We consider three graphs, G 7,3 , G 7,4 , and G 7,6 , related to Keller's conjecture in dimension 7. The conjecture is true for this dimension if and only if the size of the largest clique in at least one of the graphs is less than 2 7 = 128. We present an automated method to solve this conjecture by encoding the existence of such a clique as a propositional formula. We apply satisfiability solving combined with symmetrybreaking techniques to determine that no such clique exists. This result implies that every unit cube tiling of R 7 contains a facesharing pair of cubes. Since a faceshare-free unit cube tiling of R 8 exists (which we also verify), this completely resolves Keller's conjecture.
It is important to understand how the outcome of an election can be modified by an agent with control over the structure of the election. Electoral control has been studied for many election systems, but for all studied systems the winner problem is in P, and so control is in NP. There are election systems, such as Kemeny, that have many desirable properties, but whose winner problems are not in NP. Thus for such systems control is not in NP, and in fact we show that it is typically complete for Σ p 2 (i.e., NP NP , the second level of the polynomial hierarchy). This is a very high level of complexity. Approaches that perform quite well for solving NP problems do not necessarily work for Σ p 2 -complete problems. However, answer set programming is suited to express problems in Σ p 2 , and we present an encoding for Kemeny control.
In the context of SAT solvers, Shatter is a popular tool for symmetry breaking on CNF formulas. Nevertheless, little has been said about its use in the context of AllSAT problems: problems where we are interested in listing all the models of a Boolean formula. AllSAT has gained much popularity in recent years due to its many applications in domains like model checking, data mining, etc. One example of a particularly transparent application of AllSAT to other fields of computer science is computational Ramsey theory. In this paper we study the effect of incorporating Shatter to the workflow of using Boolean formulas to generate all possible edge colorings of a graph avoiding prescribed monochromatic subgraphs. Generating complete sets of colorings is an important building block in computational Ramsey theory. We identify two drawbacks in the naïve use of Shatter to break the symmetries of Boolean formulas encoding Ramsey-type problems for graphs: a "blow-up" in the number of models and the generation of incomplete sets of colorings. The issues presented in this work are not intended to discourage the use of Shatter as a preprocessing tool for AllSAT problems in combinatorial computing but to help researchers properly use this tool by avoiding these potential pitfalls. To this end, we provide strategies and additional tools to cope with the negative effects of using Shatter for AllSAT. While the specific application addressed in this paper is that of Ramsey-type problems, the analysis we carry out applies to many other areas in which highly-symmetrical Boolean formulas arise and we wish to find all of their models.
A backbone of a boolean formula F is a collection S of its variables for which there is a unique partial assignment aS such that F[aS] is satisfiable (Monasson et al. 1999; Williams, Gomes, and Selman 2003). This paper studies the nontransparency of backbones. We show that, under the widely believed assumption that integer factoring is hard, there exist sets of boolean formulas that have obvious, nontrivial backbones yet finding the values, aS, of those backbones is intractable. We also show that, under the same assumption, there exist sets of boolean formulas that obviously have large backbones yet producing such a backbone S is intractable. Further, we show that if integer factoring is not merely worst-case hard but is frequently hard, as is widely believed, then the frequency of hardness in our two results is not too much less than that frequency.
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