Optimal sparse matrix partitioning

A matrix a day will keep the doctor away

This page presents a data base of optimal partitionings which can be used to benchmark heuristic sparse matrix partitioners for their quality in terms of communication volume of a parallel sparse matrix-vector multiplication. This page is under perpetual construction. New matrices will be added as we go. Currently we have a batch of 839 optimally bipartitioned matrices (and a few more). Comments are welcome. The main focus of the page is bipartitioning, but there is a section with results for small matrices partitioned into p=3 or p=4 parts.

Papers

The data base accompanies the following papers for p=2:

An exact algorithm for sparse matrix bipartitioning, by Daniel M. Pelt and Rob H. Bisseling, Journal of Parallel and Distributed Computing, 85 (2015) pp. 79-90. Postprint version
An improved exact algorithm and an NP-completeness proof for sparse matrix bipartitioning, by Timon E. Knigge and Rob H. Bisseling, Parallel Computing, 96 (2020) Article 102640. Postprint version

The data base accompanies the following paper for p=3, 4:

Engelina L. Jenneskens and Rob H. Bisseling, Exact k-way sparse matrix partitioning, Proceedings 12th IEEE Workshop Parallel/Distributed Combinatorics and Optimization (PDCO'22), 2022, pp. 754-763.

Programs

The initial set of 355 solutions for p=2 has been obtained by the program MondriaanOpt, developed by Daan Pelt and Rob Bisseling, which is included as a tool in the Mondriaan package, but is a separate program with a separate version number. It uses the matrix reading functions of Mondriaan, but is otherwise self-contained (so that its correctness can be verified independently). MondriaanOpt v1.0 has been included in Mondriaan v4.1, released November 7, 2016.

A larger set of 839 solutions has been obtained for p=2 by an improved program called MP (MatrixPartitioner) with stronger bounds developed by Timon Knigge, which is available from GitHub, matrix partitioner MP. The optimal communication volumes computed by both programs are the same. (This has been verified for all 368 matrices that both could solve.)

A set of 114 small matrices has been partitioned for p=3 and a set of 100 matrices for p=4 using an Integer Linear Programming (ILP) method, by Engelina Jenneskens, who also developed the General Matrix Partitoner (GMP) based on the branch-and-bound method for this purpose. Her software is available from GitHub, General Matrix Partitoner GMP. The optimal communication volumes computed by both methods are the same, as verified for the matrices that both could solve.

Errata and known issues

Erratum: the last sentence of section 3.4 of An improved exact algorithm and an NP-completeness proof for sparse matrix bipartitioning contains an error: the results of A-adjacent subgraphs and B-adjacent subgraphs cannot be added together, but either the maximum of the two results must be taken or all subgraphs must be grown together, enforcing disjointness. In the program MatrixPartitioner, disjointness is enforced for all subgraphs, so it computes the correct bounds. Also all obtained results are correct. Error found by Katerina Karanasiou, Huawei European Research Institute, Zurich (March 2022).

Erratum: the paragraph before the last in section II.B of Exact k-way sparse matrix partitioning contains an error and should be deleted. Thus, we need to use the original local lower bound L4 and not the improved lower bound L4. This is due to a problem with cycles in the bipartite graph, as pointed out to us in July 2022 by Pál András Papp from Huawei European Research Institute, Zurich. This problem can only occur for k>2. A related problem occurs for the global lower bound GL4, in the fifth paragraph of section II.C. Here, the solution is to allow the creation of only one conflict path from every vertex corresponding to a partially assigned row or column. The corrections of the problems will be be incorporated in the August 2023 version of the program GMP. Despite the error, all the experimental results in the paper are correct. We have verified them by running the corrected branch-and-bound program and also by the ILP program. The explanation for this good fortune is that the bounds L4 and GL4 are invoked after other bounds have been tried, and hence not very often, and that the occurrence of the problem itself seems to be rare.

Splitting into multiple parts

The partitioning results in the table and picture gallery below are for bipartitionings, i.e. for p=2 processors. The main reason is that heuristic partitioners are often based on recursive bipartitioning, which thus determines the overall quality of the partitioning. Furthermore, exact algorithms can only solve small problems to optimality within a reasonable time, and problems for 2 processors are easier to solve than those for more processors. Still, a limited set of results has also been computed for p=3, 4. Symmetric matrices stored in lower triangular form are first expanded to the full sparse matrix, which also includes the nonzeros of the upper triangle. Symmetry is not imposed in the partitioning.

Imbalance

The allowed imbalance is ε = 3%, which is a value commonly used in the literature for testing partitioners of sparse matrices, graphs, and hypergraphs. Having a single value keeps the database limited in size. Results for any other desired value of ε can be obtained by running the MondriaanOpt program or the program MP with that value as input parameter. The allowed imbalance is defined by the constraint

nz(A[i]) ≤ (1+ ε) ceil(nz(A)/p), for i=0,1,...,p-1.

Here, nz(A) is the number of nonzeros of the matrix A and nz(A)[i] the number of nonzeros assigned to processor P(i). The ceiling is taken to guarantee that a feasible solution always exists, also for ε= 0. Note that this differs slightly from the formula without the ceiling commonly used in heuristic solvers. To compare solutions, the input value of ε has to be adapted for the heuristic solvers.

