Bounded-Degree TSP Path Problem (BDTSPP)

• The paper demonstrates that BDTSPP seeks a minimum-cost Eulerian s–t path in 2G while closely adhering to given vertex degree bounds.
• It employs LP relaxations, iterative rounding, and bounded-degree T-joins to achieve a cost ratio of 5/3 with an additive degree violation of +4.
• The framework bridges approximation methods and exact algorithm techniques, addressing NP-hard feasibility and opening questions on tighter degree constraints.

Searching arXiv for the specified papers to ground the article in current preprints. Search (Lee et al., 30 Jun 2026) The bounded-degree traveling salesman path problem (BDTSPP) is a bicriteria network-design variant of path TSP in which one is given a connected undirected graph $G=(V,E)$ with nonnegative edge costs, two distinct endpoints $s,t\in V$, and integral degree bounds $\{b_v\}_{v\in V}$, and seeks a minimum-cost connected spanning subgraph $H\subseteq 2G$ that admits an Eulerian $s$–$t$ path while keeping $\deg_H(v)$ close to $b_v$. Here $2G$ denotes the multigraph obtained by duplicating every edge of $E$, so each original edge may be used at most twice. The parity requirements are exact: $s,t\in V$0 must be even for $s,t\in V$1 and odd for $s,t\in V$2 and $s,t\in V$3. A central recent result shows that BDTSPP admits a polynomial-time bicriteria approximation with cost at most $s,t\in V$4 and additive degree violation $s,t\in V$5, thereby answering in the affirmative a prior open question of whether additive, rather than multiplicative, degree violation is possible (Lee et al., 30 Jun 2026).

1. Formal model and scope

In BDTSPP, the objective is not a shortcut metric tour but the cost of the constructed Eulerian subgraph itself. The analysis therefore does not rely on the triangle inequality or on shortcutting. Costs may come from a metric or from general nonnegative edge costs, and the output may contain duplicated edges because it is chosen from $s,t\in V$6 rather than from $s,t\in V$7 alone (Lee et al., 30 Jun 2026).

The problem is an “all-vertices” variant: every vertex must lie in the connected spanning subgraph. This distinguishes it from subset variants, where only a designated terminal set $s,t\in V$8 is required to be connected and visited. It also distinguishes it from exact Hamiltonian path formulations on bounded-degree graphs, where “bounded degree” refers to the maximum degree of the input graph rather than explicit vertexwise degree bounds on the output structure. That distinction matters algorithmically, because the approximation framework for BDTSPP is built around output-degree control and Eulerian augmentation, whereas exact algorithms in bounded-degree graphs typically operate through branch-and-reduce or dynamic programming on the input graph structure.

A common source of confusion is the role of parity. BDTSPP does not ask directly for a simple Hamiltonian $s,t\in V$9–$\{b_v\}_{v\in V}$0 path; it asks for a connected spanning Eulerian subgraph with odd degrees at $\{b_v\}_{v\in V}$1 and even degrees elsewhere. When the degree bounds are tight enough, this collapses to Hamiltonian path structure, but in general the feasible object is an Eulerian multigraph supporting an $\{b_v\}_{v\in V}$2–$\{b_v\}_{v\in V}$3 walk.

2. Hardness and the bicriteria viewpoint

Deciding feasibility is already NP-hard for BDTSPP. The standard reduction starts from Hamiltonian path with prescribed endpoints $\{b_v\}_{v\in V}$4, sets $\{b_v\}_{v\in V}$5 and $\{b_v\}_{v\in V}$6 for $\{b_v\}_{v\in V}$7, and observes that any connected spanning Eulerian $\{b_v\}_{v\in V}$8–$\{b_v\}_{v\in V}$9 subgraph respecting these bounds must satisfy $H\subseteq 2G$0 and $H\subseteq 2G$1 for all other vertices, which enforces a single spanning $H\subseteq 2G$2–$H\subseteq 2G$3 path. Thus feasibility is NP-complete. The circuit analogue, with all degrees even and bounds $H\subseteq 2G$4, reduces similarly from Hamiltonian cycle (Lee et al., 30 Jun 2026).

Because even feasibility is NP-hard, approximation for BDTSPP is necessarily bicriteria: one controls the cost ratio while allowing bounded violation of the degree constraints. Earlier work for the path variant achieved only multiplicative violation, specifically $H\subseteq 2G$5 with cost $H\subseteq 2G$6. The 2026 result replaces this with additive degree violation and simultaneously improves the cost ratio to $H\subseteq 2G$7, matching Hoogeveen’s analysis of the Christofides-Serdyukov paradigm for path TSP (Lee et al., 30 Jun 2026).

