Half-Integral Cycle Cut Instances

• The paper presents half-integral cycle cut instances as a subclass of symmetric TSP problems where every nontrivial tight set splits into two nonempty parts, yielding a 4-regular support graph.
• It details a hierarchical structure with cycle partner edges that underpins a specialized 4/3-approximation algorithm and enhanced maximum entropy analysis.
• The study establishes these instances as extremal examples, precisely matching the LP integrality gap of 4/3, with broader implications for combinatorial optimization.

Half-integral cycle cut instances are a structured class of symmetric metric Traveling Salesman Problem instances in which a feasible Subtour LP solution $x$ is half-integral and the family of tight subtour cuts admits a recursive cycle-like decomposition. In one formulation, every nontrivial tight set $S$ with $x(\delta(S))=2$ can be partitioned into two nonempty tight sets; after splitting each edge with $x_e=1$ into two parallel $1/2$-edges, every support edge has value $1/2$ and the support becomes 4-regular. Related work presents an equivalent viewpoint through the hierarchy of critical cuts, where every non-singleton critical set is a cycle cut rather than a degree cut (Jin et al., 1 Jul 2026, Jin et al., 2022).

1. Subtour-LP setting and formal definition

The ambient optimization problem is the symmetric metric TSP on a complete graph $G=(V,E)$ with costs $c_e=c_{ij}\ge 0$ satisfying the triangle inequality. The standard relaxation is the Subtour LP: $$\begin{aligned} \min \quad & \sum_{e \in E} c_e x_e \ \text{s.t.}\quad & x(\delta(v)) = 2, && \forall v \in V, \ & x(\delta(S)) \ge 2, && \forall S \subset V,\ S \neq \emptyset, \ & 0 \le x_e \le 1, && \forall e \in E. \end{aligned}$$ A pair $(G,x)$ is half-integral when $S$0 for all $S$1. The literature treats the LP solution as part of the instance: a half-integral cycle cut instance is not merely a metric graph, but a metric graph equipped with a particular half-integral feasible Subtour-LP solution (Jin et al., 1 Jul 2026).

A direct definition used in recent work is that a half-integral instance is a cycle cut instance if every tight set $S$2 with $S$3 can be written as a partition $S$4 into two nonempty tight sets. In the equivalent multigraph formulation, one replaces every $S$5 edge by two parallel edges of value $S$6; then every edge has LP value $S$7, every vertex has four incident support edges, and every non-singleton tight set can be decomposed into smaller tight sets. Related formulations describe the same class by requiring that, in the hierarchy of critical cuts rooted at a vertex $S$8, every non-singleton critical set is a cycle cut (Jin et al., 1 Jul 2026, Jin et al., 2022).

2. Tight-set hierarchy and cycle-cut structure

Once 1-edges are split, every tight set $S$9 satisfies $x(\delta(S))=2$0. Fixing a root vertex $x(\delta(S))=2$1, one obtains a binary laminar hierarchy of tight sets: $x(\delta(S))=2$2 and $x(\delta(S))=2$3 are tight, and every tight set $x(\delta(S))=2$4 with $x(\delta(S))=2$5 splits into two tight children $x(\delta(S))=2$6. For each sibling pair $x(\delta(S))=2$7, exactly two edges lie in $x(\delta(S))=2$8; these are the cycle partners of the split. This yields a rigid global picture: a 4-regular support graph together with a binary tight-set tree whose internal nodes are joined by partner-edge pairs (Jin et al., 1 Jul 2026).

The hierarchy-of-critical-cuts formulation emphasizes a complementary aspect. A tight set is critical if it does not cross any other tight set, and the family of critical sets is laminar. For a non-singleton critical set $x(\delta(S))=2$9 with children $x_e=1$0, contracting each child and the complement of $x_e=1$1 produces a 4-regular, 4-edge-connected multigraph on $x_e=1$2 nodes. When this contracted graph is a simple cycle $x_e=1$3 with exactly two parallel edges between consecutive vertices, $x_e=1$4 is a cycle cut; otherwise it is a degree cut. Half-integral cycle cut instances are exactly those for which no non-singleton critical set is a degree cut (Jin et al., 2022).

