duality for Mixed-Integer Semidefinite Programming: Theory and Algorithms

This paper presents the Lagrangian duality theory for mixed-integer semidefinite programming (MISDP). We derive the Lagrangian dual problem and prove that the resulting Lagrangian dual bound dominates the bound obtained from the continuous relaxation of the MISDP problem. We present a hierarchy of Lagrangian dual bounds by exploiting the theory of integer positive semidefinite matrices and propose three algorithms for obtaining those bounds. Our algorithms are variants of well-known algorithms for minimizing non-differentiable convex functions. The numerical results on the max-$k$-cut problem show that the Lagrangian dual bounds are substantially stronger than the semidefinite programming bound obtained by relaxing integrality, already for lower levels in the hierarchy.

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arXiv2025

SDP bounds on the stability number via ADMM and intermediate levels of the Lasserre hierarchy

We consider the Lasserre hierarchy for computing bounds on the stability number of graphs. The semidefinite programs (SDPs) arising from this hierarchy involve large matrix variables and many linear constraints, which makes them difficult to solve using interior-point methods. We propose solving these SDPs using the alternating direction method of multipliers (ADMM). When the second level of the Lasserre hierarchy for a given graph is intractable for the ADMM, we consider an intermediate-level relaxation of the hierarchy. To warm-start the ADMM, we use an optimal solution from the first level of the Lasserre hierarchy, which is equivalent to the well-known Lovász theta function. Additionally, we use this solution to determine which degree two monomials to add in the Lasserre hierarchy relaxation to obtain an intermediate level between 1 and 2. Computational results demonstrate that our approach yields strong bounds on the stability number, which are computable within reasonable running times. We provide the best-known bounds on the stability number of various graphs from the literature.

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arXiv2025

Improved approximation ratios for the Quantum Max-Cut problem on general, triangle-free and bipartite graphs

We study polynomial-time approximation algorithms for the Quantum Max-Cut (QMC) problem. Given an edge-weighted graph $G$ on n vertices, the QMC problem is to determine the largest eigenvalue of a particular $2^n \times 2^n$ matrix that corresponds to $G$. We provide a sharpened analysis of the currently best-known QMC approximation algorithm for general graphs. This algorithm achieves an approximation ratio of $0.599$, which our analysis improves to $0.603$. Additionally, we propose two new approximation algorithms for the QMC problem on triangle-free and bipartite graphs, that achieve approximation ratios of $0.61383$ and $0.8162$, respectively. These are the best-known approximation ratios for their respective graph classes.

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arXiv2024

On convergence of a $q$-random coordinate constrained algorithm for non-convex problems

We propose a random coordinate descent algorithm for optimizing a non-convex objective function subject to one linear constraint and simple bounds on the variables. Although it is common use to update only two random coordinates simultaneously in each iteration of a coordinate descent algorithm, our algorithm allows updating arbitrary number of coordinates. We provide a proof of convergence of the algorithm. The convergence rate of the algorithm improves when we update more coordinates per iteration. Numerical experiments on large scale instances of different optimization problems show the benefit of updating many coordinates simultaneously.

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arXiv2024

Cuts and semidefinite liftings for the complex cut polytope

We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices $xx^{\mathrm{H}}$, where the elements of $x \in \mathbb{C}^n$ are $m$th unit roots. These polytopes have applications in ${\text{MAX-3-CUT}}$, digital communication technology, angular synchronization and more generally, complex quadratic programming. For ${m=2}$, the complex cut polytope corresponds to the well-known cut polytope. We generalize valid cuts for this polytope to cuts for any complex cut polytope with finite $m>2$ and provide a framework to compare them. Further, we consider a second semidefinite lifting of the complex cut polytope for $m=\infty$. This lifting is proven to be equivalent to other complex Lasserre-type liftings of the same order proposed in the literature, while being of smaller size. Our theoretical findings are supported by numerical experiments on various optimization problems.

