Theoretical Computer Science

Motivation

The traveling salesman problem — visit $n$ cities by the cheapest round trip — is the most widely known problem in combinatorial optimization, and its central open question concerns a linear program. The subtour-elimination relaxation (the Held–Karp bound) replaces tours by fractional edge weights, and both in theory and in practice (it powers the lower bounds inside the Concorde solver) it is remarkably close to the true optimum. How close, in the worst case, is the integrality gap of the relaxation: the supremum of $\mathrm{OPT}/\mathrm{LP}$ over metric instances. Explicit instance families push the gap up to $4/3$; the best proven upper bound sits just barely below $3/2$. The 4/3 conjecture — the gap is exactly $4/3$ — has been the benchmark question of approximation algorithms for four decades.

Timeline

• 1954. Dantzig, Fulkerson, and Johnson solve a 49-city instance by hand with the cutting planes that become the subtour-elimination LP.
• 1970–1971. Held and Karp introduce the 1-tree/Lagrangian bound and show it equals the subtour LP value — since then, "the Held–Karp bound".
• 1976/1978. Christofides, and independently Serdyukov, give the $3/2$-approximation: minimum spanning tree plus a matching on odd-degree vertices.
• 1980. Wolsey (Math. Prog. Study 13) shows Christofides' analysis goes through against the LP: $\mathrm{OPT} \le \frac{3}{2}\,\mathrm{LP}$, so the integrality gap is at most $3/2$. Shmoys and Williamson (IPL 1990) rediscover this via a monotonicity property.
• 1995. Goemans (Math. Programming 69) analyzes the worst-case ratios of TSP relaxations and states the $4/3$ conjecture explicitly; the $4/3$ lower-bound families (three parallel paths) are by then folklore.
• 2011–2014. For graph metrics (shortest-path metrics of unweighted graphs) the barrier breaks: Oveis Gharan–Saberi–Singh and Mömke–Svensson beat $3/2$, and Sebő–Vygen (Combinatorica 2014) reach $7/5$ — the conjectured-optimal shape of progress, but only for a special class.
• 2020–2022. Karlin, Klein, and Oveis Gharan prove a $3/2 - \varepsilon$ approximation for general metric TSP (STOC 2021) and then an integrality-gap bound $\gamma \le 3/2 - \varepsilon$ with $\varepsilon > 10^{-36}$ (FOCS 2022), via max-entropy sampling of spanning trees and strongly Rayleigh distributions — the first general improvement over Wolsey in forty years, by an astronomically small margin.
• Today. The gap between the $4/3$ lower bound and the $3/2 - 10^{-36}$ upper bound is the conjecture. For half-integral LP solutions — where the conjectured extremal instances live — the bound has been pushed to $1.4983$ (Gupta, Lee, Li, Mucha, Newman, and Sarkar, via matroid-based rounding).

Setting

An instance on $n \ge 3$ cities is a cost function $c$ assigning to each ordered pair of cities $u, v$ a real cost $c(u,v)$, required to be a metric cost: symmetric ($c(u,v) = c(v,u)$), zero on the diagonal ($c(v,v) = 0$), and satisfying the triangle inequality $c(u,w) \le c(u,v) + c(v,w)$. Nonnegativity follows; distinct cities at distance zero are allowed, as usual for metric TSP.

A tour visits every city exactly once and returns to its start. Formally a tour is given by an ordering: a permutation $\pi$ of the cities, traversed as $\pi(0), \pi(1), \dots, \pi(n-1)$ and back to $\pi(0)$; its cost $\mathrm{tourCost}(c, \pi)$ is the sum of the costs of consecutive steps, and $\mathrm{OPT}(c)$ — written tspOpt c — is the minimum over all orderings.

The subtour-elimination (Held–Karp) relaxation replaces the tour by a fractional edge weight $x(u,v)$ for each pair of cities. A weight vector $x$ is feasible (IsHeldKarp x) when it is symmetric with zero diagonal, has entries in $[0,1]$, gives every city fractional degree two ($\sum_u x(v,u) = 2$), and crosses every nontrivial cut at least twice: for every set $S$ of cities other than $\emptyset$ and all cities, $\sum_{u \in S} \sum_{v \notin S} x(u,v) \ge 2$. The Held–Karp bound hkValue c is the infimum of $\frac{1}{2}\sum_u \sum_v c(u,v)\,x(u,v)$ over feasible $x$ (the double sum counts each edge twice, hence the $\frac12$). The incidence vector of any tour is feasible, so $\mathrm{LP} \le \mathrm{OPT}$ always.

Formalization targets

Goal — the 4/3 conjecture

$$\mathrm{OPT}(c) \;\le\; \tfrac{4}{3}\,\mathrm{LP}(c) \qquad \text{for every } n \ge 3 \text{ and every metric cost } c.$$

Together with the known lower-bound families this says the integrality gap is exactly $4/3$. The goal carries no algorithm and no constant to improve: it is the terminal statement of the ladder, open in both directions (a proof or a counterexample instance would each settle it).