This structure is strict. Not all half-integral instances are cycle cut instances: $x_e=1$5 with every edge assigned $x_e=1$6 is half-integral but not cycle cut. At the same time, the class is broad enough to contain canonical extremal examples such as the envelope graph and the known bad cases for the four-thirds conjecture (Jin et al., 1 Jul 2026, Jin et al., 2022).

3. Extremal role in the four-thirds program

The class is important because it sits at the center of the Subtour-LP integrality-gap program. The four-thirds conjecture asserts that

$x_e=1$7

Half-integral instances have long been viewed as likely worst cases, and half-integral cycle cut instances sharpen that perspective further. They contain examples where the Subtour LP has integrality gap at least $x_e=1$8, including the envelope graph, and earlier work proved that on this class the integrality gap is at most $x_e=1$9. Taken together, these facts show that the integrality gap on half-integral cycle cut instances is exactly $1/2$0 (Jin et al., 1 Jul 2026).

This makes the class extremal in a precise sense. The 2022 approximation result observes that the known bad cases for the integrality gap, including the classical three-arm examples and the $1/2$1-donuts of Boyd–Sebő, are half-integral and have a hierarchy in which all critical tight sets are cycle cuts. The resulting $1/2$2 guarantee is therefore tight for the class (Jin et al., 2022).

The class is also informative for algorithm-specific analysis. The 2026 maximum-entropy paper records two algorithmic benchmarks on the same family: the maximum entropy algorithm is proved to be a $1/2$3-approximation on half-integral cycle cut instances, while examples in the class show that its performance can be no better than $1/2$4. This leaves a nontrivial gap between the class’s true LP integrality gap $1/2$5 and the worst currently understood behavior of max entropy on the class (Jin et al., 1 Jul 2026).

4. The specialized $1/2$6-approximation algorithm

A dedicated randomized $1/2$7-approximation algorithm is known for half-integral cycle cut instances. Rather than following a Christofides-type tree-plus-$1/2$8-join template, it constructs directly a distribution over connected Eulerian multigraphs. The multigraph is built on the 4-regular support obtained from the half-integral LP solution, and shortcutting then yields a TSP tour. The expected cost is bounded by $1/2$9 (Jin et al., 2022).

The analysis is organized around cut patterns. For each cycle cut $1/2$0, the four edges of $1/2$1 are classified into four parity states $1/2$2, and the algorithm propagates these states top-down through the hierarchy. The state evolution is described by Markov chains, with separate transitions for cycle cuts having an even or odd number of children. The caterpillar drawing of the hierarchy induces a left/right decomposition of each boundary cut and a distinction between straight and twisted cycle cuts; twisted cuts swap the roles of states $1/2$3 and $1/2$4 (Jin et al., 2022).

The expected edge-usage bound is encoded in a feasible region of state distributions: $1/2$5 The condition $1/2$6 is exactly what yields expected usage $1/2$7 for each physical support edge, hence expected cost $1/2$8. A concrete fixed distribution used in the construction is

$1/2$9

together with its swapped version

$G=(V,E)$0

The hierarchy can be processed so that every straight cut carries distribution $G=(V,E)$1 and every twisted cut carries $G=(V,E)$2, while remaining inside $G=(V,E)$3 throughout the recursion (Jin et al., 2022).