This bicriteria setting is not a technical artifact. It reflects the structural barrier created by parity and hardness: exact degree compliance can already encode Hamiltonicity, so controlled additive relaxation is the natural approximation target.

3. LP relaxations and bounded-degree $H\subseteq 2G$8-joins

The natural cut-based LP relaxation for BDTSPP is

$H\subseteq 2G$9

where $s$0 is the cut induced by $s$1 and $s$2 (Lee et al., 30 Jun 2026).

A key augmentation primitive is the bounded-degree $s$3-join problem: for an even-cardinality set $s$4, find a minimum-cost edge multiset $s$5 such that $s$6 is odd iff $s$7 and $s$8. Its cut-based LP is

$s$9

This LP is integral whenever $t$0 is odd iff $t$1. Under that parity alignment, one can compute an optimal bounded-degree $t$2-join in polynomial time, and any feasible fractional solution upper-bounds the optimal augmentation cost (Lee et al., 30 Jun 2026).

This integrality is the technical bridge between parity correction and degree control. Rather than repairing parity with an unconstrained matching or join and accounting for degrees afterward, the framework solves parity correction directly under residual degree budgets.

4. Eulerian-to-tree extraction and the main approximation algorithm

The central structural lemma states that from any feasible Eulerian solution one can extract a spanning tree with degrees approximately half as large. If $t$3 is a connected multigraph in which either all degrees are even (circuit case) or exactly two vertices $t$4 have odd degree (path case), then there exists a spanning tree $t$5 such that

$t$6

Consequently, if $t$7 respects $t$8, then there exists a spanning tree $t$9 with

$\deg_H(v)$0

The proof traverses an Eulerian $\deg_H(v)$1–$\deg_H(v)$2 walk, roots a tree at $\deg_H(v)$3, and adds an edge whenever the walk first reaches a previously unvisited vertex; each vertex contributes at most $\deg_H(v)$4 child edges plus possibly one parent edge (Lee et al., 30 Jun 2026).

This lemma enables the use of the Singh–Lau iterative-rounding algorithm for bounded-degree spanning trees. For any feasible bounded-degree spanning tree instance $\deg_H(v)$5, Singh–Lau returns in polynomial time a spanning tree $\deg_H(v)$6 with

$\deg_H(v)$7

In the BDTSPP setting, the extraction lemma guarantees feasibility for bounds $\deg_H(v)$8, so the computed tree satisfies

$\deg_H(v)$9

The resulting algorithm follows the Christofides/Hoogeveen template adapted to bounded degrees:

1. Set $b_v$0 and compute a bounded-degree spanning tree $b_v$1.
2. Define the parity set $b_v$2.
3. For each vertex $b_v$3, set $b_v$4 to be the smallest integer satisfying $b_v$5 and $b_v$6 iff $b_v$7.
4. Solve the bounded-degree $b_v$8-join LP integrally with bounds $b_v$9 to obtain $2G$0.
5. Output $2G$1.

The augmentation analysis uses an optimal LP solution $2G$2 to BD-Path-LP and defines

$2G$3

A path cut-feasibility lemma implies that $2G$4 satisfies every $2G$5-odd cut constraint, and the degree calculation

$2G$6

shows feasibility for BD-TJoin. By integrality, $2G$7, so

$2G$8

Hence

$2G$9

For degrees,

$E$0

Since $E$1 and $E$2 have the same parity, this implies $E$3 (Lee et al., 30 Jun 2026).

5. Proven guarantees, related variants, and lower bounds

The formal bicriteria guarantee for BDTSPP is:

$E$4

The same structural program also yields improved results for closely related bounded-degree traveling salesman variants (Lee et al., 30 Jun 2026).

Variant	Guarantee	Note
BDTSPP	$E$5	additive violation; matches Hoogeveen cost
BDTSP	$E$6	$E$7 is best possible
BDSTSP	$E$8	improved subset-circuit cost
BDSTSPP	$E$9 or $s,t\in V$00	first additive guarantees for subset path

For the circuit version BDTSP, the algorithm outputs an Eulerian circuit $s,t\in V$01 with $s,t\in V$02 and $s,t\in V$03. The additive $s,t\in V$04 bound is best possible because allowing $s,t\in V$05 is equivalent to allowing $s,t\in V$06 under even-degree parity, and feasibility with $s,t\in V$07 is NP-hard.

For subset variants, the starting structure is a bounded-degree Steiner object rather than a spanning tree. The analysis introduces a connected subgraph $s,t\in V$08 of $s,t\in V$09 spanning the terminal set $s,t\in V$10 and an inclusionwise minimal feasible subgraph $s,t\in V$11, together with the redundant edge set $s,t\in V$12. In both BDSTSP and BDSTSPP, heavier use of the redundant edges $s,t\in V$13 is the key to improving the cost bounds for the $s,t\in V$14-join stage.

