- The paper presents additive bicriteria approximations for both circuit and path versions of bounded-degree TSP, achieving guarantees of (3/2, +2) and (5/3, +4) respectively.
- The methodology leverages a novel link between low-degree Eulerian multigraphs and spanning trees to enable efficient rounding and improved approximation ratios.
- The work extends to subset TSP variants and provides tight integrality gap constructions, demonstrating that the additive degree violations are nearly optimal under current methods.
Improved Bicriteria Algorithms for Bounded-Degree (Subset) TSP
Problem Setting and Motivation
The paper studies algorithmic advances for the Traveling Salesman Problem (TSP) and its subset variants under explicit degree constraints at each vertex—an important setting in network design where vertices can have strict traversal limits. Both the classical circuit (cycle) and path TSP are considered, as well as the “subset” versions, where only a given subset of nodes must be visited. These bounded-degree variants are motivated by domains such as traffic planning and sensor network design, where operational or physical constraints restrict node degrees.
Previous work established the computational hardness of merely finding feasible solutions due to the superposition of TSP and degree constraints. Consequently, the focus is on bicriteria approximation algorithms, which relax either the cost or the degree constraints by a small additive or multiplicative margin, seeking near-optimal solutions on both axes.
Main Contributions
Additive Bicriteria Approximation for Bounded-Degree TSP
The paper resolves a central open question regarding the “bounded-degree traveling salesman path” problem (BDTSPP): whether an additive guarantee on degree violation is attainable, as opposed to the previously known multiplicative bounds. The work introduces bicriteria algorithms with additive degree violations for both path and circuit variants.
For the standard BDTSP (circuit version), the authors present a (3/2,+2)-approximation: the solution cost is at most $3/2$ times optimal, and the degree of any node exceeds its bound by at most $2$. For BDTSPP (path version), a new (5/3,+4)-approximation is given, providing the first additive violation guarantee.
Improved Algorithms for Subset Bounded-Degree TSP
For the subset TSP (only a specified subset X of required vertices must be traversed), new algorithms improve both cost ratios and degree violations relative to previous literature. Key results include:
- BDSTSP (circuit, subset): (14/9,+6)-approximation.
- BDSTSPP (path, subset): (11/5,+6)- and (2,+8)-approximations, where the two trade-off cost against degree violation for different regimes.
Notably, the path subset case achieves the first additive (rather than multiplicative) degree guarantees.
Integrality Gap Constructions
The paper also characterizes the tightness of these guarantees by exhibiting instances where every integral solution exceeds degree bounds by at least $2$ and the optimum integral cost is at least $3/2$ times the LP relaxation value. This proves the additive $3/2$0 violation is the best possible under current techniques.
Algorithmic Innovations
The major technical advance is a novel connection between low-degree Eulerian multigraphs (arising from optimal TSP solutions) and the existence of low-degree spanning trees. Rather than relying on bounded-degree Steiner trees (used in older work, which only match the fractional LP optimum), the paper shows the stronger result that one can extract a spanning tree with degree at most $3/2$1 from any feasible solution. This structural property underlies the improved additive control over the degree violations in the rounding and augmentation steps commonly used in Christofides-type algorithms.
For subset TSP, the approach leverages parity-constrained Steiner trees but more efficiently utilizes “redundant” edges when bounding the cost of the $3/2$2-join required for Eulerian augmentation, leading to better overall approximation ratios.
Numerical Guarantees and Theoretical Implications
The core theorems provide the following guarantees for cost approximation ratio and degree violation:
| Problem |
Approximation (Cost, Degree Violation) |
Previous Best |
| BDTSP (circuit) |
(3/2, +2) |
(3/2, +4) |
| BDTSPP (path) |
(5/3, +4) |
(8/3, multiplicative) |
| BDSTSP (subset) |
(14/9, +6) |
(5/3, +4) |
| BDSTSPP (subset) |
(11/5, +6), (2, +8) |
(8/3, multiplicative) |
These bounds match or surpass prior algorithms in all settings, always achieving additive (and provably tight) degree violations.
Practical and Theoretical Impacts
These improvements have both practical and theoretical significance. For real-world networks with hard degree constraints, such as in transportation or deployment of limited-access infrastructure (e.g., in sensor networks), these results provide the first means to ensure total cost is kept close to optimal while guaranteeing that operational limits are violated only by a minimal, controlled amount, irrespective of the actual bound sizes.
Theoretically, the new structural lemmas point to a general principle: that one can, in the bounded-degree TSP framework, avoid reliance on the LP—typically a weaker relaxation due to cut constraints—and instead ground the rounding process in properties of integral optima. This insight may percolate to other constrained network design problems.
The LP integrality gap constructions further indicate that, up to constant additive violation, no better approximation is possible under known methods. Thus, the results nearly close the approximability frontier for these variants as long as the TSP approximation ratio itself holds.
Directions for Future Work
Future research may pursue:
- Extensions to more general network design problems with mixed degree/connectivity/Steiner requirements.
- Improved constant-factor cost approximations for TSP itself, which would immediately translate into better ratios for these bounded-degree variants.
- Robust or stochastic versions, where degree constraints are not deterministic but sampled or adversarial.
Conclusion
The paper provides the definitive additive bicriteria approximation guarantees for bounded-degree (subset) TSP and related path problems, combining structural graph-theoretical lemmas with refined rounding analyses. The results both improve practical feasibility for deployed networks and clarify the theoretical limits for algorithmic approximability in constrained combinatorial optimization settings.