Fast FPT algorithms for vertex subset and vertex partitioning problems using neighborhood unions

We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods across a cut of a graph. Boolean-width is similar to rank-width, which is related to the number of $GF[2]$-sums (1+1=0) of neighborhoods instead of the boolean-sums (1+1=1) used for boolean-width. We give algorithms for a large class of NP-hard vertex subset and vertex partitioning problems that are FPT when parameterized by either boolean-width, rank-width or clique-width, with runtime single exponential in either parameter if given the pertinent optimal decomposition. To compare boolean-width versus rank-width or clique-width, we first show that for any graph, the square root of its boolean-width is never more than its rank-width. Next, we exhibit a class of graphs, the Hsu-grids, for which we can solve NP-hard problems in polynomial time, if we use the right parameter. An $n \times \frac{n}{10}$ Hsu-grid on ${1/10}n^2$ vertices has boolean-width $\Theta(\log n)$ and rank-width $\Theta(n)$. Moreover, any optimal rank-decomposition of such a graph will have boolean-width $\Theta(n)$, i.e. exponential in the optimal boolean-width. A main open problem is to approximate the boolean-width better than what is given by the algorithm for rank-width [Hlin\v{e}n'y and Oum, 2008]

💡 Research Summary

The paper introduces a novel graph width parameter called boolean‑width, which measures the logarithm of the number of distinct unions of neighborhoods that can be formed across a cut of a graph. This contrasts with the well‑studied rank‑width, which counts the number of distinct vectors obtainable by GF(2) (binary) sums of the same neighborhoods. By replacing the GF(2) sum with the Boolean sum (set union), the authors define a cut‑function β_G and the corresponding width β_w(G).

The authors first establish tight relationships between boolean‑width, rank‑width, and clique‑width. Using linear‑algebraic arguments they prove that for any graph G,
 log rw(G) ≤ β_w(G) ≤ ¼ rw(G)² + 5⁄4 rw(G) + log rw(G).
Thus, boolean‑width can be as small as the square‑root of rank‑width, but never larger than a quadratic function of rank‑width. Moreover, the known bounds between rank‑width and clique‑width (rw ≤ cw ≤ 2^{rw+1}−1) imply analogous bounds for boolean‑width (β_w ≤ cw ≤ 2^{β_w+1}−1).

To illustrate the practical impact of these bounds, the paper presents the Hsu‑grid family. An n × n/10 Hsu‑grid has Θ(n²) vertices, boolean‑width Θ(log n) and rank‑width Θ(n). Any optimal rank‑decomposition of such a grid, however, has boolean‑width Θ(n), i.e., exponential in the optimal boolean‑width. This shows that choosing the “right” width parameter can turn an otherwise exponential‑time algorithm into a polynomial‑time one for certain graph classes.

The core algorithmic contribution is a dynamic‑programming framework that works on a given decomposition tree (T,δ). For each edge of T the algorithm enumerates all possible Boolean unions of neighborhoods across the corresponding cut; the number of such unions is at most 2^{β_w(G)}. Using these as DP states, the authors solve a broad class of vertex subset problems (expressed as (σ,ρ)-sets, encompassing Dominating Set, Independent Set, etc.) and vertex partitioning problems (expressed as D_q‑partitions, covering H‑Coloring, H‑Covering, etc.). The running time of these algorithms is O*(2^{O(β_w(G))}), i.e., single‑exponential in boolean‑width. In contrast, rank‑width‑based algorithms for the same problems typically run in O*(2^{O(rw(G)²)}), which can be far larger when β_w(G) ≪ rw(G).

A practical obstacle is that computing an optimal boolean‑width decomposition is currently unknown. The paper overcomes this by showing that any optimal rank‑width decomposition provides a β‑width at most O(rw(G)²). Since optimal rank‑width decompositions can be found in FPT time (O(n³) for a given bound k, per Hliněný and Oum 2008), this yields a polynomial‑time approximation scheme for boolean‑width sufficient for the DP algorithms.

Finally, the authors discuss connections to Boolean matrix theory: β_w(G) corresponds to the logarithm of the number of distinct row spaces of a Boolean matrix derived from the adjacency matrix. They point out that improving bounds on the size of Boolean row spaces would directly improve the exponent in their algorithms. The paper concludes by highlighting open problems, notably the design of algorithms that approximate boolean‑width more tightly than the current rank‑width based approach, and the exploration of Boolean‑matrix techniques to further tighten FPT bounds for a wide range of NP‑hard graph problems.