Minimum Perfect Editing in Graphs

• Minimum perfect editing is a graph modification problem where the objective is to flip the minimum number of edges to transform a graph into a perfect graph free of induced odd holes and antiholes.
• The integer-programming formulation uses binary variables and a cutting-plane approach to iteratively add constraints for eliminating forbidden odd cycle configurations.
• Heuristic repair methods, like the IterativeModificationHeuristic, selectively flip edges involved in many forbidden structures to improve computational performance and solution quality.

Minimum perfect editing is a graph modification problem in which an undirected graph is altered by the smallest possible number of edge additions and deletions so that the resulting graph is perfect. In the formulation studied in "Perfect Graph Modification Problems: An Integer Programming Approach," the input is a graph $G=(V,E)$, the output is a graph $G'=(V,E')$ on the same vertex set, and the objective is to minimize $|E\triangle E'|$ subject to the condition that $G'$ is perfect (Erdem et al., 29 Jul 2025). The problem belongs to a broader family of graph modification problems that also includes minimum perfect completion, which permits only edge additions, and the perfect sandwich problem, in which only a prescribed subset of non-edges may be turned into edges. The term "perfect editing" in this setting is graph-theoretic and differs from its use in string edit scripts and deletion–insertion coding.

1. Graph-theoretic definition and constraint structure

Let $G=(V,E)$ be an undirected graph on $n$ vertices. A graph $G'=(V,E')$ is perfect if neither it nor its complement contains an induced odd cycle of length at least $5$, called an odd hole, or the complement of such a cycle, called an odd antihole. The minimum perfect-editing problem asks for $E'\subseteq \binom{V}{2}$ minimizing

$$|E\triangle E'|\;=\;|\, (E\setminus E')\cup(E'\setminus E)\,|$$

subject to $G'=(V,E')$0 being perfect (Erdem et al., 29 Jul 2025).

This definition places perfection in the role of a global structural constraint rather than a local adjacency condition. The forbidden configurations are induced odd holes and odd antiholes, so the editing objective is naturally expressed as a search for the smallest perturbation that destroys every such obstruction. A common point of confusion is to conflate minimum perfect editing with minimum perfect completion. The former allows both additions and deletions of edges, whereas the latter allows only additions; the perfect sandwich problem is more restrictive still, since only a given subset of non-edges may be changed.

2. Integer-programming formulation

The exact approach introduces binary variables

$G'=(V,E')$1

Let $G'=(V,E')$2. Let $G'=(V,E')$3 denote the family of all labeled odd-hole configurations $G'=(V,E')$4 of odd cardinality at least $G'=(V,E')$5, and $G'=(V,E')$6 the family of all labeled odd-antihole configurations. The model is

$G'=(V,E')$7

The objective counts exactly the number of edge-status flips between $G'=(V,E')$8 and $G'=(V,E')$9. Constraint (2.1) forbids any induced odd hole $|E\triangle E'|$0, and constraint (2.2) forbids any induced odd antihole $|E\triangle E'|$1. Whenever all such inequalities hold, the Strong Perfect Graph Theorem implies that the edited graph is perfect. The formulation therefore converts a structural graph class recognition condition into a 0–1 optimization model with obstruction-elimination constraints.

3. Cutting planes and odd-hole separation

Because $|E\triangle E'|$2 grows exponentially in $|E\triangle E'|$3, the model is not solved by enumerating all constraints in advance. Instead, the method uses a cutting-plane scheme (Erdem et al., 29 Jul 2025). It initializes

$|E\triangle E'|$4

solves the restricted integer program, builds the candidate graph

$|E\triangle E'|$5

and then runs a separation subroutine, Algorithm FindOddHoles, on both $|E\triangle E'|$6 and its complement $|E\triangle E'|$7. If no odd hole or odd antihole is found, the candidate graph is perfect and optimal for the full model; otherwise, the newly discovered configurations are added as cuts and the process repeats.

The separation routine uses recursive backtracking to build all chordless cycles of odd length at least $|E\triangle E'|$8. The expected number of odd holes in an Erdős–Rényi graph $|E\triangle E'|$9 is analyzed as

$G'$0

with an analogous expression for odd antiholes. This expectation is used to guide a stopping rule in which the search may terminate after finding a prescribed fraction of the expected number of forbidden structures. The reported experiments nevertheless found that, for $G'$1, fully enumerating all violated holes at each iteration produced the best gaps. The resulting methodology exemplifies a standard exact-optimization trade-off: exponential families of constraints are handled by iterative discovery rather than complete a priori generation.

