This document summarizes a solution to Problem J from the ICPC Asia::Tokyo 2014 competition. The problem involves minimizing the cost of choosing products for an exhibition by considering the costs with and without choosing a specific product 1. The objective function can be computed in O(n5logn) time by considering combinations of points in XYZ-space and finding the minimum on the convex hull. For fixed parameters, the objective function can also be minimized by considering all combinations and sorting. The overall minimizer is found by trying all edges of the feasible domain cube [0,1]3.

1. ICPC Asia::Tokyo 2014
Problem J – Exhibition

2. J: Exhibition – Solution (1/3)
Objective function can be written as follows:
★ min cost when we
choose product 1
★ min cost when we do
not choose product 1
(★) does not depend on α,β,γ. How to compute it?

3. J: Exhibition – Solution (2/3)
★
• Let F be combin(n-1, k) points in the XYZ-space
consisting of (Σi∈S xi, Σi∈S yi, Σi∈S zi) for each S.
• Minimizer of (★) is a vertex of convex hull of F.
• Let S* be minimizer. Then there exists p,q ∈ (0,1)
with p+q < 1 s.t.
• If we know p,q, ↑ can be minimized by sorting.
• There are at most O(n4) candidates for such p,q.
So we can compute (★) in O(n5logn) time.

4. J: Exhibition – Solution (3/3)
★
• For fixed α,β,γ, (★) can be minimized in the similar
manner. (We consider at most O(n4) ways for S.)
• Feasible domain is complement of
a union of convex sets..
• Minimizer is achieved at the edge of cube [0, 1]3.
Try all of edges and take the best one.

5. J: Exhibition – Summary
Acceptance:
• !#$%&()*+-./:;<=>?@[]^_`{|}~ (The University of Tokyo)
• H2O(Peking University)