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arXiv2024

The quadratic minimum spanning tree problem: lower bounds via extended formulations

The quadratic minimum spanning tree problem (QMSTP) is the problem of finding a spanning tree of a graph such that the total interaction cost between pairs of edges in the tree is minimized. We first show that most of the bounding approaches for the QMSTP are closely related. Then, we exploit an extended formulation for the minimum spanning tree problem to derive a sequence of relaxations for the QMSTP with increasing complexity. The resulting relaxations differ from the relaxations in the literature. Namely, our relaxations have a polynomial number of constraints and can be efficiently solved by a cutting plane algorithm. Moreover our bounds outperform most of the bounds from the literature.

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arXiv2023

On Integrality in Semidefinite Programming for Discrete Optimization

It is well-known that by adding integrality constraints to the semidefinite programming (SDP) relaxation of the max-cut problem, the resulting integer semidefinite program is an exact formulation of the problem. In this paper we show similar results for a wide variety of discrete optimization problems for which SDP relaxations have been derived. Based on a comprehensive study on discrete positive semidefinite matrices, we introduce a generic approach to derive mixed-integer semidefinite programming (MISDP) formulations of binary quadratically constrained quadratic programs and binary quadratic matrix programs. Applying a problem-specific approach, we derive more compact MISDP formulations of several problems, such as the quadratic assignment problem, the graph partition problem and the integer matrix completion problem. We also show that several structured problems allow for novel compact MISDP formulations through the notion of association schemes. Complementary to the recent advances on algorithmic aspects related to MISDP, this work opens new perspectives on solution approaches for the here considered problems.

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arXiv2023

The Chvátal-Gomory Procedure for Integer SDPs with Applications in Combinatorial Optimization

In this paper we study the well-known Chvátal-Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of spectrahedra is derived by exploiting total dual integrality for SDPs. Moreover, we show how to exploit (strengthened) CG cuts in a branch-and-cut framework for ISDPs. Different from existing algorithms in the literature, the separation routine in our approach exploits both the semidefinite and the integrality constraints. We provide separation routines for several common classes of binary SDPs resulting from combinatorial optimization problems. In the second part of the paper we present a comprehensive application of our approach to the quadratic traveling salesman problem (QTSP). Based on the algebraic connectivity of the directed Hamiltonian cycle, two ISDPs that model the QTSP are introduced. We show that the CG cuts resulting from these formulations contain several well-known families of cutting planes. Numerical results illustrate the practical strength of the CG cuts in our branch-and-cut algorithm, which outperforms alternative ISDP solvers and is able to solve large QTSP instances to optimality.

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arXiv2023

On solving the MAX-SAT using sum of squares

We consider semidefinite programming (SDP) approaches for solving the maximum satisfiability problem (MAX-SAT) and the weighted partial MAX-SAT. It is widely known that SDP is well-suited to approximate the (MAX-)2-SAT. Our work shows the potential of SDP also for other satisfiability problems, by being competitive with some of the best solvers in the yearly MAX-SAT competition. Our solver combines sum of squares (SOS) based SDP bounds and an efficient parser within a branch & bound scheme. On the theoretical side, we propose a family of semidefinite feasibility problems, and show that a member of this family provides the rank two guarantee. We also provide a parametric family of semidefinite relaxations for the MAX-SAT, and derive several properties of monomial bases used in the SOS approach. We connect two well-known SDP approaches for the (MAX)-SAT, in an elegant way. Moreover, we relate our SOS-SDP relaxations for the partial MAX-SAT to the known SAT relaxations.