Results for p=3, 4

The results for p=3, 4 are presented as two sets of files containing the solutions obtained by the ILP method: ILP matrices for p=3 and ILP matrices for p=4. The solutions are given in Matrix Market format, with a header, a line giving the number of rows, columns, and nonzeros of the matrix, followed by the nonzeros, one per line, giving the row index i, column index j, and the processor number of the owner of the nonzero. Pictures of the solutions are also provided in LaTeX/Tikz format: Tikz pictures for p=3 and Tikz pictures for p=4. These should be included in a Latex document to view them.

The solutions given were obtained using an ILP method implemented in IBM ILOG CPLEX 20.1 software and running on a 4.3 GHz AMD 3800 XT processor with 8 cores and 16 threads and 16 GB RAM.

Two Comma Separated Values files ilp_solutions_p=3.csv and ilp_solutions_p=4.csv present all the statistics of the solutions by the ILP method The CSV file can easily be handled by a spreadsheet program for further analysis. The columns are as follows: matrix name, number of rows m, number of columns n, number of nonzeros nz, number of nonempty rows, number of nonempty columns, communication volume CV, imbalance, time of GMP (in s), time of ILP (in s).

Results for p=2

The results for p=2 are presented in two different ways: a set of files containing the solution by MP of a batch of 839 matrices, and a picture gallery presented as a table containing the initial set of 355 matrices obtained by using MondriaanOpt.

MP matrices

This set of solutions has been obtained by running the MP program for 24 hours on all sparse matrices with at most 100,000 nonzeros from the SuiteSparse Matrix collection (formerly the University of Florida sparse matrix collection). This succeeded in 839 out of 1602 cases (cases with duplicate names were removed). The hardware used for each run of MP is a single core of an Intel Ivy Bridge E5-2695 v2 processor running at 2.4 GHz, part of a 24-core node of the Dutch National Supercomputer Cartesius at SURFsara in Amsterdam.

A Comma Separated Values file mp_solutions.csv presents all the statistics of the solutions by MP in the same format as the table of the picture galley, see below. The CSV file can easily be handled by a spreadsheet program for further analysis. The number of rows and columns in the CSV file is the number after removal of empty rows and columns. Thus these numbers may be less than the original numbers. The timings in the CSV file are given in seconds.

The solutions themselves, i.e., the bipartitioned matrices, are collected in a single file mp_matrices.tar.gz in gzipped tar format. The file size is 24 Mbyte, but it expands to a directory with a total size of 845 Mbyte when uncompressed. The directory contains the matrices as a textfile (with extension .mtx-I2f) giving the partitioning. Images of the bipartitioned matrices are collected in a single file mp_SVG.tar.gz in gzipped tar format. The file size is 59 Mbyte, but it expands to a directory with a total size of 1.4 Gbyte when uncompressed. The images are given in .svg format. The .mtx-I2f and .svg formats are described below.

Picture gallery: table rows

Each row of the table represents a sparse matrix. The rows are sorted by increasing number of matrix nonzeros (nz).

Picture gallery: table columns

The table contains 10 columns. The meaning of the headers is as follows.

Name: matrix name in the SuiteSparse sparse matrix collection. If the same matrix name (like `football') appears in two different matrix groups, the group name is also given. If you click on the name you get the details from the SuiteSparse page.
Rows: number of matrix rows (m).
Columns: number of matrix columns (n).
Nonzeros: number of nonzero matrix elements (nz). Note that explicitly stored zeros are also treated as nonzeros. See e.g. bcsstm01 which has 24 explicitly stored zeros.
Volume: communication volume in a parallel sparse matrix-vector multiplication (in data words).
Imbalance: achieved imbalance, expressed as the difference between the maximum number of nonzeros per processor and the minimum.
Solution: link to the solution given as an integer matrix in Matrix-Market format, with a header line, one line of the form m, n, nz, and then nz lines of the form

i j proc

where i is the row number of a nonzero, j the column number, and proc the processor number assigned by the solution, either 1 for processor P(1), or 2 for processor P(2), or a special value 3 meaning the nonzero is free, so that it can be assigned either way, to P(1) or P(2), without affecting communication. Still, their assignment must satisfy the load imbalance constraint (e.g. for the matrix jgl009, at least one of the free nonzeros in row 4 must be assigned to P(2), blue, as here the constraint requires perfect balance). Free nonzeros can be assigned such as to give the best possible load balance. The matrix name is extended by the postfix -I2f, to denote that the matrix has been partitioned into 2 parts, and that free nonzeros are displayed. The numerical values of the original matrix are discarded. The processor numbering is 1-based, in line with the Matrix-Market conventions, and to facilitate plotting by Matlab.
Date: date when the solution has been added to the table, in dd-mm-yyyy format. Keeps track of history.
Mond means MondriaanOpt was used to produce the solution. MP means the MatrixPartitioner program MP was used. Mond/MP means MondriaanOpt was used, and MP was used to verify the volume bound.
Time: time needed to solve the problem, either less than a second, minute, hour, day, week, month, year, decade on the current computer used. This is only a rough indication, as it depends on the initial upper bound on the volume given (for MondriaanOpt, the volume obtained by a run of the medium-grain method from Mondriaan v4.0 +1; for MP, trying increasing upper bounds, each time multiplying a failed bound by a factor 1.25) and the computer used (for MondriaanOpt, 1 core of an Apple iMac with 3.4 GHz Intel Core i5 processor; for MP, 1 core of a 2.4 GHZ Intel Ivy Bridge E5-2695 v2 processor).
Picture: a picture in Scalable Vector Graphics (SVG) format with red nonzeros corresponding to P(1), blue nonzeros to P(2), and yellow ones to the free nonzeros. For matrices that are too large (nz >1000) to show within the table in SVG format, you have to click on a link to see the picture. For matrices that are very large (nz > 100000), the picture will most likely not display on your browser.

