arXiv subjects

Sven Polak

Publications and source records attributed to Sven Polak.

At least 19 recordsLinked to original sources

arXiv2026

Lean-verified lower bounds for the Shannon capacity of odd cycles

We give new lower bounds for the Shannon capacities of small odd cycles: $\Theta(C_7)\geq3.258805369885\ldots$, $\Theta(C_{11})\geq5.294502522149\ldots$, $\Theta(C_{13})\geq6.302455083464\ldots$, $\Theta(C_{15})\geq7.301600534487\ldots$, $\Theta(C_{19})\geq9.357192705918\ldots$, $\Theta(C_{21})\geq10.342455853338\ldots$, and $\Theta(C_{23})\geq11.328224257774\ldots$. The bounds are obtained by an iterative procedure due to Gao (2026) which is based on a method by Itty, Rosin, Carstensen and Reichman (2026). The bounds are fully formalised in Lean.

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arXiv2026

Sum-of-squares certificates for symmetric polynomials on the hypercube: a counterexample to a conjecture of De Klerk and Laurent

This paper studies sum-of-squares (SoS) representations of nonnegative polynomials over the hypercube $[0,1]^n$. De Klerk and Laurent (SIAM J. Optim., 2010) conjectured that the smallest constant $C_n$ such that the polynomial $x_1\cdots x_n +C_n$ is contained in the degree-$n$ truncated quadratic module $M_{n,n}(x_1-x_1^2,\ldots,x_n-x_n^2)$ of the hypercube is $C_n=1/(n(n+2))$, for $n$ even. We specialize symmetry reduction techniques for finding sum-of-squares certificates to the hypercube, where the generators $x_i-x_i^2$ are not individually invariant under the symmetric group but form an invariant set, and apply them to this conjecture. Combining this reduction with a further (heuristic) sparsity reduction, a rational rounding step, and an exact verification over $\mathbb{Q}$, we prove the bound $C_8\leq 11/1000 <1/80$. In particular, this disproves the conjectured optimal value for $n=8$.

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arXiv2025

A group-theoretic approach to Shannon capacity of graphs and a limit theorem from lattice packings

We develop a group-theoretic approach to the Shannon capacity problem. Using this approach we extend and recover, in a structured and unified manner, various families of previously known lower bounds on the Shannon capacity. Bohman (2003) proved that, in the limit $p\to\infty$, the Shannon capacity of cycle graphs $\Theta(C_p)$ converges to the fractional clique covering number, that is, $\lim_{p \to \infty} p/2 - \Theta(C_p) = 0$. We strengthen this result by proving that the same is true for all fraction graphs: $\lim_{p/q \to \infty} p/q - \Theta(E_{p/q}) = 0$. Here the fraction graph $E_{p/q}$ is the graph with vertex set $\mathbb{Z}/p\mathbb{Z}$ in which two distinct vertices are adjacent if and only if their distance mod $p$ is strictly less than $q$. We obtain the limit via the group-theoretic approach. In particular, the independent sets we construct in powers of fraction graphs are subgroups (and, in fact, lattices). Our approach circumvents known barriers for structured ("linear") constructions of independent sets of Calderbank-Frankl-Graham-Li-Shepp (1993) and Guruswami-Riazanov (2021).

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arXiv2025

Semidefinite lower bounds for covering codes

Let $K_q(n,r)$ denote the minimum size of a $q$-ary covering code of word length $n$ and covering radius $r$. In other words, $K_q(n,r)$ is the minimum size of a set of $q$-ary codewords of length $n$ such that the Hamming balls of radius $r$ around the codewords cover the Hamming space $\{0,\ldots,q-1\}^n$. The special case $K_3(n,1)$ is often referred to as the football pool problem, as it is equivalent to finding a set of forecasts on $n$ football matches that is guaranteed to contain a forecast with at most one wrong outcome. In this paper, we build and expand upon the work of Gijswijt (2005), who introduced a semidefinite programming lower bound on $K_q(n,r)$ via matrix cuts. We develop techniques that strengthen this bound, by introducing new semidefinite constraints inspired by Lasserre's hierarchy for 0-1 programs and symmetry reduction methods, and a more powerful objective function. The techniques lead to sharper lower bounds, setting new records across a broad range of values of $q$, $n$, and $r$.