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arXiv2022

Partitioning through projections: strong SDP bounds for large graph partition problems

The graph partition problem (GPP) aims at clustering the vertex set of a graph into a fixed number of disjoint subsets of given sizes such that the sum of weights of edges joining different sets is minimized. This paper investigates the quality of doubly nonnegative (DNN) relaxations, i.e., relaxations having matrix variables that are both positive semidefinite and nonnegative, strengthened by additional polyhedral cuts for two variations of the GPP: the $k$-equipartition and the graph bisection problem. After reducing the size of the relaxations by facial reduction, we solve them by a cutting-plane algorithm that combines an augmented Lagrangian method with Dykstra's projection algorithm. Since many components of our algorithm are general, the algorithm is suitable for solving various DNN relaxations with a large number of cutting planes. We are the first to show the power of DNN relaxations with additional cutting planes for the GPP on large benchmark instances up to 1,024 vertices. Computational results show impressive improvements in strengthened DNN bounds.

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arXiv2022

Facial Reduction for Symmetry Reduced Semidefinite Doubly Nonnegative Programs

We consider both facial reduction, \FRp, and symmetry reduction, \SRp, techniques for semidefinite programming, \SDPp. We show that the two together fit surprisingly well in an alternating direction method of multipliers, \ADMMp, approach. In fact, this approach allows for simply adding on nonnegativity constraints, and solving the doubly nonnegative, \DNN, relaxation of many classes of hard combinatorial problems. We also show that the singularity degree remains the same after \SRp, and that the \DNN relaxations considered here have singularity degree one, that is reduced to zero after \FRp. The combination of \FR and \SR leads to a significant improvement in both numerical stability and running time for both the \ADMM and interior point approaches. We test our method on various \DNN relaxations of hard combinatorial problems including quadratic assignment problems with sizes of more than $n=500$. This translates to a semidefinite constraint of order $250,000$ and $625\times 10^8$ nonnegative constrained variables, before applying the reduction techniques.

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arXiv2021

On the generalized $\vartheta$-number and related problems for highly symmetric graphs

This paper is an in-depth analysis of the generalized $\vartheta$-number of a graph. The generalized $\vartheta$-number, $\vartheta_k(G)$, serves as a bound for both the $k$-multichromatic number of a graph and the maximum $k$-colorable subgraph problem. We present various properties of $\vartheta_k(G)$, such as that the sequence $(\vartheta_k(G))_k$ is increasing and bounded from above by the order of the graph $G$. We study $\vartheta_k(G)$ when $G$ is the strong, disjunction or Cartesian product of two graphs. We provide closed form expressions for the generalized $\vartheta$-number on several classes of graphs including the Kneser graphs, cycle graphs, strongly regular graphs and orthogonality graphs. Our paper provides bounds on the product and sum of the $k$-multichromatic number of a graph and its complement graph, as well as lower bounds for the $k$-multichromatic number on several graph classes including the Hamming and Johnson graphs.

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arXiv2021

Lower Bounds for the Bandwidth Problem

The Bandwidth Problem seeks for a simultaneous permutation of the rows and columns of the adjacency matrix of a graph such that all nonzero entries are as close as possible to the main diagonal. This work focuses on investigating novel approaches to obtain lower bounds for the bandwidth problem. In particular, we use vertex partitions to bound the bandwidth of a graph. Our approach contains prior approaches for bounding the bandwidth as special cases. By varying sizes of partitions, we achieve a trade-off between quality of bounds and efficiency of computing them. To compute lower bounds, we derive a Semidefinite Programming relaxation. We evaluate the performance of our approach on several data sets, including real-world instances.

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arXiv2021

SDP-based bounds for the Quadratic Cycle Cover Problem via cutting plane augmented Lagrangian methods and reinforcement learning

We study the Quadratic Cycle Cover Problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the reduction and the structure of the graph, which is exploited in an efficient algorithm that constructs this matrix for any instance of the problem. To solve our relaxations, we propose an algorithm that incorporates an augmented Lagrangian method into a cutting plane framework by utilizing Dykstra's projection algorithm. Our algorithm is suitable for solving SDP relaxations with a large number of cutting planes. Computational results show that our SDP bounds and our efficient cutting plane algorithm outperform other QCCP bounding approaches from the literature. Finally, we provide several SDP-based upper bounding techniques, among which a sequential Q-learning method that exploits a solution of our SDP relaxation within a reinforcement learning environment.

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