4. Heuristic repair and upper bounds

To improve practical performance, the framework supplements exact cutting planes with a greedy repair heuristic called IterativeModificationHeuristic (Erdem et al., 29 Jul 2025). For an integer-feasible candidate graph $G'$2, the method first identifies the present odd holes and odd antiholes. It then counts, for each edge or non-edge pair $G'$3, the number of induced odd holes or antiholes containing that pair. While forbidden structures remain, it selects a pair $G'$4 participating in the maximum number of such structures, flips its status, and updates the hole set only in subgraphs containing $G'$5. If the flip does not strictly reduce the total count, it is reverted.

This heuristic is used to obtain improved upper bounds for both the editing and completion problems. It is also applied to fractional solutions by rounding $G'$6 to $G'$7, setting the remaining variables to $G'$8, and then repairing the resulting infeasibilities. A plausible implication is that the heuristic serves two roles simultaneously: it produces feasible perfect graphs quickly and also supplies incumbent solutions that sharpen the branch-and-cut search.

5. Computational profile and scalability

The computational experiments used CPLEX 22.1 on a $G'$9-core machine with $G=(V,E)$0 GB RAM. The test set comprised $G=(V,E)$1 random graphs $G=(V,E)$2 with

$G=(V,E)$3

with five instances for each parameter pair and a $G=(V,E)$4-minute time limit (Erdem et al., 29 Jul 2025).

Configuration	Reported outcome
No cuts, all holes pre-added	Solves only very small $G=(V,E)$5
Cutting planes alone	Solved all $G=(V,E)$6 in under one minute; for $G=(V,E)$7 mean gap $G=(V,E)$8; for $G=(V,E)$9 gaps $n$0
Add heuristic on integer candidates	Mean gap reduced from $n$1 to $n$2
Also call heuristic at every tenth fractional node	Mean gap reduced to $n$3
Pre-add all input-graph holes and use both heuristic strategies	Best performance: mean gap $n$4 in $n$5 s

For the minimum perfect-completion problem and the perfect-sandwich decision problem, similar trends were reported. Sandwich experiments extended to $n$6, and feasibility was most likely when both $n$7 and the optional-edge density were large. The complexity picture remains restrictive: minimum perfect editing is NP-hard, the full integer program has exponentially many hole constraints, and detecting odd holes is itself NP-hard when constrained to pass through a given vertex. Even so, the backtracking-based separation was reported to work well for $n$8, while the heuristic ran in a fraction of a second for $n$9, though its solution gap rose to $G'=(V,E')$0 by $G'=(V,E')$1.

6. Terminological disambiguation: edit scripts and perfect codes

Outside graph theory, the expression "perfect" appears in distinct edit-related contexts. In string similarity, "Algorithm to derive shortest edit script using Levenshtein distance algorithm" defines a string edit script

$G'=(V,E')$2

with operations in $G'=(V,E')$3, and calls the script "perfect" or "minimum" when

$G'=(V,E')$4

where $G'=(V,E')$5 is the Levenshtein distance between strings $G'=(V,E')$6 and $G'=(V,E')$7 (Liju, 2022). The dynamic-programming matrix $G'=(V,E')$8 satisfies

$G'=(V,E')$9

and, for $5$0, $5$1,

$5$2

A traceback from $5$3 to $5$4 recovers a shortest edit script in $5$5 time and $5$6 space, with traceback cost $5$7. The paper contrasts this Levenshtein-DP approach with Myers’ $5$8 diff algorithm, which is framed through longest common subsequence and emits insert/delete operations rather than directly modeling substitution.

In coding theory, "Optimal codes in deletion and insertion metric" addresses yet another notion. It improves the upper bound of Levenshtein for the cardinality of a code of length $5$9 capable of correcting single deletions over an alphabet of even size, shows that the new upper bound is sharp, and constructs an optimal perfect code capable of correcting single deletions for the same parameters (0810.3729). This usage concerns deletion and insertion metrics and perfect error-correcting codes, not graph modification or edit-script extraction.

These distinctions matter because the shared vocabulary of edits, optimality, and perfection can obscure materially different mathematical objects: edge sets in graphs, operation sequences on strings, and code families in combinatorial coding theory. A plausible implication is that "minimum perfect editing" should be interpreted by domain, not by terminology alone.