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arXiv2024

Mutually unbiased bases: polynomial optimization and symmetry

A set of $k$ orthonormal bases of $\mathbb C^d$ is called mutually unbiased if $|\langle e,f\rangle |^2 = 1/d$ whenever $e$ and $f$ are basis vectors in distinct bases. A natural question is for which pairs $(d,k)$ there exist~$k$ mutually unbiased bases in dimension $d$. The (well-known) upper bound $k \leq d+1$ is attained when~$d$ is a power of a prime. For all other dimensions it is an open problem whether the bound can be attained. Navascués, Pironio, and Acín showed how to reformulate the existence question in terms of the existence of a certain $C^*$-algebra. This naturally leads to a noncommutative polynomial optimization problem and an associated hierarchy of semidefinite programs. The problem has a symmetry coming from the wreath product of $S_d$ and $S_k$. We exploit this symmetry (analytically) to reduce the size of the semidefinite programs making them (numerically) tractable. A key step is a novel explicit decomposition of the $S_d \wr S_k$-module $\mathbb C^{([d]\times [k])^t}$ into irreducible modules. We present numerical results for small $d,k$ and low levels of the hierarchy. In particular, we obtain sum-of-squares proofs for the (well-known) fact that there do not exist $d+2$ mutually unbiased bases in dimensions~$d=2,3,4,5,6,7,8$. Moreover, our numerical results indicate that a sum-of-squares refutation, in the above-mentioned framework, of the existence of more than $3$ MUBs in dimension $6$ requires polynomials of total degree at least~$12$.

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arXiv2024

Semidefinite approximations for bicliques and biindependent pairs

We investigate some graph parameters dealing with biindependent pairs $(A,B)$ in a bipartite graph $G=(V_1\cup V_2,E)$, i.e., pairs $(A,B)$ where $A\subseteq V_1$, $B\subseteq V_2$ and $A\cup B$ is independent. These parameters also allow to study bicliques in general graphs. When maximizing the cardinality $|A\cup B|$ one finds the stability number $α(G)$, well-known to be polynomial-time computable. When maximizing the product $|A|\cdot |B|$ one finds the parameter $g(G)$, shown to be NP-hard by Peeters (2003), and when maximizing the ratio $|A|\cdot |B|/|A\cup B|$ one finds $h(G)$, introduced by Vallentin (2020) for bounding product-free sets in finite groups. We show that $h(G)$ is an NP-hard parameter and, as a crucial ingredient, that it is NP-complete to decide whether a bipartite graph $G$ has a balanced maximum independent set. These hardness results motivate introducing semidefinite programming bounds for $g(G)$, $h(G)$, and $α_{\text{bal}}(G)$ (the maximum cardinality of a balanced independent set). We show that these bounds can be seen as natural variations of the Lovász $\vartheta$-number, a well-known semidefinite bound on $α(G)$. In addition we formulate closed-form eigenvalue bounds and we show relationships among them as well as with earlier spectral parameters by Hoffman, Haemers (2001) and Vallentin (2020).

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arXiv2023

New lower bounds on crossing numbers of $K_{m,n}$ from semidefinite programming

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph $K_{m,n}$, extending a method from de Klerk et al. [SIAM J. Discrete Math. 20 (2006), 189--202] and the subsequent reduction by De Klerk, Pasechnik and Schrijver [Math. Prog. Ser. A and B, 109 (2007) 613--624]. We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that $\text{cr}(K_{10,n}) \geq 4.87057 n^2 - 10n$, $\text{cr}(K_{11,n}) \geq 5.99939 n^2-12.5n$, $\text{cr}(K_{12,n}) \geq 7.25579 n^2 - 15n$, $\text{cr}(K_{13,n}) \geq 8.65675 n^2-18n$ for all $n$. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite.

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arXiv2023

A note on the computational complexity of the moment-SOS hierarchy for polynomial optimization

The moment-sum-of-squares (moment-SOS) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the hierarchy is that, at a fixed level, it can be formulated as a semidefinite program of size polynomial in the number of variables n. Although this suggests that it may therefore be computed in polynomial time, this is not necessarily the case. Indeed, as O'Donnell (2017) and later Raghavendra & Weitz (2017) show, there exist examples where the sos-representations used in the hierarchy have exponential bit-complexity. We study the computational complexity of the moment-SOS hierarchy, complementing and expanding upon earlier work of Raghavendra & Weitz (2017). In particular, we establish algebraic and geometric conditions under which polynomial-time computation is guaranteed to be possible.

