Abstract
The Grothendieck constant κ ( G ) of a graph G = ( [ n ] , E ) is the integrality gap of the canonical semidefinite relaxation of the integer program max x ∈ { ± 1 } n ∑ i j ∈ E w i j x i ⋅ x j , replacing ± 1 variables by unit vectors. We show that κ ( G ) = g / ( g − 2 ) cos ( π / g ) ≤ 3 / 2 when G has no K 5 -minor and girth g ; moreover, κ ( G ) ≤ κ ( K k ) if the cut polytope of G is defined by inequalities supported by at most k points; lastly the worst case ratio of clique-web inequalities is bounded by 3.
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