481 research outputs found

    Ising formulations of many NP problems

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    We provide Ising formulations for many NP-complete and NP-hard problems, including all of Karp's 21 NP-complete problems. This collects and extends mappings to the Ising model from partitioning, covering and satisfiability. In each case, the required number of spins is at most cubic in the size of the problem. This work may be useful in designing adiabatic quantum optimization algorithms.Comment: 27 pages; v2: substantial revision to intro/conclusion, many more references; v3: substantial revision and extension, to-be-published versio

    Completion and deficiency problems

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    Given a partial Steiner triple system (STS) of order nn, what is the order of the smallest complete STS it can be embedded into? The study of this question goes back more than 40 years. In this paper we answer it for relatively sparse STSs, showing that given a partial STS of order nn with at most r≀Ρn2r \le \varepsilon n^2 triples, it can always be embedded into a complete STS of order n+O(r)n+O(\sqrt{r}), which is asymptotically optimal. We also obtain similar results for completions of Latin squares and other designs. This suggests a new, natural class of questions, called deficiency problems. Given a global spanning property P\mathcal{P} and a graph GG, we define the deficiency of the graph GG with respect to the property P\mathcal{P} to be the smallest positive integer tt such that the join Gβˆ—KtG\ast K_t has property P\mathcal{P}. To illustrate this concept we consider deficiency versions of some well-studied properties, such as having a KkK_k-decomposition, Hamiltonicity, having a triangle-factor and having a perfect matching in hypergraphs. The main goal of this paper is to propose a systematic study of these problems; thus several future research directions are also given

    Minimum degrees for powers of paths and cycles

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    We study minimum degree conditions under which a graph GG contains kthk^{th} power of paths and cycles of arbitrary specified lengths. We determine precise thresholds, assuming that the order of G is large. This extends a result of Allen, B\"ottcher and Hladk\'y concerning the containment of squares of paths and squares of cycles of arbitrary specified lengths and settles a conjecture of theirs in the affirmative.Comment: 69 pages, 3 figures. arXiv admin note: text overlap with arXiv:0906.3299 by other author
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