5. Maximum entropy on the same class

The same structural class admits a markedly sharper analysis of the Karlin–Klein–Oveis Gharan maximum entropy algorithm. On general metric instances, the max-entropy framework samples a 1-tree from a maximum entropy distribution with prescribed marginals and then adds a minimum-cost perfect matching on the odd-degree vertices. On half-integral cycle cut instances, the maximum entropy distribution simplifies dramatically: at the root it selects exactly one edge from each of two designated pairs, and at every internal tight set with cycle partners $G=(V,E)$4 it selects exactly one of $G=(V,E)$5, uniformly at random; the choices at different nodes of the tight-set tree are independent (Jin et al., 1 Jul 2026).

The main theorem states that if $G=(V,E)$6 is a half-integral cycle cut instance, $G=(V,E)$7 is the max entropy distribution over 1-trees with marginals $G=(V,E)$8, $G=(V,E)$9, and $c_e=c_{ij}\ge 0$0 is the cheapest perfect matching on the odd-degree vertices of $c_e=c_{ij}\ge 0$1, then

$c_e=c_{ij}\ge 0$2

The proof does not analyze the matching directly. Instead it constructs a distribution on Eulerian multi-subgraphs that extends the sampled 1-tree and is stationary on every 4-edge tight cut. The stationarity conditions force each edge to be odd with probability $c_e=c_{ij}\ge 0$3, each edge to be doubled with probability $c_e=c_{ij}\ge 0$4, and hence each edge to have expected multiplicity $c_e=c_{ij}\ge 0$5. Since every support edge has LP value $c_e=c_{ij}\ge 0$6, this yields expected cost $c_e=c_{ij}\ge 0$7 (Jin et al., 1 Jul 2026).

This guarantee is algorithmic rather than polyhedral. The paper explicitly notes that it does not improve the $c_e=c_{ij}\ge 0$8 upper bound on the integrality gap of the class, because that bound was already obtained by a different algorithm. Its significance is instead methodological: the analysis shows that a very general algorithm behaves substantially better than $c_e=c_{ij}\ge 0$9 on a structurally rich and extremal family, which suggests possible directions for improved analyses beyond the class (Jin et al., 1 Jul 2026).

6. Broader half-integral cycle–cut phenomena

The TSP notion belongs to a wider half-integral packing-versus-cut landscape. This suggests a broader cycle–cut paradigm in which cycles are packed with congestion $$\begin{aligned} \min \quad & \sum_{e \in E} c_e x_e \ \text{s.t.}\quad & x(\delta(v)) = 2, && \forall v \in V, \ & x(\delta(S)) \ge 2, && \forall S \subset V,\ S \neq \emptyset, \ & 0 \le x_e \le 1, && \forall e \in E. \end{aligned}$$0 and dualized by a small hitting set. For non-null cycles in group-labeled graphs, one has a half-integral Erdős–Pósa theorem: either there are $$\begin{aligned} \min \quad & \sum_{e \in E} c_e x_e \ \text{s.t.}\quad & x(\delta(v)) = 2, && \forall v \in V, \ & x(\delta(S)) \ge 2, && \forall S \subset V,\ S \neq \emptyset, \ & 0 \le x_e \le 1, && \forall e \in E. \end{aligned}$$1 non-null cycles such that every vertex appears in at most two of them, or there is a set of at most $$\begin{aligned} \min \quad & \sum_{e \in E} c_e x_e \ \text{s.t.}\quad & x(\delta(v)) = 2, && \forall v \in V, \ & x(\delta(S)) \ge 2, && \forall S \subset V,\ S \neq \emptyset, \ & 0 \le x_e \le 1, && \forall e \in E. \end{aligned}$$2 vertices meeting every non-null cycle (Lokshtanov et al., 2017). For directed graphs, if there is no family of $$\begin{aligned} \min \quad & \sum_{e \in E} c_e x_e \ \text{s.t.}\quad & x(\delta(v)) = 2, && \forall v \in V, \ & x(\delta(S)) \ge 2, && \forall S \subset V,\ S \neq \emptyset, \ & 0 \le x_e \le 1, && \forall e \in E. \end{aligned}$$3 directed cycles in which every vertex lies on at most two cycles, then there exists a feedback vertex set of size $$\begin{aligned} \min \quad & \sum_{e \in E} c_e x_e \ \text{s.t.}\quad & x(\delta(v)) = 2, && \forall v \in V, \ & x(\delta(S)) \ge 2, && \forall S \subset V,\ S \neq \emptyset, \ & 0 \le x_e \le 1, && \forall e \in E. \end{aligned}$$4; for directed odd cycles, there is a function $$\begin{aligned} \min \quad & \sum_{e \in E} c_e x_e \ \text{s.t.}\quad & x(\delta(v)) = 2, && \forall v \in V, \ & x(\delta(S)) \ge 2, && \forall S \subset V,\ S \neq \emptyset, \ & 0 \le x_e \le 1, && \forall e \in E. \end{aligned}$$5 such that every digraph has either $$\begin{aligned} \min \quad & \sum_{e \in E} c_e x_e \ \text{s.t.}\quad & x(\delta(v)) = 2, && \forall v \in V, \ & x(\delta(S)) \ge 2, && \forall S \subset V,\ S \neq \emptyset, \ & 0 \le x_e \le 1, && \forall e \in E. \end{aligned}$$6 directed odd cycles with every vertex used at most twice or a vertex set of size at most $$\begin{aligned} \min \quad & \sum_{e \in E} c_e x_e \ \text{s.t.}\quad & x(\delta(v)) = 2, && \forall v \in V, \ & x(\delta(S)) \ge 2, && \forall S \subset V,\ S \neq \emptyset, \ & 0 \le x_e \le 1, && \forall e \in E. \end{aligned}$$7 meeting all directed odd cycles (Masařík et al., 2019, Kawarabayashi et al., 2020).

