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- W1844342971 abstract "For the problem of maximizing a quadratic function f(x) over the binary n-cube Bn, we define a set Π of “upper planes”, i.e. linear functions π(x) such that π(x) ≥ f(x) for all x ∈ Bn. A best upper plane π* is a π ∈ Π for which the maximum value of π(x) is minimal. After introducing several naturally defined sets Π, it is shown that the set of best upper planes for all these Π′ s is the same, and can be efficiently obtained by solving a continuous vertex packing problem in a graph G having a special structure. The gap between the maxima over Bn of π*(x) and of f(x) is shown to be equal to the integrality gap for the vertex packing problem of G; a class of gap-free functions (properly including supermodular ones) is exhibited. Among the best upper planes there is a “master plane” μ(x) with the property that a variable takes the value 1 (0) in all optimal solutions of the continuous vertex packing problem of G if and only if the corresponding coefficient of μ is strictly positive (strictly negative). If this happens, then the same variable is seen also to take the above constant value in all the maximizing points of f(x) in Bn. As an application, the special class of quadratic pseudo-boolean functions describing the stable sets of a graph is examined. It is shown that the method detects at least one vertex belonging to, or absent from, all maximum stable sets (i.e. the master plane does not reduce to a constant) if and only if the graph has no perfect 2-matching." @default.
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- W1844342971 date "1981-01-01" @default.
- W1844342971 modified "2023-10-14" @default.
- W1844342971 title "UPPER PLANES OF QUADRATIC 0–1 FUNCTIONS AND STABILITY IN GRAPHS" @default.
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- W1844342971 doi "https://doi.org/10.1016/b978-0-12-468662-5.50019-7" @default.
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