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arXiv2022

The maximum cardinality of trifferent codes with lengths 5 and 6

A code $\mathcal{C} \subseteq \{0, 1, 2\}^n$ is said to be trifferent with length $n$ when for any three distinct elements of $\mathcal{C}$ there exists a coordinate in which they all differ. Defining $\mathcal{T}(n)$ as the maximum cardinality of trifferent codes with length $n$, $\mathcal{T}(n)$ is unknown for $n \ge 5$. In this note, we use an optimized search algorithm to show that $\mathcal{T}(5) = 10$ and $\mathcal{T}(6) = 13$.

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arXiv2021

Symmetry reduction to optimize a graph-based polynomial from queueing theory

For given integers $n$ and $d$, both at least 2, we consider a homogeneous multivariate polynomial $f_d$ of degree $d$ in variables indexed by the edges of the complete graph on $n$ vertices and coefficients depending on cardinalities of certain unions of edges. Cardinaels, Borst and Van Leeuwaarden (arXiv:2111.05777, 2021) asked whether $f_d$, which arises in a model of job-occupancy in redundancy scheduling, attains its minimum over the standard simplex at the uniform probability vector. Brosch, Laurent and Steenkamp [SIAM J. Optim. 31 (2021), 2227--2254] proved that $f_d$ is convex over the standard simplex if $d=2$ and $d=3$, implying the desired result for these $d$. We give a symmetry reduction to show that for fixed $d$, the polynomial is convex over the standard simplex (for all $n\geq 2$) if a constant number of constant matrices (with size and coefficients independent of $n$) are positive semidefinite. This result is then used in combination with a computer-assisted verification to show that the polynomial $f_d$ is convex for $d\leq 9$.

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arXiv2020

New Methods in Coding Theory: Error-Correcting Codes and the Shannon Capacity

In this thesis we present several results in coding theory, concerning error-correcting codes and the Shannon capacity. 1. We give a general symmetry reduction of matrices occuring in semidefinite programs in coding theory. 2. We apply the symmetry reduction to efficiently compute semidefinite programming upper bounds for nonbinary error-correcting codes equipped with the Hamming distance (joint work with Bart Litjens and Lex Schrijver), with the Lee distance and for binary constant weight codes. 3. We explore other methods to find new upper bounds for nonbinary codes with the Hamming distance, based on combinatorial divisibility arguments. 4. We study uniqueness and classification of codes, using the output of the semidefinite programming solver. Most of our classification results are related to subcodes of the binary Golay code. This is joint work with Andries Brouwer. 5. We consider the Shannon capacity of circular graphs. The circular graph $C_{d,q}$ is the graph with vertex set $\mathbb{Z}_q$ (the cyclic group of order q) in which two distinct vertices are adjacent if and only if their distance (mod $q$) is strictly less than $d$. The Shannon capacity of $C_{d,q}$ can be seen to only depend on the fraction q/d. We prove that the function which assigns to each rational number $q/d$ the Shannon capacity of $C_{d,q}$ is continuous at integer points $q/d$. Moreover, we give a new lower bound on the Shannon capacity of the $7$-cycle. This is joint work with Lex Schrijver. This thesis was written at the University of Amsterdam under supervision of Lex Schrijver.

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arXiv2019

Semidefinite programming bounds for Lee codes

For $q,n,d \in \mathbb{N}$, let $A_q^L(n,d)$ denote the maximum cardinality of a code $C \subseteq \mathbb{Z}_q^n$ with minimum Lee distance at least $d$, where $\mathbb{Z}_q$ denotes the cyclic group of order $q$. We consider a semidefinite programming bound based on triples of codewords, which bound can be computed efficiently using symmetry reductions, resulting in several new upper bounds on $A_q^L(n,d)$. The technique also yields an upper bound on the independent set number of the $n$-th strong product power of the circular graph $C_{d,q}$, which number is related to the Shannon capacity of $C_{d,q}$. Here $C_{d,q}$ is the graph with vertex set $\mathbb{Z}_q$, in which two vertices are adjacent if and only if their distance (mod $q$) is strictly less than $d$. The new bound does not seem to improve significantly over the bound obtained from Lovász theta-function, except for very small $n$.

