Consider the following “gold-digger problem” (GDP). You are given a map of a territory

consisting of a set of towns connected by trails. Each trail (u, v) connecting towns u and v

is labeled with a dollar value w(u, v), which is the value of the gold you will find along that

trail. You can traverse a trail as often as you want, but you only get the value of the trail

the first time you traverse it; subsequent traversals have no value. Each town v has a lodging

cost c(v), which you pay each time you enter the town.

An expedition is a cyclic path that starts and ends at a given town. An expedition has profit

k if the total value of gold found, minus the cost of the all the town visits, is k. The goal is

to find an expedition of maximum profit.

Either show that there exists a polynomial-time algorithm for GDP, or show that the corresponding

decision problem is NP-complete. If you want to show that the problem is hard, you

may reduce from any of: SAT, 3-CNF-SAT, CLIQUE, VERTEX-COVER, SUBSET-SUM,

PARTITION, HAM-CYCLE, TSP.

consisting of a set of towns connected by trails. Each trail (u, v) connecting towns u and v

is labeled with a dollar value w(u, v), which is the value of the gold you will find along that

trail. You can traverse a trail as often as you want, but you only get the value of the trail

the first time you traverse it; subsequent traversals have no value. Each town v has a lodging

cost c(v), which you pay each time you enter the town.

An expedition is a cyclic path that starts and ends at a given town. An expedition has profit

k if the total value of gold found, minus the cost of the all the town visits, is k. The goal is

to find an expedition of maximum profit.

Either show that there exists a polynomial-time algorithm for GDP, or show that the corresponding

decision problem is NP-complete. If you want to show that the problem is hard, you

may reduce from any of: SAT, 3-CNF-SAT, CLIQUE, VERTEX-COVER, SUBSET-SUM,

PARTITION, HAM-CYCLE, TSP.

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