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By T-H. Hubert Chan, Minming Li, Lusheng Wang

This e-book constitutes the refereed complaints of the tenth overseas convention on Combinatorial Optimization and purposes, COCOA 2016, held in Hong Kong, China, in December 2016.

The 60 complete papers integrated within the publication have been conscientiously reviewed and chosen from 122 submissions. The papers are prepared in topical sections resembling graph thought, geometric optimization, complexity and knowledge constitution, combinatorial optimization, and miscellaneous.

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Extra info for Combinatorial Optimization and Applications: 10th International Conference, COCOA 2016, Hong Kong, China, December 16–18, 2016, Proceedings

Example text

In what follows, we prove that umeac is NPcomplete. Notice that if |S ∪ T | ≤ C log2 n for some constant C, then it follows from Theorems 1, 4, 5, and 6 that umeac can be solved in polynomial time by enumerating all subsets of S ∪ T . We will prove the NP-completeness of umeac by reduction from Partition. In this problem, we are given a finite set I and a vector π in ZI+ such that π(I) is even. Then, the goal is to decide whether there exists a subset J of I such that π(J) = π(I \J). It is well known [14] that Partition is NP-complete.

Then, Theorem 1 implies that d ∈ Fi (w) if and only if w ow i (X) − d(X) ≥ 0 for a subset X of S ∪ T minimizing oi (X) − d(X) among w all subsets of S ∪ T . 2 Theorem 7. The problem ime is NP-complete. Proof. Theorems 1, 4, 5, and 6 imply that ime is in NP. We prove that ime is NP-complete by reduction from dpdc. Assume that we are given an instance of dpdc such that h = 0 and i (a) ∈ {0, 1} for every integer i in {1, 2} and every arc a in L. Then, we construct an instance of ime as follows. Define V := N ∪ {s∗ }, where s∗ is a new vertex.

A vector d in RS∪T is called an allocation, if d(v) ≥ 0 for every vertex v in S and d(v) ≤ 0 for every vertex v in T . For each integer i in {1, 2}, each allocation d in RS∪T , and each vector w in RA + , d is said to be (i, w)-feasible, if there exists a function f : A × [Θ] → R+ satisfying the following conditions. (D1) Let a and θ be an arc in A and an integer in [Θ], respectively. - If θ ≤ Θ − τi (a), then f (a, θ) ≤ qi · w(a). - If θ > Θ − τi (a), then f (a, θ) = 0. (D2) Let v and θ be a vertex in V and an integer in [Θ], respectively.

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