The paper also gives complementary lower-bound evidence for the path case. For every $s,t\in V$15, there exists a cycle instance $s,t\in V$16 with $s,t\in V$17 antipodal, $s,t\in V$18, and $s,t\in V$19 for $s,t\in V$20 such that the LP value is $s,t\in V$21, any integral solution must violate some degree bound by at least $s,t\in V$22, and even with arbitrary degree violation the minimum integral cost is $s,t\in V$23. The resulting integrality gap is at least

$s,t\in V$24

which approaches $s,t\in V$25. This shows that additive $s,t\in V$26 is unavoidable in some BDTSPP instances, while still leaving open whether the algorithmic upper bound can be tightened from $s,t\in V$27 to $s,t\in V$28.

6. Relation to exact algorithms on bounded-degree graphs

A separate line of work studies exact TSP on undirected weighted graphs whose maximum input degree is bounded. The paper "Quantum speedup of the Travelling Salesman Problem for bounded-degree graphs" analyzes the Hamiltonian cycle variant and proves quadratic quantum speedups over Xiao–Nagamochi’s branch-and-reduce algorithms. For degree-$s,t\in V$29 graphs it gives classical time $s,t\in V$30 and quantum time $s,t\in V$31; for degree-$s,t\in V$32 graphs it gives classical time $s,t\in V$33 and quantum time $s,t\in V$34; for degree-$s,t\in V$35 and degree-$s,t\in V$36 graphs, via splitting reductions and quantum minimum finding, it gives $s,t\in V$37 with bounded error at most $s,t\in V$38 and polynomial space (Moylett et al., 2016).

That paper does not explicitly treat the path variant. However, the supplied technical details state that the same framework extends to Hamiltonian path problems, including BDTSPP understood as minimum-cost Hamiltonian path, through two standard adaptations. The first is predicate-level endpoint enforcement: modify the branch-and-reduce predicate so that $s,t\in V$39 and $s,t\in V$40 have degree $s,t\in V$41 and all other vertices degree $s,t\in V$42, while adjusting the parity condition so that the $s,t\in V$43-components containing $s,t\in V$44 and $s,t\in V$45 are incident to an odd number of forced edges and all other $s,t\in V$46-components to an even number. The second is reduction to cycle TSP by adding a zero-cost forced edge $s,t\in V$47, which converts an $s,t\in V$48–$s,t\in V$49 Hamiltonian path into a Hamiltonian cycle containing that edge, at the cost of increasing the endpoints’ degrees by one.

Under the predicate-level adaptation, the quoted asymptotic bounds carry over to the path setting without changing the maximum degree. Under the forced-edge reduction, the applicable bound depends on the inflated degree $s,t\in V$50. For unspecified endpoints, trying all $s,t\in V$51 endpoint pairs adds only polynomial overhead in the $s,t\in V$52 notation.

This exact-algorithm literature is related to BDTSPP only in a qualified sense. It concerns simple Hamiltonian structures in bounded-degree input graphs, whereas the approximation results for BDTSPP concern Eulerian multigraphs under explicit output-degree bounds. The connection is strongest in the feasibility reductions and in very tight degree regimes, where parity and bounded degree force Hamiltonian path behavior.

7. Limitations and open directions

The 2026 approximation results leave several quantitative questions unresolved. For BDTSPP, the algorithm achieves additive degree violation $s,t\in V$53, while the gap construction shows that $s,t\in V$54 is necessary in some instances. Whether the path constant can be reduced to $s,t\in V$55, matching the circuit case, remains open (Lee et al., 30 Jun 2026).

On the cost side, the all-vertices bounds match the classic Christofides and Hoogeveen constants, but the data explicitly notes that in unconstrained path TSP, recent reductions approach the circuit constant. This suggests a natural open direction: determining whether analogous improvements are possible under bounded-degree constraints.

For subset problems, the improved constants $s,t\in V$56 for BDSTSP and $s,t\in V$57 for BDSTSPP arise from a more aggressive use of redundant Steiner edges in the augmentation analysis. A plausible implication is that further refinements of those compositions, or stronger LP formulations coupling degree and parity more tightly, could yield smaller constants.

Finally, the contrast between the bicriteria approximation framework and exact bounded-degree-graph algorithms highlights two distinct meanings of “bounded degree” in the TSP literature. In BDTSPP proper, the essential difficulty is the interaction of degree budgets, parity, and connectivity in an Eulerian multigraph; in exact Hamiltonian-path algorithms on bounded-degree input graphs, the essential difficulty is combinatorial search under sparse local structure. The current state of the art treats these two regimes with markedly different tools: iterative rounding and bounded-degree $s,t\in V$58-join integrality on one side, and branch-and-reduce with quantum backtracking on the other.