Optimal Cuts and Bisections on the Real Line in Polynomial Time

The exact complexity of geometric cuts and bisections is the longstanding open problem including even the dimension one. In this paper, we resolve this problem for dimension one (the real line) by designing an exact polynomial time algorithm. Our results depend on a new technique of dealing with metric equalities and their connection to dynamic programming. The method of our solution could be also of independent interest.

💡 Research Summary

The paper tackles two classic geometric optimization problems—minimum cut and bisection—restricted to points on the real line. While in higher dimensions both problems are known to be NP‑hard, the exact computational status in one dimension has remained open. The authors close this gap by presenting a deterministic polynomial‑time algorithm that finds optimal solutions for both problems on the line.

The key technical contribution is the introduction of “metric equalities,” a notion that captures when the distance between two points equals a specific value. On the real line, after sorting the points, equal distances form contiguous intervals (called equal‑intervals). Within each such interval the cost of a cut does not change, which allows the algorithm to treat many potential cut positions as a single aggregated state.

The algorithm proceeds as follows. First, the input points are sorted in O(n log n) time. A preprocessing step builds a hash‑based index of all equal‑intervals, which can be done in linear time because each distance appears at most O(n) times. The dynamic programming (DP) table DP