Combinatorial optimization with 2-joins

A 2-join is an edge cutset that naturally appears in decomposition of several classes of graphs closed under taking induced subgraphs, such as perfect graphs and claw-free graphs. In this paper we construct combinatorial polynomial time algorithms for finding a maximum weighted clique, a maximum weighted stable set and an optimal coloring for a class of perfect graphs decomposable by 2-joins: the class of perfect graphs that do not have a balanced skew partition, a 2-join in the complement, nor a homogeneous pair. The techniques we develop are general enough to be easily applied to finding a maximum weighted stable set for another class of graphs known to be decomposable by 2-joins, namely the class of even-hole-free graphs that do not have a star cutset. We also give a simple class of graphs decomposable by 2-joins into bipartite graphs and line graphs, and for which finding a maximum stable set is NP-hard. This shows that having holes all of the same parity gives essential properties for the use of 2-joins in computing stable sets.

💡 Research Summary

The paper investigates the use of 2‑joins—specific edge cutsets that appear naturally in the decomposition of several hereditary graph classes—to design purely combinatorial polynomial‑time algorithms for three fundamental optimization problems: maximum weighted clique, maximum weighted stable set, and optimal coloring. The authors focus on two families of graphs. The first is the class of perfect (Berge) graphs that lack three structural obstacles: a balanced skew partition, a 2‑join in the complement, and a homogeneous pair. The second is the class of even‑hole‑free graphs without a star cutset. Both classes are known to be decomposable by 2‑joins, but prior work had only used 2‑joins for recognition, not for optimization, largely because strong cutsets (star cutsets, balanced skew partitions, etc.) were difficult to handle algorithmically.

The core technical contribution is the development of “extreme 2‑joins,” a refined notion of 2‑join where at least one side of the decomposition is already a basic graph (bipartite, line graph of a bipartite graph, path‑cobipartite, or path‑double‑split, together with their complements). The authors prove that any non‑basic graph in the considered classes possesses an extreme 2‑join. This property guarantees that a decomposition tree can be built where every internal node has a child that is a leaf (i.e., a basic graph). Consequently, the recursion depth is linear in the size of the input, preventing exponential blow‑up.

For the maximum weighted clique problem, the authors show how to assign weights to the vertices of the two sides of an extreme 2‑join so that the optimal clique in the whole graph can be reconstructed from optimal cliques in the two subgraphs. Since cliques can be found in polynomial time in each basic graph (e.g., via maximum matching in bipartite graphs or line‑graph techniques), the overall algorithm runs in polynomial time for the whole class.

The stable‑set problem is more delicate because the parity of holes plays a crucial role. The paper proves two lemmas that describe a very special overlap between a maximum weighted stable set and a 2‑join when all holes have the same parity (either all even or all odd). Using these lemmas, the authors construct “extensions of basic graphs,” which are not themselves basic but retain the Berge property while possibly introducing balanced skew partitions. To avoid breaking the class, they employ a two‑phase decomposition: first they use classical class‑preserving blocks to build the tree, then they replace certain nodes with the new extension blocks, carefully preserving the ability to compute optimal stable sets. This yields a polynomial‑time algorithm for maximum weighted stable sets in both target classes.

For coloring, the authors invoke the classic method for perfect graphs: given subroutines for maximum cliques and maximum stable sets, one can iteratively color the graph in polynomial time. Since both subroutines are now available for the considered classes, optimal coloring follows.

To illustrate the limits of the approach, the authors present a simple graph class that decomposes via extreme 2‑joins into bipartite graphs and line graphs of cycles with a single chord. Despite its structural similarity to Berge graphs, they prove that finding a maximum stable set in this class is NP‑hard. This demonstrates that the parity condition on holes (all holes having the same parity) is essential for the success of the 2‑join based algorithms.

In summary, the paper provides the first purely combinatorial polynomial‑time algorithms for maximum weighted clique, maximum weighted stable set, and optimal coloring in (i) Berge graphs without balanced skew partitions, complement 2‑joins, or homogeneous pairs, and (ii) even‑hole‑free graphs without star cutsets. The techniques hinge on extreme 2‑joins, careful weight redistribution, and a two‑stage decomposition that respects the structural constraints of the graph classes. The work also clarifies why parity of holes is a critical ingredient, by exhibiting an NP‑hardness result when this condition is dropped.