Inapproximability of Treewidth, One-Shot Pebbling, and Related Layout Problems

We study the approximability of a number of graph problems: treewidth and pathwidth of graphs, one-shot black (and black-white) pebbling costs of directed acyclic graphs, and a variety of different graph layout problems such as minimum cut linear arrangement and interval graph completion. We show that, assuming the recently introduced Small Set Expansion Conjecture, all of these problems are hard to approximate within any constant factor.

💡 Research Summary

This paper investigates the approximability of several fundamental graph problems—treewidth, pathwidth, one‑shot black and black‑white pebbling, Minimum Cut Linear Arrangement (MCLA), and Interval Graph Completion (IGC)—under the assumption of the Small Set Expansion (SSE) Conjecture. The authors show that, if the SSE Conjecture holds, none of these problems admit a polynomial‑time approximation algorithm with any constant‑factor guarantee; in other words, they are SSE‑hard to approximate within any constant factor.

The work begins by reviewing the status of treewidth and pathwidth. Both parameters measure how close a graph is to a tree or a path, respectively, and are central to many algorithmic applications (e.g., dynamic programming on bounded‑width graphs, probabilistic inference). Exact computation is NP‑hard, and the best known polynomial‑time approximations achieve ratios of O(√log n · log n). Prior to this paper, only additive‑error hardness results were known, and no relative‑approximation hardness was established. By constructing instances where either the pathwidth is very small (≤ c·|V|) or the treewidth is very large (≥ α·c·|V|) and proving that distinguishing these cases is SSE‑hard, the authors obtain the first constant‑factor inapproximability for both treewidth and pathwidth.

Next, the paper turns to one‑shot pebbling on directed acyclic graphs (DAGs). In the black version, a pebble may be placed on a node only when all its immediate predecessors already contain pebbles; in the black‑white version, white pebbles can be placed freely but removed only under the same predecessor condition. The “one‑shot” restriction forces each vertex to receive a pebble at most once, dramatically increasing the required pebble count. While O(√log n · log n) approximation algorithms are known for these problems, no hardness results existed. The authors prove that even for single‑sink DAGs with maximum indegree 2, approximating the one‑shot black pebbling cost within any constant factor is SSE‑hard; the same holds for the black‑white variant without degree restrictions. The proof exploits the equivalence between one‑shot pebbling and a certain graph layout ordering problem: the optimal pebbling order directly yields a vertex permutation that minimizes a layout objective.

The third contribution addresses layout problems. MCLA asks for a vertex permutation that minimizes the maximum number of edges crossing any cut point; IGC seeks the smallest supergraph that is an interval graph, which can be expressed as minimizing, over permutations, the sum of the longest forward edge from each vertex. Both problems have O(log n · √log n) (MCLA) or O(√log n · log log n) (IGC) approximation algorithms, but no hardness results were known. By reducing from the SSE‑hard Minimum Linear Arrangement (MLA) instance and carefully preserving expansion properties, the authors show that both MCLA and IGC are SSE‑hard to approximate within any constant factor.

In fact, the paper defines a family of eight related layout problems generated by three natural variations (different objective functions, directed vs. undirected graphs, and topological ordering constraints). Using a unified reduction framework, it establishes super‑constant SSE‑hardness for all of them simultaneously.

The authors also explain how treewidth corresponds to “elimination width,” a layout measure, and how pathwidth corresponds to the layout problem underlying one‑shot black‑white pebbling. These connections allow the hardness results for layout problems to be transferred to the width parameters and pebbling problems, completing the chain of reductions.

Finally, the paper situates its results within the broader literature on approximation algorithms for separator and layout problems. Most existing algorithms rely on approximating the c‑balanced separator (via Leighton‑Rao, ARV SDP, etc.), achieving O(√log n) or O(√log n · log log n) ratios. The SSE‑hardness results demonstrate that, assuming the SSE Conjecture, these ratios are essentially optimal: no constant‑factor improvement is possible. The work thus provides a comprehensive hardness landscape for a suite of graph problems that were previously lacking strong inapproximability evidence.

In summary, under the Small Set Expansion Conjecture, treewidth, pathwidth, one‑shot black/black‑white pebbling, Minimum Cut Linear Arrangement, and Interval Graph Completion are all hard to approximate within any constant factor. This unifies and extends previous hardness results, explains why current algorithms achieve only polylogarithmic guarantees, and suggests that any breakthrough in approximation would require either disproving the SSE Conjecture or developing fundamentally new algorithmic techniques beyond current separator‑based methods.