Name	Rows	Columns	Nonzeros	Volume	Imbalance	Solution	Date	Version	Time	Picture
Trec3	1	2	1	0	1	solution	24-06-2015	Mond/MP	second
Trec4	2	3	3	1	1	solution	24-06-2015	Mond/MP	second
GL7d10	1	60	8	1	0	solution	04-08-2015	Mond/MP	second
Trec5	3	7	12	2	0	solution	06-08-2015	Mond/MP	second
b1_ss	7	7	15	3	1	solution	23-06-2015	Mond/MP	second
rel3	12	5	18	3	0	solution	23-06-2015	Mond/MP	second
ch3-3-b2	6	18	18	0	0	solution	15-12-2015	Mond/MP	second
cage3	5	5	19	4	1	solution	19-06-2015	Mond/MP	second
lpi_galenet	8	14	22	2	0	solution	10-08-2015	Mond/MP	second
relat3	12	5	24	3	0	solution	12-08-2015	Mond/MP	second
lpi_itest2	9	13	26	3	0	solution	13-08-2015	Mond/MP	second
lpi_itest6	11	17	29	2	1	solution	14-08-2015	Mond/MP	second
Tina_AskCal	11	11	29	3	1	solution	17-08-2015	Mond/MP	second
n3c4-b1	15	6	30	5	0	solution	18-08-2015	Mond/MP	second
n3c4-b4	6	15	30	5	0	solution	15-12-2015	Mond/MP	second
Tina_AskCog	11	11	36	4	0	solution	21-08-2015	Mond/MP	second
ch3-3-b1	18	9	36	5	0	solution	21-08-2015	Mond/MP	second
GD01_b	18	18	37	1	1	solution	25-08-2015	Mond/MP	second
Trec6	6	15	40	5	0	solution	25-08-2015	Mond/MP	second
farm	7	17	41	4	1	solution	25-08-2015	Mond/MP	second
Tina_DisCal	11	11	41	5	1	solution	26-08-2015	Mond/MP	second
kleemin	8	16	44	6	0	solution	26-08-2015	Mond/MP	second
LFAT5	14	14	46	4	0	solution	27-08-2015	Mond/MP	second
Tina_DisCog	11	11	48	6	0	solution	27-08-2015	Mond/MP	second
bcsstm01	48	48	48	0	0	solution	11-08-2015	Mond/MP	second
cage4	9	9	49	9	1	solution	07-08-2015	Mond/MP	second
jgl009	9	9	50	5	0	solution	28-08-2015	Mond/MP	second
GD98_a	38	38	50	0	0	solution	28-08-2015	Mond/MP	second
GD95_a	36	36	57	1	1	solution	28-08-2015	Mond/MP	second
klein-b1	30	10	60	5	0	solution	28-08-2015	Mond/MP	second
klein-b2	20	30	60	6	0	solution	31-08-2015	Mond/MP	second
n3c4-b2	20	15	60	9	0	solution	02-09-2015	Mond/MP	second
n3c4-b3	15	20	60	9	0	solution	02-09-2015	Mond/MP	second
Ragusa18	23	23	64	5	0	solution	03-09-2015	Mond/MP	second
bcsstm02	66	66	66	0	0	solution	07-09-2015	Mond/MP	second
lpi_bgprtr	20	40	70	4	0	solution	07-09-2015	Mond/MP	second
wheel_3_1	21	25	74	8	2	solution	10-09-2015	Mond/MP	second
rgg010	10	10	76	8	0	solution	10-09-2015	Mond/MP	second
jgl011	11	11	76	7	0	solution	10-09-2015	Mond/MP	second
Ragusa16	24	24	81	7	1	solution	11-09-2015	Mond/MP	second
LF10	18	18	82	4	0	solution	13-09-2015	Mond/MP	second
problem	12	46	86	2	2	solution	13-09-2015	Mond/MP	second
GD02_a	23	23	87	7	1	solution	15-09-2015	Mond/MP	second
n3c5-b1	45	10	90	8	2	solution	17-09-2015	Mond/MP	second
Stranke94	10	10	90	10	0	solution	17-09-2015	Mond/MP	second
ch4-4-b3	24	96	96	0	0	solution	20-09-2015	Mond/MP	second
GD95_b	73	73	96	2	0	solution	20-09-2015	Mond/MP	second
Hamrle1	32	32	98	5	2	solution	20-09-2015	Mond/MP	second
lp_afiro	27	51	102	5	0	solution	20-09-2015	Mond/MP	second
rel4	66	12	104	5	0	solution	25-09-2015	Mond/MP	second
bcsstm03	112	112	112	0	0	solution	07-09-2015	Mond/MP	second
p0033	15	48	113	5	3	solution	25-09-2015	Mond/MP	second
Pajek/football	35	35	118	8	0	solution	25-09-2015	Mond	second
n4c5-b11	10	120	120	0	0	solution	30-09-2015	Mond/MP	second
wheel_4_1	36	41	122	12	2	solution	30-09-2015	Mond/MP	hour
GlossGT	72	72	122	5	2	solution	30-09-2015	Mond/MP	second
ibm32	32	32	126	13	0	solution	05-10-2015	Mond/MP	minute
bcspwr01	39	39	131	6	1	solution	05-10-2015	Mond/MP	second
bcsstm04	132	132	132	0	0	solution	07-09-2015	Mond/MP	second
p0040	23	63	133	3	1	solution	11-09-2015	Mond/MP	second
GD01_c	33	33	135	7	1	solution	05-10-2015	Mond/MP	second
bcsstm22	138	138	138	0	0	solution	06-10-2015	Mond/MP	second
lpi_woodinfe	35	89	140	0	2	solution	06-10-2015	Mond/MP	second
ch4-4-b1	72	16	144	11	2	solution	06-10-2015	Mond/MP	minute
Trec7	11	36	147	8	3	solution	09-10-2015	Mond/MP	second
Trefethen_20b	19	19	147	16	5	solution	14-10-2015	Mond/MP	second
lp_sc50b	50	78	148	5	4	solution	14-10-2015	Mond/MP	minute
d_ss	53	53	149	4	3	solution	09-10-2015	Mond/MP	second
GD99_c	105	105	149	0	1	solution	15-10-2015	Mond/MP	second
