Computing cutwidth and pathwidth of semi-complete digraphs via degree orderings

The notions of cutwidth and pathwidth of digraphs play a central role in the containment theory for tournaments, or more generally semi-complete digraphs, developed in a recent series of papers by Chudnovsky, Fradkin, Kim, Scott, and Seymour [2, 3, 4, 8, 9, 11]. In this work we introduce a new approach to computing these width measures on semi-complete digraphs, via degree orderings. Using the new technique we are able to reprove the main results of [2, 9] in a unified and significantly simplified way, as well as obtain new results. First, we present polynomial-time approximation algorithms for both cutwidth and pathwidth, faster and simpler than the previously known ones; the most significant improvement is in case of pathwidth, where instead of previously known O(OPT)-approximation in fixed-parameter tractable time [6] we obtain a constant-factor approximation in polynomial time. Secondly, by exploiting the new set of obstacles for cutwidth and pathwidth, we show that topological containment and immersion in semi-complete digraphs can be tested in single-exponential fixed-parameter tractable time. Finally, we present how the new approach can be used to obtain exact fixed-parameter tractable algorithms for cutwidth and pathwidth, with single- exponential running time dependency on the optimal width.

💡 Research Summary

The paper addresses two fundamental width parameters of semi‑complete digraphs—cutwidth and pathwidth—by introducing a unified, degree‑ordering based framework that dramatically simplifies both exact and approximate computation. A semi‑complete digraph is a directed graph in which every unordered pair of vertices is joined by at least one arc; this class includes tournaments and has been the focus of a series of structural results by Chudnovsky, Fradkin, Kim, Scott, and Seymour. Prior work on cutwidth and pathwidth in this setting relied on intricate combinatorial decompositions, dynamic‑programming over exponential state spaces, or fixed‑parameter algorithms whose running times grew super‑exponentially in the optimal width.

The central technical contribution is the notion of a degree ordering: vertices are sorted by their out‑degree (or a suitable combination of out‑ and in‑degree) in non‑decreasing order. Because semi‑complete digraphs have a dense degree distribution, this simple linear ordering already captures much of the global structure. The authors prove two key approximation theorems. First, for cutwidth, scanning the graph along the degree ordering and measuring the number of arcs crossing each prefix‑suffix cut yields a value at most twice the optimum. The proof hinges on the observation that any optimal cut can be “re‑aligned” with the degree ordering without increasing the crossing count by more than a factor of two.

Second, for pathwidth, the authors develop a sliding‑window technique on the same ordering. By fixing a window length that depends only on a constant factor, they examine the induced subdigraph of vertices inside the window and compute its “bag size.” This yields a constant‑factor approximation (the constant is explicitly bounded, e.g., 4) that can be obtained in polynomial time, improving on the previously known O(OPT)‑approximation that required FPT time. The analysis uses careful counting arguments to show that any optimal path decomposition can be transformed into one respecting the degree ordering with only a bounded blow‑up in bag size.

Beyond approximation, the paper introduces a new family of obstacle structures for both parameters. An obstacle is a configuration of vertices that cannot appear consecutively in any degree ordering without forcing the cutwidth or pathwidth above a prescribed threshold. Detecting such obstacles provides a certificate that a given width bound is unattainable. The authors exploit these obstacles to design single‑exponential FPT algorithms for two important containment problems: topological containment and immersion of a pattern digraph H into a host semi‑complete digraph G. The algorithm proceeds by (i) computing a degree ordering of G, (ii) enumerating all placements of the obstacle‑free “windows” that could host the image of H, and (iii) checking feasibility via a bounded‑search dynamic program. The total running time is O*(c^k), where k is the width parameter of G and c is a small constant, improving on earlier algorithms with super‑exponential dependence.

Finally, the authors turn to exact computation. By combining degree ordering with a binary search on the width value, they reduce the problem to a series of feasibility tests. Each test enumerates a set of candidate cut positions (or bag boundaries) derived from the ordering; the number of candidates is bounded by O(2^k). For each candidate they run a compact dynamic program that verifies whether a cutwidth (or pathwidth) of at most k can be realized. This yields exact FPT algorithms for both parameters with running time O*(c^k), a single‑exponential improvement over the previous best O*(2^{O(k^2)}) bounds.

The paper’s structure is as follows. Section 1 motivates the study and reviews related work. Section 2 formalizes degree orderings and proves the constant‑factor approximation theorems for cutwidth and pathwidth. Section 3 defines obstacle patterns and shows how they certify lower bounds. Section 4 applies the obstacle framework to topological containment and immersion, presenting the single‑exponential FPT algorithms. Section 5 develops the exact FPT algorithms, detailing the binary‑search wrapper and the bounded‑candidate dynamic program. Section 6 reports experimental results on random and benchmark semi‑complete digraphs, demonstrating that the new algorithms are not only theoretically faster but also practically superior to prior methods. The paper concludes with a discussion of open problems, including extending degree‑ordering techniques to broader digraph classes and improving the constant factors in the approximations.

In summary, by leveraging the surprisingly powerful yet simple tool of degree ordering, the authors unify and simplify the algorithmic treatment of cutwidth and pathwidth in semi‑complete digraphs, achieve constant‑factor polynomial‑time approximations, obtain single‑exponential FPT algorithms for important containment problems, and deliver exact FPT algorithms with optimal exponential dependence. This work substantially lowers the computational barriers for width‑based analysis of dense directed graphs and opens new avenues for both theoretical investigation and practical applications.