Optimizing alphabet reduction pairs of arrays
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866909224765751296 |
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| author | Culus, Jean-François Toulouse, Sophie |
| author_facet | Culus, Jean-François Toulouse, Sophie |
| contents | In [1], we introduced a family of combinatorial designs, which we call "alphabet reduction pairs of arrays", ARPAs for short. These designs depend on three integer parameters $q, p \leq q, k\leq p$: $q$ is the size of the symbol set $\{0, 1 ,\ldots, q -1\}$ in which the coefficients of the arrays take their values; $p$ is the maximum number of distinct symbols allowed in a row of the second array of the pair; $k$ is the larger integer for which the two arrays of the pair coincide -- up to the order of their rows -- on any $k$-ary subset of their columns. The first array must contain at least one occurrence of the word $0\ 1\ \ldots\ q -1$. Intuitively, the idea is to cover "as many as possible" occurrences of this word of $q$ symbols with "as few as possible" words of at most $p$ different symbols.
These designs are related to the approximability of "Constraint Satisfaction Problems with bounded constraint arity", known as $k\,$CSPs. In this context, we are particularly interested in ARPAs in which the frequency of the word $0\ 1\ \ldots\ q -1$ is maximal.
We introduce a seemingly simpler family of combinatorial designs as "Cover pairs of arrays" (CPAs). The arrays of a CPA take Boolean coefficients, and must still coincide on any $k$-ary subset of their columns. The purpose is, as it were, to cover "as many as possible" occurrences of the word of $q$ ones using "as few as possible" $q$-length Boolean words of weight at most $p$. We show that, when it comes to maximizing the frequency of the words $0\ 1\ \ldots\ q -1$ in ARPAs and $1\ 1\ \ldots\ 1$ in CPAs, ARPAs and CPAs are equivalent. We prove the optimality of the ARPAs given in [1] for the case $p =k$. In addition, we provide optimal ARPAs for the cases $k =1$ and $k =2$. We emphasize the fact that both families of combinatorial designs are related to the approximability of $k\,$CSPs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10930 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Optimizing alphabet reduction pairs of arrays Culus, Jean-François Toulouse, Sophie Combinatorics Discrete Mathematics In [1], we introduced a family of combinatorial designs, which we call "alphabet reduction pairs of arrays", ARPAs for short. These designs depend on three integer parameters $q, p \leq q, k\leq p$: $q$ is the size of the symbol set $\{0, 1 ,\ldots, q -1\}$ in which the coefficients of the arrays take their values; $p$ is the maximum number of distinct symbols allowed in a row of the second array of the pair; $k$ is the larger integer for which the two arrays of the pair coincide -- up to the order of their rows -- on any $k$-ary subset of their columns. The first array must contain at least one occurrence of the word $0\ 1\ \ldots\ q -1$. Intuitively, the idea is to cover "as many as possible" occurrences of this word of $q$ symbols with "as few as possible" words of at most $p$ different symbols. These designs are related to the approximability of "Constraint Satisfaction Problems with bounded constraint arity", known as $k\,$CSPs. In this context, we are particularly interested in ARPAs in which the frequency of the word $0\ 1\ \ldots\ q -1$ is maximal. We introduce a seemingly simpler family of combinatorial designs as "Cover pairs of arrays" (CPAs). The arrays of a CPA take Boolean coefficients, and must still coincide on any $k$-ary subset of their columns. The purpose is, as it were, to cover "as many as possible" occurrences of the word of $q$ ones using "as few as possible" $q$-length Boolean words of weight at most $p$. We show that, when it comes to maximizing the frequency of the words $0\ 1\ \ldots\ q -1$ in ARPAs and $1\ 1\ \ldots\ 1$ in CPAs, ARPAs and CPAs are equivalent. We prove the optimality of the ARPAs given in [1] for the case $p =k$. In addition, we provide optimal ARPAs for the cases $k =1$ and $k =2$. We emphasize the fact that both families of combinatorial designs are related to the approximability of $k\,$CSPs. |
| title | Optimizing alphabet reduction pairs of arrays |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2406.10930 |