- The paper identifies a six-job, four-machine K4 instance whose configuration linear program has value 2 while every integral schedule requires makespan 3, achieving the 3/2 gap.
- Exhaustive, exact-arithmetic enumeration proves that no instance with five or fewer jobs works and that this six-job witness is unique up to machine relabeling, while finding 13 seven-job, 154 eight-job, and 1,662 nine-job witnesses.
- The paper combines connectivity and rounding lemmas, Burnside-checked canonical enumeration, and complexity results showing that recognizing threshold-2 witnesses is coNP-complete, making exhaustive classification essential.
The problem and its setting
In restricted assignment—makespan minimization where each job has one size and a set of allowed machines—the configuration linear program (CLP) is the tightest relaxation in common use, and its integrality gap is open: it is known only to lie in [3/2,11/6] (2608.13318). On the subclass of two-weight graph balancing, where every job is allowed on at most two machines and sizes lie in {1,2}, the value was settled by Jansen, Land, and Maack: a seven-job instance attains $3/2$, and their Corollary 11 upper bound of $2 - s/b$ meets it at sizes {1,2}. The paper under review does not revisit that value. It asks a different question: how small an instance—a witness—must be to realize the gap, and how many such instances exist at each size.
The six-job witness
The paper exhibits an instance on four machines forming the complete graph K4: four unit jobs on a Hamiltonian cycle and two weight-2 jobs on the complementary perfect matching. Its integral optimum is 3 while the CLP is feasible at threshold 2, giving gap exactly $3/2$. The lower-bound argument is a parity trap: any orientation of the perfect matching saturates one vertex from each bipartition class at load 2, and since the cycle alternates between those classes, some unit job has both endpoints already saturated and forces load 3. Feasibility at threshold 2 is certified in exact rational arithmetic: each machine splits weight $1/2$–$1/2$ between its matching job alone and its two cycle jobs.
The witness is one job smaller than the smallest previously published instance for this class, and the paper is careful to attribute the unconditional $3/2$ lower bound itself to Jansen–Land–Maack rather than claim novelty for the gap value. It also documents that the two instances are genuinely distinct—the same degree sequence but different numbers of weight-2 jobs and different support graphs—with no embedding existing under any relabeling.
Minimality, uniqueness, and classification
The central results are exhaustive:
- Minimality: no instance with at most five jobs attains gap {1,2}0, on any number of machines.
- Uniqueness: at six jobs, again on any number of machines, the witness above is unique up to machine relabeling and unused machines. At four machines the redundant enumeration returns three raw hits, which are exactly the orbit of the witness under relabeling; the count matches the orbit–stabilizer computation ({1,2}1).
- Classification: at seven jobs uniqueness fails completely—exactly thirteen witnesses exist, including the Jansen–Land–Maack instance among three connected witnesses on four machines, five more on five machines, one on six, and four disconnected composites of the six-job core with a single extra job. At eight jobs there are exactly 154 witnesses (97 connected), and at nine exactly 1,662 (978 connected), with counts growing 1, 13, 154, 1,662 across six through nine jobs.
Two structural lemmas carry much of the weight. A connectivity reduction shows both optima decompose over connected components, so every witness contains a connected witness with at most as many jobs plus one machine. A weight bound shows a feasible relaxation at threshold {1,2}2 on {1,2}3 machines has total job weight at most {1,2}4, bounding job counts. A strict rounding lemma—reproving, with proof, a strict form of the Lenstra–Shmoys–Tardos bound that Verschae and Wiese assert without proof—shows every witness has {1,2}5, which pins all witnesses to a single cell of the threshold landscape. Notably, this cell emptiness depends on integer sizes in {1,2}6: doubling the witness to sizes {1,2}7 produces an instance with relaxation value 4 and integral optimum 6.
A separate theorem shows three machines never suffice, at any number of jobs, so the witness is minimum in machines as well as jobs. This is proved twice over: once via the rounding machinery plus a sweep, and once via an elementary pool-rounding argument using only subset sums and Eulerian orientations, which also yields independent bounds of {1,2}8 on three machines and {1,2}9 on four.
Grading by machine count instead of job count makes each level finite: witnesses on at most $3/2$0 machines number 0, 4, 111, and 3,293 for $3/2$1, at any number of jobs.
Structure at threshold 2 and computational hardness
Since every known witness has relaxation value 2, the paper characterizes feasibility at that threshold combinatorially. Fractionally, two inequality families per machine suffice—one of which (the big-job constraint $3/2$2) shows the CLP is strictly stronger than the assignment relaxation even at threshold 2. Integrally, schedulability reduces to choosing "servants" for weight-2 jobs and checking Hakimi's orientation condition via max-flow.
Against this, deciding whether $3/2$3 is NP-complete—and remains so restricted to instances fractionally feasible at threshold 2, via a reduction from bounded-occurrence satisfiability. Consequently recognizing witnesses is coNP-complete, and no min–max characterization of threshold-2 witnesses exists unless $3/2$4. The classification theorems therefore proceed by bounded exhaustion rather than certificate, and Lemma's integral characterization is necessarily one-sided: an NP certificate with no coNP twin.
Verification methodology
Every feasibility decision behind the exhaustive claims was made twice—once in floating point and once by an exact rational phase-1 simplex with Bland's rule—agreeing on all decisions and, critically, on hit sets rather than merely counts. The canonical enumerator (which quotients by machine relabeling) is validated against Burnside's lemma at every level, against brute-force canonicalization on small ranges, and against the redundant enumeration where both are feasible. Two positive controls anchor the negatives: the pipeline must rediscover the six-job witness, and at seven jobs must recover the Jansen–Land–Maack instance, which it was not designed around. An independently written enumerator sharing no code confirms the six-job uniqueness result from scratch. The paper is candid about what remains shared across pipelines (the configuration enumeration, brute-force optima, pruning) and about which direction errors bias; it also proposes, without implementing, a Farkas-certificate fast path that would make future exact tiers far cheaper.
Limitations and open questions
The results are confined to sizes $3/2$5 and jobs on at most two machines; enlarging either parameter can move per-instance gaps in either direction, and only class-level containment survives. The nine-job layer's disconnected half rests on a composite calculus whose completeness argument leans on the strict rounding corollary, though an independent route avoiding it is supplied. The exploratory search above $3/2$6 (51,812 candidates, none exceeding $3/2$7) is floating-point throughout and reported as evidence, not proof. The paper leaves open several specific questions: whether the recurring structural pattern observed among seven-job witnesses holds as a theorem; whether the witness family generalizes to larger cycles with chorded matchings or three sizes; and—outside the class—where the restricted-assignment configuration-LP gap actually lies within $3/2$8, which this work does not narrow.
Conclusion
This note converts a settled integrality-gap value into a precise finite combinatorial object: a minimum, unique six-job witness for the $3/2$9 configuration-LP gap in two-weight graph balancing, together with complete classifications at seven through nine jobs and a finished landscape up to six machines. Its methodological contribution—double-decided exact arithmetic, Burnside-checked canonical enumeration, and mandatory rediscovery of known positives—is arguably as durable as its enumerative results, particularly given the coNP-completeness of witness recognition, which forecloses any shortcut to these classifications.