Minimum cell connection and separation in line segment arrangements

We study the complexity of the following cell connection and separation problems in segment arrangements. Given a set of straight-line segments in the plane and two points $a$ and $b$ in different cells of the induced arrangement: (i) compute the minimum number of segments one needs to remove so that there is a path connecting $a$ to $b$ that does not intersect any of the remaining segments; (ii) compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell; (iii) compute the minimum number of segments one needs to retain so that any path connecting $a$ to $b$ intersects some of the retained segments. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a linear-time algorithm for a variant of problem (i) where the path connecting $a$ to $b$ must stay inside a given polygon $P$ with a constant number of holes, the segments are contained in $P$, and the endpoints of the segments are on the boundary of $P$. For problem (iii) we provide a cubic-time algorithm.

💡 Research Summary

The paper investigates three natural optimization problems defined on an arrangement of straight‑line segments in the plane, with two distinguished points a and b that lie in different cells of the arrangement.

1. 2‑Cells‑Connection (Problem i) – Find a minimum‑cardinality subset of segments whose removal makes a and b belong to the same cell, i.e., a path from a to b crosses as few segments as possible.
2. All‑Cells‑Connection (Problem ii) – Find a minimum‑cardinality subset of segments whose removal leaves the arrangement with a single cell.
3. 2‑Cells‑Separation (Problem iii) – Find a minimum‑cardinality subset of segments that must be kept so that every a‑b path intersects at least one of the kept segments.

The authors first prove that Problems (i) and (ii) are NP‑hard even when the segments are in general position. The hardness of (i) is shown via a careful reduction from Max‑2‑SAT, which also yields APX‑hardness, indicating that no PTAS exists unless P=NP. The hardness of (ii) follows from a reduction to the feedback‑vertex‑set problem on the segment intersection graph; this holds even when the segments do not intersect each other at all. Moreover, when any three segments intersect only at a common endpoint, the authors obtain a fixed‑parameter tractable (FPT) algorithm for (i) with the number of proper crossings as the parameter.

For Problem (iii) the paper presents a polynomial‑time exact algorithm. The key idea is to build the intersection graph G = (S, E) of the segments, assign to each edge e = {s, s′} the weight w(s)+w(s′) (weights may be zero), and then search for a minimum‑weight cycle whose induced polygonal curve separates a from b. The authors prove that the family of cycles defined by fundamental cycles of shortest‑path trees satisfies the “3‑path condition” (a homological property originally due to Thomassen). This condition guarantees that a shortest separating cycle must be a fundamental cycle τ(T_r, e) for some root segment r and some non‑tree edge e. By constructing a shortest‑path tree T_r for each r and evaluating all non‑tree edges, the optimal cycle can be found in O(nk + n² log n) time, where n = |S| and k is the number of intersecting segment pairs. When segment weights are 0/1, the running time improves to O(nk + n²).

The paper also studies a restricted version of (i) where all segments lie inside a polygon P that may contain a constant number of holes, and each segment’s endpoints lie on the boundary of P. Although the unrestricted version remains NP‑hard, the authors give a linear‑time algorithm for this restricted case. The algorithm groups segments into clusters using homotopy classes; each cluster is either entirely contained in an optimal solution or completely excluded. Because the number of holes is constant, the number of distinct homotopy classes (and thus clusters) is also constant, allowing a simple sweep‑line / union‑find based procedure to run in O(n) time.

Finally, for Problem (iii) the authors provide a cubic‑time exact algorithm based on a reduction to a minimum cut in the segment intersection graph, which can be solved by standard max‑flow techniques.

The paper situates its contributions within a broader literature on geometric barrier problems, sensor‑network coverage, and related work on unit disks and unit segments. It highlights that while many barrier‑construction problems admit polynomial algorithms in restricted domains (e.g., simple polygons, unit‑disk graphs), the segment‑arrangement variants considered here exhibit a richer complexity landscape, with both hardness results and efficient algorithms for important special cases. The work opens avenues for further research on approximation algorithms, higher‑dimensional extensions, and dynamic or online versions of the problems.