bcsstm05	153	153	153	0	1	solution	15-10-2015	Mond/MP	second
refine	29	62	153	3	5	solution	16-10-2015	Mond/MP	second
karate	34	34	156	8	0	solution	15-12-2015	Mond/MP	second
Trefethen_20	20	20	158	17	4	solution	16-10-2015	Mond/MP	second
can_24	24	24	160	8	0	solution	08-12-2015	Mond/MP	second
lp_sc50a	50	78	160	5	4	solution	10-12-2015	Mond/MP	minute
bcspwr02	49	49	167	4	5	solution	11-12-2015	Mond/MP	second
lap_25	25	25	169	10	1	solution	11-12-2015	Mond/MP	second
relat4	66	12	172	4	4	solution	14-12-2015	Mond/MP	second
pores_1	30	30	180	9	2	solution	31-08-2015	Mond/MP	second
wheel_5_1	57	61	182	13	4	solution	14-12-2015	Mond/MP	day
GD96_b	111	111	193	3	5	solution	15-12-2015	Mond/MP	second
GD98_b	121	121	207	0	1	solution	16-12-2015	Mond/MP	second
n2c6-b1	105	15	210	11	2	solution	17-12-2015	Mond/MP	second
n3c6-b1	105	105	210	11	2	solution	17-12-2015	Mond/MP	second
n4c5-b1	105	15	210	11	2	solution	17-12-2015	Mond/MP	second
can_62	62	62	218	6	6	solution	18-12-2015	Mond/MP	second
dwt_72	72	72	222	4	2	solution	18-12-2015	Mond/MP	second
divorce	50	9	225	8	1	solution	25-06-2015	Mond/MP	second
GD96_d	180	180	229	0	1	solution	06-01-2016	Mond/MP	second
GD02_b	80	80	232	5	4	solution	25-06-2015	Mond/MP	minute
cage5	37	37	233	14	1	solution	06-07-2015	Mond/MP	second
Maragal_1	32	14	234	14	0	solution	08-01-2016	Mond/MP	second
d_dyn	87	87	238	5	4	solution	07-01-2016	Mond/MP	minute
d_dyn1	87	87	238	5	4	solution	07-01-2016	Mond/MP	minute
n3c5-b7	30	120	240	14	0	solution	12-01-2016	Mond/MP	hour
lpi_forest6	66	131	246	5	2	solution	13-01-2016	Mond/MP	second
Sandi_authors	86	86	248	4	0	solution	21-08-2015	Mond/MP	second
GD96_c	65	65	250	5	6	solution	13-01-2016	Mond/MP	minute
IG5-6	30	77	251	18	5	solution	14-01-2016	Mond/MP	second
bibd_9_3	36	84	252	22	0	solution	25-01-2016	Mond/MP	week
cat_ears_2_1	85	85	254	14	4	solution	01-02-2016	Mond/MP	week
dwt_59	59	59	267	8	1	solution	05-02-2016	Mond/MP	minute
GD97_b	47	47	264	11	0	solution	19-08-2015	Mond/MP	second
ex5	27	27	279	9	9	solution	05-02-2016	Mond/MP	second
will57	57	57	281	4	7	solution	29-01-2016	Mond/MP	second
GD95_c	62	62	287	6	1	solution	05-02-2016	Mond/MP	second
curtis54	54	54	291	7	7	solution	05-02-2016	Mond/MP	second
west0067	67	67	294	12	8	solution	02-09-2015	Mond/MP	hour
pivtol	102	102	306	4	8	solution	03-09-2015	Mond/MP	minute
mesh1e1	48	48	306	18	6	solution	21-08-2015	Mond/MP	hour
mesh1em1	48	48	306	18	6	solution	09-02-2016	Mond/MP	hour
mesh1em6	48	48	306	18	6	solution	09-02-2016	Mond/MP	hour
impcol_b	59	59	312	10	8	solution	14-08-2015	Mond/MP	hour
lp_kb2	43	68	313	14	9	solution	09-02-2016	Mond/MP	hour
dolphins	62	62	318	18	6	solution	06-07-2015	Mond/MP	hour
dwt_66	66	66	320	4	0	solution	29-01-2016	Mond/MP	second
n2c6-b10	30	306	330	0	0	solution	17-09-2015	Mond/MP	second
wheel_7_1	114	113	338	17	10	solution	25-06-2018	MP	minute
chesapeake	39	39	340	16	6	solution	21-08-2015	Mond/MP	second
lp_sc105	105	163	340	5	4	solution	10-02-2016	Mond/MP	hour
bfwb62	62	62	342	8	2	solution	09-02-2016	Mond/MP	second
west0156	156	156	371	5	5	solution	10-02-2016	Mond/MP	day
can_73	73	73	377	18	1	solution	09-02-2016	Mond/MP	minute
GD06_theory	101	101	380	0	0	solution	18-01-2016	Mond/MP	second
olm100	100	100	396	2	0	solution	21-09-2015	Mond/MP	second
tub100	100	100	396	4	10	solution	29-01-2016	Mond/MP	minute
odepb400	400	400	399	0	1	solution	18-01-2016	Mond/MP	second
bcsstk01	48	48	400	24	0	solution	10-02-2016	Mond/MP	hour
ch5-5-b1	200	25	400	17	0	solution	10-02-2016	Mond/MP	hour
gams10a	114	171	407	2	5	solution	22-09-2015	Mond/MP	second
gams10am	114	171	407	2	5	solution	22-09-2015	Mond/MP	second
impcol_c	137	137	411	14	13	solution	15-02-2016	Mond/MP	day
west0132	132	132	414	6	12	solution	17-02-2016	Mond/MP	minute
n4c6-b1	210	21	420	15	4	solution	17-02-2016	Mond/MP	minute
cis-n4c6-b1	210	21	420	15	4	solution	17-02-2016	Mond/MP	minute
bcsstm06	420	420	420	0	0	solution	19-01-2016	Mond/MP	second
lp_adlittle	56	138	424	9	0	solution	17-02-2016	Mond/MP	second
