# A 3/4-Approximation Algorithm for Multiple Subset Sum by Caprara A. PDF

By Caprara A.

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A bipartition of an undirected graph G is a partition of the vertex set V (G) = . A ∪ B such that the subgraphs induced by A and B are both empty. A graph is called bipartite if it has a bipartition. The simple bipartite graph G with V (G) = . A ∪ B, |A| = n, |B| = m and E(G) = {{a, b} : a ∈ A, b ∈ B} is denoted by . K n,m (the complete bipartite graph). When we write G = (A ∪ B, E(G)), we mean that G[A] and G[B] are both empty. 27. (K¨onig [1916]) An undirected graph is bipartite if and only if it contains no circuit of odd length.

4. (b)⇒(c): We have that |δ − (v)| = 1 for all v ∈ V (G) \ {r }. So for any v we have an r -v-path (start at v and always follow the entering edge until r is reached). 3(b). (e)⇒(f): The minimality in (e) implies δ − (r ) = ∅. 3(b) there is an r -v-path for all v. Suppose there are two r -v-paths P and Q for some v. Let e be the last edge of P that does not belong to Q. Then after deleting e, every vertex is still reachable from r . 3(b) this contradicts the minimality in (e). (f)⇒(g)⇒(a): trivial ✷ A cut in an undirected graph G is an edge set of type δ(X ) for some ∅ = X ⊂ V (G).

Let G be a connected planar undirected graph with arbitrary embedding. The edge set of any circuit in G corresponds to a minimal cut in G ∗ , and any minimal cut in G corresponds to the edge set of a circuit in G ∗ . Proof: Let = (ψ, (Je )e∈E(G) ) be a ﬁxed planar embedding of G. Let C be a circuit in G. 30, R2 \ e∈E(C) Je splits into exactly two connected regions. Let A and B be the set of. faces of in the inner and outer region, ∗ ∗ respectively. We have V (G ) = A ∪ B and E G (A, B) = {e∗ : e ∈ E(C)}.

### A 3/4-Approximation Algorithm for Multiple Subset Sum by Caprara A.

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