In the reduction from 3-SAT to Hamiltonian Path, how many nodes are in the graph we construct, assuming the 3-SAT problem has n variables and k clauses?

Question

In the reduction from 3-SAT to Hamiltonian Path, how many nodes are in the graph we construct, assuming the 3-SAT problem has n variables and k clauses?

BlackTom AI · Accepted Answer

The question asks about the size of the constructed graph in the reduction from 3-SAT to Hamiltonian Path, in terms of the input parameters n (number of variables) and k (number of clauses).

First, consider typical gadget-based reductions: you create a fixed-size gadget for each variable and a fixed-size gadget for each clause, and connect these gadgets in a way that preserves satisfiability. Since each gadget has constant size (i.e., its number of nodes does not depend on n or k beyond the count of gadgets), the total number of nodes is proportional to the number of gadgets, which is linear in n plus linear in k.

Therefore, the total node count is O(n + k). This reflects the standard result that the size of the constructed graph in such reductions grows linearly with the input size of the 3-SAT instance (the sum of variables and clauses), rather than as their product.

Option analysis:
- If the option were O(nk): this would imply the graph size grows with the product of the number of variables and the number of clauses. That would typically occur if there were a dense or interdependent construction where each variable–clause pair added unique nodes. However, in the canonical gadget-based reductions to Hamiltonian Path, each gadget is constant-sized, and gadgets are connected in a way that yields a total size linear in n + k, not their product. Hence O(nk) overestimates the size in the usual construction.

In short, the conventional reduction yields a graph with Theta(n + k) nodes, not Theta(nk). The given option O(n*k) is not correct for the standard reduction framework. The reasoning hinges on the fact that each variable gadget and each clause gadget contribute a constant amount of nodes, and the totals add up rather than multiply.

CPSC_V 320 201/202/203 2024W2 Reading Quiz #11 (NP-Completeness 2)

In the reduction from 3-SAT to Hamiltonian Path, how many nodes are in the graph we construct, assuming the 3-SAT problem has n variables and k clauses?

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