ash219	219	85	438	7	8	solution	18-02-2016	Mond/MP	hour
bfwa62	62	62	450	11	2	solution	17-02-2016	Mond/MP	second
lpi_ex73a	193	211	457	4	3	solution	24-02-2016	Mond/MP	month
GD00_a	352	352	458	0	0	solution	26-01-2016	Mond/MP	second
GD97_c	452	452	460	1	8	solution	17-02-2016	Mond/MP	second
lp_scagr7	129	185	465	6	13	solution	17-02-2016	Mond/MP	second
bcspwr03	118	118	476	8	12	solution	17-02-2016	Mond/MP	hour
bcsstm20	485	485	485	0	1	solution	26-01-2016	Mond/MP	second
lp_stocfor1	117	165	501	9	15	solution	26-02-2016	Mond/MP	hour
west0167	167	167	507	7	15	solution	25-02-2016	Mond/MP	minute
lesmis	77	77	508	10	0	solution	26-02-2016	Mond/MP	second
lp_blend	74	114	522	11	14	solution	26-02-2016	Mond/MP	minute
ash85	85	85	523	14	1	solution	25-02-2016	Mond/MP	hour
lns_131	131	131	536	22	16	solution	04-03-2016	Mond/MP	day
lnsp_131	131	131	536	22	16	solution	03-03-2016	Mond/MP	day
dwt_87	87	87	541	12	1	solution	25-02-2016	Mond/MP	minute
IG5-7	62	150	549	26	1	solution	25-02-2016	Mond/MP	minute
Trec8	23	84	549	17	1	solution	25-02-2016	Mond/MP	second
can_61	61	61	557	10	1	solution	29-02-2016	Mond/MP	second
impcol_a	207	207	572	7	16	solution	04-05-2016	Mond/MP	day
08blocks	300	300	592	0	0	solution	21-09-2015	Mond/MP	second
nos4	100	100	594	14	16	solution	04-05-2016	Mond/MP	hour
ch5-5-b4	120	600	600	0	10	solution	28-01-2016	Mond/MP	second
Sandi_sandi	314	360	613	0	1	solution	29-02-2016	Mond/MP	second
bcsstk03	112	112	640	0	0	solution	02-02-2016	Mond/MP	second
GD01_a	311	311	645	22	19	solution	17-05-2016	Mond/MP	week
lpi_box1	231	261	651	4	11	solution	04-03-2016	Mond/MP	minute
gent113	113	113	655	17	19	solution	04-03-2016	Mond/MP	minute
rel5	340	35	656	10	16	solution	04-03-2016	Mond/MP	minute
ash331	331	104	662	9	6	solution	07-03-2016	Mond/MP	day
zed	116	142	666	10	2	solution	08-03-2016	Mond/MP	second
lp_recipe	91	204	687	11	1	solution	08-03-2016	Mond/MP	hour
bcsstk22	138	138	696	8	20	solution	07-03-2016	Mond/MP	day
rotor1	100	100	708	14	20	solution	04-03-2016	Mond/MP	minute
n3c6-b11	60	675	720	0	0	solution	03-02-2016	Mond/MP	second
lpi_klein1	54	108	750	33	0	solution	08-03-2016	Mond/MP	hour
lp_share2b	96	162	777	9	1	solution	20-08-2015	Mond/MP	day
bwm200	200	200	796	4	20	solution	08-03-2016	Mond/MP	minute
rajat11	135	135	812	8	22	solution	07-03-2016	Mond/MP	hour
bcsstm19	817	817	817	0	1	solution	02-02-2016	Mond/MP	second
dwt_234	234	234	834	4	22	solution	09-03-2016	Mond/MP	minute
grid1_dual	224	224	840	0	0	solution	08-02-2016	Mond/MP	second
robot	120	120	870	12	12	solution	09-03-2016	Mond/MP	minute
polbooks	105	105	882	18	0	solution	09-03-2016	Mond/MP	hour
spaceStation_1	99	99	927	42	9	solution	29-06-2016	Mond	week
steam3	80	80	928	8	0	solution	25-06-2015	Mond/MP	hour
grid1	252	252	952	0	0	solution	04-02-2016	Mond/MP	second
cis-n4c6-b15	60	920	960	0	0	solution	03-02-2016	Mond/MP	second
ck104	104	104	992	0	0	solution	03-02-2016	Mond/MP	second
SmallW	396	396	994	12	28	solution	08-03-2016	Mond/MP	minute
CAG_mat72	72	72	1012	13	6	solution	14-03-2016	Mond/MP	second	click here (large picture)
GD00_c	638	638	1041	15	29	solution	25-07-2016	Mond/MP	week	click here (large picture)
lp_vtp_base	198	346	1051	16	1	solution	15-03-2016	Mond/MP	hour	click here (large picture)
relat5	340	35	1058	10	18	solution	14-03-2016	Mond/MP	second	click here (large picture)
bcsstm08	1074	1074	1074	0	0	solution	04-02-2016	Mond/MP	second	click here (large picture)
bcsstm09	1083	1083	1083	0	1	solution	08-02-2016	Mond/MP	second	click here (large picture)
lp_share1b	117	253	1179	7	35	solution	15-03-2016	Mond/MP	minute	click here (large picture)
ww_36_pmec_36	66	66	1194	29	4	solution	15-03-2016	Mond/MP	hour	click here (large picture)
odepa400	400	400	1201	3	31	solution	11-09-2015	Mond/MP	hour	click here (large picture)
lpi_cplex2	224	378	1215	15	35	solution	25-07-2016	Mond/MP	week	click here (large picture)
bibd_9_5	36	126	1260	34	24	solution	28-04-2016	Mond/MP	day	click here (large picture)
arc130	130	130	1282	13	12	solution	14-03-2016	Mond/MP	second	click here (large picture)