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arXiv2018

New lower bound on the Shannon capacity of C7 from circular graphs

We give an independent set of size $367$ in the fifth strong product power of $C_7$, where $C_7$ is the cycle on $7$ vertices. This leads to an improved lower bound on the Shannon capacity of $C_7$: $Θ(C_7)\geq 367^{1/5} > 3.2578$. The independent set is found by computer, using the fact that the set $\{t \cdot (1,7,7^2,7^3,7^4) \,\, | \,\, t \in \mathbb{Z}_{382}\} \subseteq \mathbb{Z}_{382}^5$ is independent in the fifth strong product power of the circular graph $C_{108,382}$. Here the circular graph $C_{k,n}$ is the graph with vertex set $\mathbb{Z}_{n}$, the cyclic group of order $n$, in which two distinct vertices are adjacent if and only if their distance (mod $n$) is strictly less than $k$.

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arXiv2018

On the circular chromatic number of a subgraph of the Kneser graph

Let $n,k,r$ be positive integers with $n \geq rk$ and $r \geq 2$. Consider a circle $C$ with~$n$ points~$1,\ldots,n$ in clockwise order. The $r$-stable \emph{interlacing graph} $\text{IG}_{n,k}^{(r)}$ is the graph with vertices corresponding to $k$-subsets $S$ of $\{1,...,n\}$ such that any two distinct points in~$S$ have distance at least~$r$ around the circle, and edges between~$k$-subsets $P$ and $Q$ if they \emph{interlace}: after removing the points in~$P$ from $C$, the points in~$Q$ are in different connected components. In this paper we prove that the circular chromatic number of $\text{IG}_{n,k}^{(r)}$ is equal to $ n/k $ (hence the chromatic number is $\lceil n/k \rceil$) and that its circular clique number is also $ n/k $. Furthermore, we show that its independence number is $\binom{n-(r-1)k-1}{k-1}$, thereby strengthening a result by Talbot.

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arXiv2017

Sum-perfect graphs

Inspired by a famous characterization of perfect graphs due to Lovász, we define a graph $G$ to be sum-perfect if for every induced subgraph $H$ of $G$, $α(H) + ω(H) \geq |V(H)|$. (Here $α$ and $ω$ denote the stability number and clique number, respectively.) We give a set of $27$ graphs and we prove that a graph $G$ is sum-perfect if and only if $G$ does not contain any of the graphs in the set as an induced subgraph.

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arXiv2017

New nonbinary code bounds based on divisibility arguments

For $q,n,d \in \mathbb{N}$, let $A_q(n,d)$ be the maximum size of a code $C \subseteq [q]^n$ with minimum distance at least $d$. We give a divisibility argument resulting in the new upper bounds $A_5(8,6) \leq 65$, $A_4(11,8)\leq 60$ and $A_3(16,11) \leq 29$. These in turn imply the new upper bounds $A_5(9,6) \leq 325$, $A_5(10,6) \leq 1625$, $A_5(11,6) \leq 8125$ and $A_4(12,8) \leq 240$. Furthermore, we prove that for $μ,q \in \mathbb{N}$, there is a 1-1-correspondence between symmetric $(μ,q)$-nets (which are certain designs) and codes $C \subseteq [q]^{μq}$ of size $μq^2$ with minimum distance at least $μq - μ$. We derive the new upper bounds $A_4(9,6) \leq 120$ and $A_4(10,6) \leq 480$ from these `symmetric net' codes.

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arXiv2017

Semidefinite programming bounds for constant weight codes

For nonnegative integers $n,d,w$, let $A(n,d,w)$ be the maximum size of a code $C \subseteq \mathbb{F}_2^n$ with constant weight $w$ and minimum distance at least $d$. We consider two semidefinite programs based on quadruples of code words that yield several new upper bounds on $A(n,d,w)$. The new upper bounds imply that $A(22,8,10)=616$ and $A(22,8,11)=672$. Lower bounds on $A(22,8,10)$ and $A(22,8,11)$ are obtained from the $(n,d)=(22,7)$ shortened Golay code of size $2048$. It can be concluded that the shortened Golay code is a union of constant weight $w$ codes of sizes $A(22,8,w)$.

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arXiv2017

Improved upper bound on A(18,8)

For nonnegative integers $n$ and $d$, let $A(n,d)$ be the maximum cardinality of a binary code of length $n$ and minimum distance at least $d$. We consider a slight sharpening of the semidefinite programming bound of Gijswijt, Mittelmann and Schrijver, and obtain that $A(18,8)\leq 70$.

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