Took me a second to even remember what the 3-partition problem was.
Start by clearly defining the 3-partition problem and its NP-complete nature, then outline a solution approach such as dynamic programming for small sums or backtracking with pruning for general cases. Emphasize the importance of considering constraints and discussing trade-offs between exact and heuristic methods.
Pro tip: Mention that while 3-partition is NP-complete, practical instances often have small sums or can be solved with pseudo-polynomial DP; also note that the problem is strongly NP-complete, so no FPTAS exists unless P=NP.
State that given a multiset of 3m positive integers, the goal is to partition it into m triples each summing to the same target T (total sum / m). Clarify that it's a decision problem.
Explain that 3-partition is NP-complete in the strong sense, meaning it remains NP-complete even when numbers are bounded by a polynomial in m. This rules out pseudo-polynomial algorithms unless P=NP.
If the target sum T is small, use dynamic programming with state (index, sum1, sum2, sum3) or a 2D DP over sums to track achievable subset sums. Complexity O(n * T^2) or O(n * T^3) depending on implementation.
For general cases, use backtracking: sort numbers descending, try to place each number into one of m bins, prune when a bin exceeds T or when remaining numbers cannot fill bins. Use symmetry breaking to avoid duplicate states.
Since exact solution is hard, discuss greedy heuristics (e.g., first-fit decreasing) or approximation algorithms, noting that no constant-factor approximation exists for 3-partition unless P=NP.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.