gams30a	354	531	1287	2	31	solution	18-09-2015	Mond/MP	second	click here (large picture)
gams30am	354	531	1287	2	31	solution	24-09-2015	Mond/MP	second	click here (large picture)
can_144	144	144	1296	12	0	solution	21-03-2016	Mond/MP	hour	click here (large picture)
n4c5-b10	120	630	1320	14	0	solution	04-08-2016	Mond/MP	week	click here (large picture)
Cities	55	46	1342	38	40	solution	28-04-2016	Mond/MP	hour	click here (large picture)
rajat05	301	301	1384	6	0	solution	21-03-2016	Mond/MP	minute	click here (large picture)
lp_bore3d	233	334	1448	21	0	solution	12-05-2016	Mond/MP	day	click here (large picture)
dwt_245	245	245	1461	8	41	solution	10-05-2016	Mond/MP	minute	click here (large picture)
bcsstm11	1473	1473	1473	0	1	solution	08-02-2016	Mond/MP	second	click here (large picture)
oscil_dcop_01	430	430	1544	2	16	solution	24-09-2015	Mond/MP	second	click here (large picture)
oscil_dcop_02	430	430	1544	2	16	solution	11-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_03	430	430	1544	2	16	solution	11-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_04	430	430	1544	2	16	solution	22-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_05	430	430	1544	2	16	solution	22-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_06	430	430	1544	2	16	solution	22-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_07	430	430	1544	2	16	solution	25-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_08	430	430	1544	2	16	solution	25-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_09	430	430	1544	2	16	solution	25-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_10	430	430	1544	2	16	solution	28-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_11	430	430	1544	2	16	solution	28-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_12	430	430	1544	2	16	solution	29-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_13	430	430	1544	2	16	solution	29-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_14	430	430	1544	2	16	solution	29-04-2016	Mond/MP	second	click here (large picture)
oscil_dcop_15	430	430	1544	2	16	solution	04-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_16	430	430	1544	2	16	solution	04-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_17	430	430	1544	2	16	solution	04-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_18	430	430	1544	2	16	solution	09-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_19	430	430	1544	2	16	solution	09-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_20	430	430	1544	2	16	solution	10-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_21	430	430	1544	2	16	solution	10-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_22	430	430	1544	2	16	solution	10-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_23	430	430	1544	2	16	solution	11-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_24	430	430	1544	2	16	solution	11-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_25	430	430	1544	2	16	solution	11-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_26	430	430	1544	2	16	solution	12-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_27	430	430	1544	2	16	solution	12-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_28	430	430	1544	2	16	solution	18-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_29	430	430	1544	2	16	solution	18-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_30	430	430	1544	2	16	solution	19-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_31	430	430	1544	2	16	solution	19-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_32	430	430	1544	2	16	solution	19-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_33	430	430	1544	2	16	solution	23-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_34	430	430	1544	2	16	solution	23-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_35	430	430	1544	2	16	solution	23-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_36	430	430	1544	2	16	solution	23-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_37	