A long-standing conjecture for the traveling salesman problem (TSP) states that the integrality gap of the standard linear programming relaxation of the TSP is at most 4/3. Despite significant efforts, the conjecture remains open. We consider the half-integral case, in which the LP has solution values in $\{0, 1/2, 1\}$. Such instances have been conjectured to be the most difficult instances for the overall four-thirds conjecture. Karlin, Klein, and Oveis Gharan, in a breakthrough result, were able to show that in the half-integral case, the integrality gap is at most 1.49993. This result led to the first significant progress on the overall conjecture in decades; the same authors showed the integrality gap is at most $1.5- 10^{-36}$ in the non-half-integral case. For the half-integral case, the current best-known ratio is 1.4983, a result by Gupta et al. With the improvements on the 3/2 bound remaining very incremental even in the half-integral case, we turn the question around and look for a large class of half-integral instances for which we can prove that the 4/3 conjecture is correct. The previous works on the half-integral case perform induction on a hierarchy of critical tight sets in the support graph of the LP solution, in which some of the sets correspond to "cycle cuts" and the others to "degree cuts". We show that if all the sets in the hierarchy correspond to cycle cuts, then we can find a distribution of tours whose expected cost is at most 4/3 times the value of the half-integral LP solution; sampling from the distribution gives us a randomized 4/3-approximation algorithm. We note that the known bad cases for the integrality gap have a gap of 4/3 and have a half-integral LP solution in which all the critical tight sets in the hierarchy are cycle cuts; thus our result is tight.

翻译：旅行商问题（TSP）的一个长期猜想是：其标准线性规划松弛的整性间隙不超过4/3。尽管付出大量努力，该猜想仍未解决。我们考虑半整情形，其中线性规划的解值属于$\{0, 1/2, 1\}$。这类实例被认为是对整个4/3猜想最困难的情形。Karlin、Klein与Oveis Gharan在突破性工作中证明了半整情形下整性间隙不超过1.49993，这一结果数十年来首次推动整体猜想的重大进展；同一作者还证明了非半整情形下整性间隙不超过$1.5-10^{-36}$。对于半整情形，当前最优比值为1.4983，该结果由Gupta等人取得。鉴于即使在半整情形下，3/2界限的改进仍极为缓慢，我们转换思路，寻找能证明4/3猜想成立的大类半整实例。此前关于半整情形的工作基于线性规划解支撑图中关键紧集层次结构进行归纳，其中部分集对应“循环割”，其余对应“度割”。我们证明：若层次结构中所有集均为循环割，则可找到一种旅游分布，其期望成本不超过半整线性规划解值的4/3倍；对该分布抽样即可得到随机化4/3近似算法。值得注意的是，已知的整性间隙反例具有4/3的间隙，且其半整线性规划解中层次结构的所有关键紧集均为循环割——因此我们的结果是紧的。