Milestones — the known ladder

Five results over the same definitions: the relaxation is valid ($\mathrm{LP} \le \mathrm{OPT}$); instance families force the gap arbitrarily close to $4/3$; tree doubling gives $\mathrm{OPT} \le 2\,\mathrm{LP}$; Wolsey's theorem gives $\mathrm{OPT} \le \frac{3}{2}\,\mathrm{LP}$, the classical upper bound; and the Karlin–Klein–Oveis Gharan record $\mathrm{OPT} \le (\frac{3}{2} - \varepsilon)\,\mathrm{LP}$ for some $\varepsilon > 10^{-36}$ (FOCS 2022). The last milestone is a statement-level target: its known proof (max-entropy sampling, strongly Rayleigh polynomials) is far beyond current formalization practice, so the mission's usable proving frontier remains Wolsey — the milestone records the state of the art as a formal statement.

Significance

The 4/3 conjecture is the reference open problem of approximation algorithms: the quality of the subtour LP calibrates every algorithmic advance on TSP, and the conjectured extremal instances guide the search for better rounding schemes. The bound is also what practical solvers actually compute — branch-and-cut on this LP solves instances with tens of thousands of cities — so the conjecture is a statement about the observed tightness of the world's most-used combinatorial lower bound.

Nothing in this circle exists in any proof assistant: Mathlib has no TSP, no LP relaxations, no polyhedral combinatorics of tours. The mission's milestones force the base layer into existence — tours over Equiv.Perm, cut constraints over Finset, and, for the upper bounds, the parity and tree arguments (spanning trees against the LP, T-joins for the $3/2$ bound) whose infrastructure is reusable for matching theory and network design far beyond TSP.

Difficulty

The naive plan — round the LP solution to a tour — has no known analysis losing less than $3/2$ in general, and the half-integral extremal instances show the hard cases are structured and simple-looking at once. Christofides' matching argument is provably stuck at $3/2$ against the LP; forty years of work moved the constant by $10^{-36}$, and that advance needed an entirely new probabilistic toolkit. On the other side, no instance family with ratio above $4/3$ has ever been found despite extensive computational search over small instances (Benoit–Boyd and successors). Both directions of the goal are genuinely open territory.

Formalization scope

The Lean model commits to: cities Fin n; costs c : Fin n → Fin n → ℝ with IsMetricCost (symmetry, zero diagonal, triangle inequality — nonnegativity is derived, and semimetrics are included as in the standard statement of the conjecture); tours as orderings π : Equiv.Perm (Fin n) traversed cyclically via finRotate, so every permutation denotes a Hamiltonian cycle and every Hamiltonian cycle is denoted; both optimal values as sInf over nonempty, bounded-below sets of reals, so they are genuine minima for n ≥ 3. The hypothesis 3 ≤ n is load-bearing: for n ≤ 2 the degree-2 constraints are infeasible, sInf ∅ = 0 by convention, and the bounds would be false — every theorem therefore carries it.

Welcome contributions: the milestones in any order — held_karp_le_opt is the natural entry point (the tour's incidence vector crosses every cut at least twice); integrality_gap_lower_bound needs the three-path instance family and a case analysis on its tours; tree_doubling_bound needs spanning trees against the LP; wolsey_bound adds the T-join/parity argument and is the summit. Reusable infrastructure — spanning tree polytopes, T-joins, Eulerian traversals, cut lemmas — is welcome as platform theorems. Graph-TSP ($7/5$), path TSP, and asymmetric TSP are deliberately left to future missions; the Karlin–Klein–Oveis Gharan bound is stated as a milestone, but its sampling machinery is expected to arrive, if ever, as shared infrastructure built over many contributions.

Selected references

• G. Dantzig, R. Fulkerson, S. Johnson, Solution of a large-scale traveling-salesman problem, Oper. Res. 2 (1954).
• M. Held, R. Karp, The traveling-salesman problem and minimum spanning trees, Oper. Res. 18 (1970); Part II, Math. Programming 1 (1971).
• N. Christofides, Worst-case analysis of a new heuristic for the travelling salesman problem, CMU report (1976); A. Serdyukov, Upravlyaemye Sistemy 17 (1978).
• L. Wolsey, Heuristic analysis, linear programming and branch and bound, Math. Prog. Study 13 (1980). doi:10.1007/BFb0120913
• D. Shmoys, D. Williamson, Analyzing the Held-Karp TSP bound: a monotonicity property with application, Inf. Process. Lett. 35 (1990). doi:10.1016/0020-0190(90)90028-V
• M. Goemans, Worst-case comparison of valid inequalities for the TSP, Math. Programming 69 (1995). doi:10.1007/BF01585563
• A. Sebő, J. Vygen, Shorter tours by nicer ears, Combinatorica 34 (2014). arXiv:1201.1870
• A. Karlin, N. Klein, S. Oveis Gharan, A (slightly) improved approximation algorithm for metric TSP, STOC 2021. arXiv:2007.01409
• A. Karlin, N. Klein, S. Oveis Gharan, A (slightly) improved bound on the integrality gap of the subtour LP for TSP, FOCS 2022. arXiv:2105.10043
• V. Traub, J. Vygen, Approximation Algorithms for Traveling Salesman Problems, Cambridge University Press, 2024. book page