430	430	1544	2	16	solution	25-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_38	430	430	1544	2	16	solution	25-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_39	430	430	1544	2	16	solution	25-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_40	430	430	1544	2	16	solution	27-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_41	430	430	1544	2	16	solution	27-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_42	430	430	1544	2	16	solution	27-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_43	430	430	1544	2	16	solution	30-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_44	430	430	1544	2	16	solution	30-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_45	430	430	1544	2	16	solution	30-05-2016	Mond/MP	second	click here (large picture)
oscil_dcop_46	430	430	1544	2	16	solution	01-06-2016	Mond/MP	second	click here (large picture)
oscil_dcop_47	430	430	1544	2	16	solution	01-06-2016	Mond/MP	second	click here (large picture)
oscil_dcop_48	430	430	1544	2	16	solution	01-06-2016	Mond/MP	second	click here (large picture)
oscil_dcop_49	430	430	1544	2	16	solution	02-06-2016	Mond/MP	second	click here (large picture)
oscil_dcop_50	430	430	1544	2	16	solution	02-06-2016	Mond/MP	second	click here (large picture)
oscil_dcop_51	430	430	1544	2	16	solution	02-06-2016	Mond/MP	second	click here (large picture)
oscil_dcop_52	430	430	1544	2	16	solution	03-06-2016	Mond/MP	second	click here (large picture)
oscil_dcop_53	430	430	1544	2	16	solution	03-06-2016	Mond/MP	second	click here (large picture)
oscil_dcop_54	430	430	1544	2	16	solution	03-06-2016	Mond/MP	second	click here (large picture)
oscil_dcop_55	430	430	1544	2	16	solution	06-06-2016	Mond/MP	second	click here (large picture)
oscil_dcop_56	430	430	1544	2	16	solution	06-06-2016	Mond/MP	second	click here (large picture)
oscil_dcop_57	430	430	1544	2	16	solution	06-06-2016	Mond/MP	second	click here (large picture)
oscil_trans_01	430	430	1614	2	20	solution	24-09-2015	Mond/MP	second	click here (large picture)
GD96_a	1096	1096	1677	0	1	solution	02-05-2016	Mond/MP	second	click here (large picture)
lp_scagr25	471	671	1725	6	13	solution	20-05-2016	Mond/MP	hour	click here (large picture)
CSphd	1882	1882	1740	0	0	solution	08-02-2016	Mond/MP	second	click here (large picture)
p0548	176	724	1887	10	57	solution	21-06-2016	Mond/MP	day	click here (large picture)
bcsstm26	1922	1922	1922	0	0	solution	03-03-2016	Mond/MP	second	click here (large picture)
olm500	500	500	1996	2	48	solution	28-09-2015	Mond/MP	second	click here (large picture)
mhdb416	416	416	2312	2	66	solution	29-09-2015	Mond/MP	second	click here (large picture)
Chebyshev1	261	261	2319	4	5	solution	09-05-2016	Mond/MP	second	click here (large picture)
lp_scsd1	77	760	2388	12	20	solution	29-08-2016	Mond/MP	hour	click here (large picture)
gams60am	714	1071	2607	2	77	solution	29-09-2015	Mond/MP	minute	click here (large picture)
Harvard500	500	500	2636	10	76	solution	21-06-2016	Mond/MP	hour	click here (large picture)
ck400	400	400	2860	0	12	solution	03-03-2016	Mond/MP	second	click here (large picture)
n2c6-b9	306	1470	3060	0	0	solution	03-03-2016	Mond/MP	second	click here (large picture)
bcsstm23	3134	3134	3134	0	0	solution	03-03-2016	Mond/MP	second	click here (large picture)
dwt_492	492	492	3156	0	70	solution	02-03-2016	Mond/MP	second	click here (large picture)
G11	800	800	3200	0	0	solution	02-03-2016	Mond/MP	second	click here (large picture)
G12	800	800	3200	0	0	solution	02-03-2016	Mond/MP	second	click here (large picture)
bibd_81_2	3240	3240	3240	0	0	solution	10-03-2016	Mond/MP	second	click here (large picture)
dwt_512	512	512	3502	0	72	solution	10-03-2016	Mond/MP	second	click here (large picture)
tols1090	1090	1090	3546	0	54	solution	10-03-2016	Mond/MP	second	click here (large picture)
bcsstm24	3562	3562	3562	0	0	solution	15-03-2016	Mond/MP	second	click here (large picture)
bcsstm21	3600	3600	3600	0	0	solution	15-03-2016	Mond/MP	second	click here (large picture)
