# MIPLIB 2017 instances with no known feasible solution

MIPLIB 2017, the standard test library for mixed-integer programming, contains six instances for which nobody has reported a single feasible point, and none of them has been proved infeasible either. One of them, `datt256`, is a model of the Eternity II edge-matching puzzle, whose 2-million-dollar prize went unclaimed.

## Problem

Each instance is a mixed-integer linear program, distributed as an MPS file:

$$\min\; c^{\top}x \quad\text{subject to}\quad Ax \le b,\;\; \ell \le x \le u,\;\; x_j \in \mathbb{Z} \text{ for } j \in I,$$

where rows may also be equalities or $\ge$ constraints. For each instance below, either find a point $x$ that satisfies every constraint, bound and integrality requirement, or prove that no such point exists.

| Instance | Variables | Constraints | What it models (submitter) |
|---|---|---|---|
| `datt256` | 262,144, all binary | 11,077 | the Eternity II puzzle (Jon Dattorro) |
| `nucorsav` | 55,410 (49,360 binary) | 1,812,472 | scheduling on a continuous steel caster (Alexandra M. Newman) |
| `fhnw-binpack4-58` | 7,550 (7,400 binary) | 9,900 | 3D bin packing with extra constraints (Simon Felix) |
| `supportcase30` | 1,024, all binary | 1,028 | unknown, from the IBM developerWorks forum (Domenico Salvagnin) |
| `neos-3603137-hoteo` | 4,003 (3,913 binary, 90 integer) | 10,510 | anonymous NEOS Server submission (Jeff Linderoth) |
| `neos-3355323-arnon` | 10,248 (10,104 binary, 144 integer) | 21,216 | anonymous NEOS Server submission (Jeff Linderoth) |

## Background

MIPLIB is maintained at the Zuse Institute Berlin; its sixth edition, MIPLIB 2017, is described by Gleixner et al. (2021). Every instance is labelled easy (solved in under an hour with at most 16 threads by an out-of-the-box solver), hard (solved only by longer or special runs) or open (not yet reported solved). Of the 212 open instances, six carry the "no solution" tag, which MIPLIB describes as instances for which no solution is known and which could also be infeasible.

Eternity II, launched in July 2007, asks for 256 square pieces to be placed on a 16 × 16 board so that all adjacent edge colours match, with one piece fixed in advance. Edge matching is NP-complete in general (Demaine and Demaine, 2007), and such puzzles serve as hard SAT and CSP benchmarks (Ansótegui et al., 2008).

## What is known

- No feasible point is known for any of the six instances, and none is known to be infeasible; MIPLIB lists their objective value as "NA" (solution file version 37, released 24 September 2026).
- MIPLIB publishes no dual bounds, so here the open question is feasibility itself.
- Four of the six (`fhnw-binpack4-58`, `supportcase30` and the two NEOS instances) have a zero objective, so any feasible point solves them outright.
- For the puzzle itself, the best rewarded partial solution matched 467 of 480 edges (Louis Verhaard, 2008), and the prize deadline passed without a complete solution.

## What would count as progress

- A solution file in MIPLIB's SOL format (one `variable value` pair per line, plus an `=obj=` line) for any of the six. MIPLIB's solution checker tests integrality, bounds and every linear constraint in exact arithmetic with an absolute tolerance of 1e-4, which takes seconds.
- For `datt256` and `nucorsav`, which have objectives, a first feasible point also sets the best-known value; proving it optimal is a separate, harder step.
- A proof of infeasibility for any instance, ideally with a certificate that an independent checker can replay.

## References

1. A. Gleixner et al. (2021). MIPLIB 2017: data-driven compilation of the 6th mixed-integer programming library. *Mathematical Programming Computation* 13, 443–490. https://doi.org/10.1007/s12532-020-00194-3
2. E. D. Demaine and M. L. Demaine (2007). Jigsaw puzzles, edge matching, and polyomino packing: connections and complexity. *Graphs and Combinatorics* 23, 195–208. https://doi.org/10.1007/s00373-007-0713-4
3. C. Ansótegui, R. Béjar, C. Fernàndez and C. Mateu (2008). Edge matching puzzles as hard SAT/CSP benchmarks. *CP 2008*, LNCS 5202, 560–565. https://doi.org/10.1007/978-3-540-85958-1_42
4. MIPLIB 2017 instance page for `datt256`: https://miplib.zib.de/instance_details_datt256.html
5. Eternity II puzzle, Wikipedia: https://en.wikipedia.org/wiki/Eternity_II_puzzle

## Source

Imported from [MIPLIB 2017, No Solution tag](https://miplib.zib.de/tag_no_solution.html), curated by Zuse Institute Berlin. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.