Abstract. In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within the framework of constrained polynomial optimization problems, show how the degree principle for the symmetric group can be computationally exploited and also propose some methods to efficiently compute in the geometric quotient.

We prove that there is a constantcsuch that, for each positive integerk, every (2k+ 1) × (2k+ 1) arrayAon the symbols (1,. . .,2k+1) with at mostc(2k+1) symbols in every cell, and each symbol repeated at mostc(2k+1) times in every row and column isavoidable; that is, there is a (2k+1) × (2k+1) Latin squareSon the symbols 1,. . .,2k+1 such that, for eachi,j∈ {1,. . .,2k+1}, the symbol in position (i,j) ofSdoes not appear in the corresponding cell inA. This settles the last open case of a conjecture by Häggkvist. Using this result, we also show that there is a constant ρ, such that, for any positive integern, if each cell in ann×narrayBis assigned a set ofm≤ ρnsymbols, where each set is chosen independently and uniformly at random from {1,. . .,n}, then the probability thatBis avoidable tends to 1 asn→ ∞.

An n × n partial Latin square P is called α-dense if each row and column has at most αn non-empty cells and each symbol occurs at most αn times in P. An n × n array A where each cell contains a subset of {1,…, n} is a (βn, βn, βn)-array if each symbol occurs at most βn times in each row and column and each cell contains a set of size at most βn. Combining the notions of completing partial Latin squares and avoiding arrays, we prove that there are constants α, β > 0 such that, for every positive integer n, if P is an α-dense n × n partial Latin square, A is an n × n (βn, βn, βn)-array, and no cell of P contains a symbol that appears in the corresponding cell of A, then there is a completion of P that avoids A; that is, there is a Latin square L that agrees with P on every non-empty cell of P, and, for each i, j satisfying 1 ≤ i, j ≤ n, the symbol in position (i, j) in L does not appear in the corresponding cell of A.

An $(m,m,m)$-array of order $n$ is an $n\times n$ array such that each cell is assigned a set of at most $m$ symbols from $\left\{1,\dots ,n\right\}$ such that no symbol occurs more than $m$ times in any row or column. An $ (m,m,m)$-array is called avoidable if there exists a Latin square such that no cell in the Latin square contains a symbol that also belongs to the set assigned to the corresponding cell in the array. We show that there is a constant $\gamma $ such that if $m\le\gamma 2^k$ and $k\ge14$, then any $(m,m,m)$-array of order $n=2^k$ is avoidable. Such a constant $\gamma$ has been conjectured to exist for all $n$ by Häggkvist.