Parameterized Complexity of Directed Traveling Salesman Problem

Abstract: The Directed Traveling Salesman Problem (DTSP) is a variant of the classical Traveling Salesman Problem in which the edges in the graph are directed and a vertex and edge can be visited multiple times. The goal is to find a directed closed walk of minimum length (or total weight) that visits every vertex of the given graph at least once. In a yet more general version, Directed Waypoint Routing Problem (DWRP), some vertices are marked as terminals and we are only required to visit all terminals. Furthermore, each edge has its capacity bounding the number of times this edge can be used by a solution. While both problems (and many other variants of TSP) were extensively investigated, mostly from the approximation point of view, there are surprisingly few results concerning the parameterized complexity. Our starting point is the result of Marx et al. [APPROX/RANDOM 2016] who proved that DTSP is W[1]-hard parameterized by distance to pathwidth 3. In this paper we aim to initiate the systematic complexity study of variants of DTSP with respect to various, mostly structural, parameters. We show that DWRP is FPT parameterized by the solution size, the feedback edge number, and the vertex integrity of the underlying undirected graph. Furthermore, the problem is XP parameterized by treewidth. On the complexity side, we show that the problem is W[1]-hard parameterized by the distance to constant treedepth.

• The paper presents FPT algorithms for DWRP under parameters like solution size, feedback edge set, and vertex integrity, achieving precise runtime guarantees such as e^k 4^k k^(O(log k)) m log m.
• The methodology employs advanced techniques including color coding, dynamic programming over tree decompositions, and N-fold IP to effectively compress the search space of directed routing problems.
• The paper also establishes W[1]-hardness for DTSP under parameters like treewidth and pathwidth, providing tight lower bounds that delineate tractable from intractable cases.

Parameterized Complexity of the Directed Traveling Salesman Problem

Introduction and Problem Definition

The Directed Traveling Salesman Problem (DTSP) is a fundamental optimization problem in graph theory defined on directed graphs. In this formulation, one seeks a closed directed walk of minimum length (or total weight) that visits every vertex at least once, with the possibility of revisiting vertices and traversing edges multiple times. The Directed Waypoint Routing Problem (DWRP) further generalizes this by asking for a walk that covers a given terminal set and abides by individual arc capacity constraints. While the undirected TSP and its variants have been extensively studied, particularly from the viewpoint of polynomial kernels and parameterized complexity, the complexity landscape for DTSP and its directed generalizations under structural and solution-size parameters remains incomplete.

Summary of Main Results

This work provides a comprehensive analysis of the parameterized complexity of DTSP and DWRP with respect to a spectrum of parameters describing the structure of the (undirected) underlying graph, as well as the solution size. Key results include:

• FPT Results:
  • DWRP is fixed-parameter tractable (FPT) parameterized by solution size, feedback edge set number, and vertex integrity (the minimum $k$ such that, after deleting at most $k$ vertices, all components have size at most $k$).
  • The authors obtain:
  • For solution size $k$, DWRP can be decided in $e^k 4^k k^{O(\log k)} m \log m$ time.
  • For feedback edge number $k$, DWRP is solvable in $k^{O(k)} + n^{O(1)}$ time.
  • For vertex integrity $k$, there is an $k^{O(k^6)} n \log^2 n (\log n + \log U)$ time algorithm.
  • DWRP is in XP (solvable in $n^{f(k)}$ time for parameter $k$) with respect to treewidth, and FPT when treewidth is combined with a bound on maximum vertex visits.
• Hardness Results:
  • DWRP is W[1]-hard parameterized by the modulator to constant treedepth, even when the arc weights are unit.
  • DTSP inherits W[1]-hardness parameterized by the distance to pathwidth 3, which strengthens prior results by giving tight bounds for both weighted and unweighted settings.

The analysis leverages advanced combinatorial and algebraic tools, notably color coding, dynamic programming over tree decompositions, and parameterized ILP (including advanced N-fold IP algorithms).

Algorithmic Techniques

The positive FPT results are facilitated by several algorithmic frameworks:

• Color Coding for Solution Size: By utilizing families of splitters as in the method of color coding, the algorithms can efficiently enumerate bounded-size solutions, extending to both solution size and the number of distinct arcs utilized.
• Decompositions for Structural Parameters: For feedback edge set number (FESN) and vertex integrity, the solutions revolve around reducing the search space or compressing instances into bounded-size core graphs. Specifically, for vertex integrity, an N-fold integer programming approach is executed, capturing both solution coverage of terminals and solution connectivity.
• Treewidth-Parameterized DP: For bounded-treewidth instances (combined with a bound on the maximum number of vertex traversals), a dynamic programming over nice tree decompositions is carried out, tracking multifacetted state information for partial covering closed walks.

Each of these approaches yields nontrivial theoretical guarantees in running time as a function of the respective parameter(s).

Hardness and Lower Bounds

The authors provide matching lower bounds for most parameters, via intricate W[1]-hardness reductions. These include:

• Reductions from capacitated dominating set and other canonical W[1]-hard problems to DWRP, involving elaborate gadget constructions enforcing specific traversal and covering properties while tightly controlling graph structure parameters.
• Careful analysis ensuring that these constructions preserve bounded parameters (e.g., the modulator to stars or to pathwidth 3) translates the hardness of the original problem to the hardness of DWRP/DTSP under the considered parameterizations.

The lower bounds also include explicit ETH-based runtime lower bounds, e.g., precluding $f(k) n^{o(k)}$-time algorithms for certain structural parameters.

Notable Numerical and Structural Claims

• The runtime $e^k 4^k k^{O(\log k)} m \log m$ for solution size $k$ provides a strong single-exponential FPT guarantee for small-size solutions.
• For vertex integrity $k$, the N-fold IP dimension is polynomial in $k$, yielding a $k^{O(k^6)}$ dependence in the running time. Although impractical for large $k$, this establishes FPT tractability for this structurally expressive and well-studied parameter.

Contrasts with Undirected TSP

In the undirected case, parameterizations by vertex cover number, distance to bounded components, and treewidth admit polynomial kernels or single-exponential FPT algorithms. For the directed setting, this work identifies several stark dichotomies: parameters like treewidth, pathwidth, and treedepth (or distance to their bounded values) fail to admit FPT algorithms for DTSP/DWRP (W[1]-hard), while feedback edge set number and vertex integrity permit FPT. This delineates clear boundaries in the landscape of tractability for directed routing problems.

Theoretical and Practical Implications

From a theoretical perspective, these results clarify the exact frontiers for exact algorithms for directed routing problems under structural parameterizations. The application of N-fold IP further highlights the reach of modern parameterized optimization techniques for covering and connectivity problems. Practically, these insights inform which structural constraints one must place on real-world instances (e.g., logistics or networking) to secure exact tractability.

Future Directions

Several open problems are posed. Notably, it remains unresolved if the FPT/XP dichotomy for treewidth, feedback vertex set, distance to stars, and other "gap" parameters can be closed. Directed-width measures adapted to cycles and arborescences may lead to sharper complexity results. Additionally, the cell between FPT and hardness in directed graphs that become acyclic after deleting few vertices or edges is open.

Conclusion

This work initiates the systematic parameterized complexity analysis of the Directed TSP and Directed Waypoint Routing problems, establishing for the first time tractability with respect to parameters such as vertex integrity and feedback edge set number, and proving tight lower bounds for many canonical structural parameters. The results provide a clear and detailed map of the computational hardness of these problems under a wide range of natural parameterizations, and introduce robust new frameworks for their algorithmic study.

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Reference:

"Parameterized Complexity of Directed Traveling Salesman Problem" (2506.22127)