Algorithmic work on graph separation problems exhibits the same half-integral pattern. In the $$\begin{aligned} \min \quad & \sum_{e \in E} c_e x_e \ \text{s.t.}\quad & x(\delta(v)) = 2, && \forall v \in V, \ & x(\delta(S)) \ge 2, && \forall S \subset V,\ S \neq \emptyset, \ & 0 \le x_e \le 1, && \forall e \in E. \end{aligned}$$8 CSP framework, half-integral packings and covers of conflicting walks guide linear-time FPT algorithms for problems including Group Feedback Vertex Set, Subset Feedback Vertex Set, Node Multiway Cut, and Non-monochromatic Cycle Transversal; the associated LP relaxations have optimal solutions in $$\begin{aligned} \min \quad & \sum_{e \in E} c_e x_e \ \text{s.t.}\quad & x(\delta(v)) = 2, && \forall v \in V, \ & x(\delta(S)) \ge 2, && \forall S \subset V,\ S \neq \emptyset, \ & 0 \le x_e \le 1, && \forall e \in E. \end{aligned}$$9 and admit persistency properties used in branching (Iwata et al., 2017). Experimental work on Odd Cycle Transversal and Multiway Cut likewise studies branching algorithms guided by half-integral relaxations and half-integral packings of conflicting paths (Pilipczuk et al., 2018).

A polyhedral analogue appears in ATSP. Every half-integer vertex $(G,x)$0 of the asymmetric subtour elimination polytope can be written as

$(G,x)$1

for two cycle covers $(G,x)$2. In that setting, the cycle side is an average of two cycle covers, while the cut side is enforced by subtour inequalities and the active-cut structure that makes the point extreme (Sosso et al., 7 Nov 2025).

Within TSP proper, the remaining open direction is clear from the current state of knowledge. Half-integral cycle cut instances are already understood at the $(G,x)$3 level, whereas the unresolved obstacle is the half-integral degree cut case. A plausible implication is that progress toward a general $(G,x)$4 theorem for half-integral instances will require a comparably sharp structural analysis for degree cuts, together with a way of interfacing that analysis with the cycle-cut machinery already available (Jin et al., 2022, Jin et al., 1 Jul 2026).