Two equivalent measures on weighted hypergraphs

Abstract

Let H = ( N , E , w ) be a hypergraph with a node set N = { 0 , 1 , … , n - 1 } , a hyperedge set E ⊆ 2 N , and real edge-weights w ( e ) for e ∈ E . Given a convex n-gon P in the plane with vertices x 0 , x 1 , … , x n - 1 which are arranged in this order clockwisely, let each node i ∈ N correspond to the vertex x i and define the area A P ( H ) of H on P by the sum of the weighted areas of convex hulls for all hyperedges in H. For 0 ⩽ i < j < k ⩽ n - 1 , a convex three-cut C ( i , j , k ) of N is { { i , … , j - 1 } , { j , … , k - 1 } , { k , … , n - 1 , 0 , … , i - 1 } } and its size c H ( i , j , k ) in H is defined as the sum of weights of edges e ∈ E such that e contains at least one node from each of { i , … , j - 1 } , { j , … , k - 1 } and { k , … , n - 1 , 0 , … , i - 1 } . We show that the following two conditions are equivalent: • A P ( H ) ⩽ A P ( H ′ ) for all convex n-gons P. • c H ( i , j , k ) ⩽ c H ′ ( i , j , k ) for all convex three-cuts C ( i , j , k ) . From this property, a polynomial time algorithm for determining whether or not given weighted hypergraphs H and H ′ satisfy “ A P ( H ) ⩽ A P ( H ′ ) for all convex n-gons P” is immediately obtained.