ck656	656	656	3884	0	12	solution	17-03-2016	Mond/MP	second	click here (large picture)
olm1000	1000	1000	3996	2	112	solution	24-09-2015	Mond/MP	second	click here (large picture)
tub1000	1000	1000	3996	4	116	solution	18-05-2016	Mond/MP	day	click here (large picture)
nos2	957	957	4137	4	123	solution	18-05-2016	Mond/MP	day	click here (large picture)
t2dal_e	4257	4257	4257	0	1	solution	17-03-2016	Mond/MP	second	click here (large picture)
ch6-6-b5	720	4320	4320	0	0	solution	18-03-2016	Mond/MP	second	click here (large picture)
lp_ship04s	402	1506	4400	4	100	solution	06-05-2016	Mond/MP	second	click here (large picture)
mk10-b4	945	4725	4725	0	45	solution	17-03-2016	Mond/MP	second	click here (large picture)
cage	366	366	5124	0	0	solution	16-03-2016	Mond/MP	second	click here (large picture)
netz4504	1961	1961	5156	0	0	solution	17-03-2016	Mond/MP	second	click here (large picture)
netscience	1589	1589	5484	0	0	solution	08-02-2016	Mond/MP	second	click here (large picture)
Pd_rhs	8081	12406	6323	0	1	solution	16-03-2016	Mond/MP	second	click here (large picture)
lp_ship04l	402	2166	6380	4	92	solution	06-05-2016	Mond/MP	minute	click here (large picture)
m3plates	11107	11107	6639	0	1	solution	17-03-2016	Mond/MP	second	click here (large picture)
lp_ship08s	778	2467	7194	7	174	solution	02-06-2016	Mond/MP	day	click here (large picture)
Chem97ZtZ	2541	2541	7361	0	1	solution	29-03-2016	Mond/MP	second	click here (large picture)
fpga_trans_01	1220	1220	7382	4	0	solution	05-05-2016	Mond/MP	minute	click here (large picture)
fpga_trans_02	1220	1220	7382	4	0	solution	05-05-2016	Mond/MP	minute	click here (large picture)
lung1	1650	1650	7419	4	213	solution	27-05-2016	Mond/MP	day	click here (large picture)
epb0	1794	1794	7764	5	184	solution	20-05-2016	Mond/MP	day	click here (large picture)
olm2000	2000	2000	7996	2	224	solution	28-09-2015	Mond/MP	second	click here (large picture)
bwm2000	2000	2000	7996	4	238	solution	17-05-2016	Mond/MP	day	click here (large picture)
G32	2000	2000	8000	0	0	solution	29-03-2016	Mond/MP	second	click here (large picture)
G34	2000	2000	8000	0	0	solution	05-04-2016	Mond/MP	second	click here (large picture)
rosen7	264	776	8034	8	22	solution	30-05-2016	Mond/MP	second	click here (large picture)
diag	2559	2559	8184	0	0	solution	30-03-2016	Mond/MP	second	click here (large picture)
lp_ship12s	1151	2869	8284	8	56	solution	14-06-2016	Mond/MP	day	click here (large picture)
iprob	3001	3001	9000	2	0	solution	02-05-2016	Mond/MP	second	click here (large picture)
laser	3002	3002	9000	4	270	solution	17-05-2016	Mond/MP	day	click here (large picture)
bcsstm38	8032	8032	10485	0	1	solution	06-04-2016	Mond/MP	second	click here (large picture)
G48	3000	3000	12000	0	0	solution	04-04-2016	Mond/MP	second	click here (large picture)
G49	3000	3000	12000	0	0	solution	04-04-2016	Mond/MP	second	click here (large picture)
grid2	3296	3296	12864	0	0	solution	30-03-2016	Mond/MP	second	click here (large picture)
lp_ship08l	778	4363	12882	7	166	solution	03-06-2016	Mond/MP	day	click here (large picture)
bcsstm25	15439	15439	15439	0	1	solution	05-04-2016	Mond/MP	second	click here (large picture)
ukerbe1	5981	5981	15704	0	0	solution	29-03-2016	Mond/MP	minute	click here (large picture)
rosen8	520	1544	16058	8	14	solution	30-05-2016	Mond/MP	second	click here (large picture)
mhd3200b	3200	3200	18316	2	542	solution	02-05-2016	Mond/MP	hour	click here (large picture)
p6000	2095	7967	19826	2	66	solution	02-05-2016	Mond/MP	minute	click here (large picture)
olm5000	5000	5000	19996	2	592	solution	28-09-2015	Mond/MP	minute	click here (large picture)
G57	5000	5000	20000	0	0	solution	05-04-2016	Mond/MP	minute	click here (large picture)
stoch_aircraft	3754	7517	20267	6	1	solution	26-08-2015	Mond/MP	minute	click here (large picture)
aircraft	3754	7517	20267	6	1	solution	04-08-2016	Mond/MP	minute	click here (large picture)
primagaz	1554	10836	21665	7	7	solution	03-06-2016	Mond/MP	day	click here (large picture)
rosen1	520	1544	23794	8	20	solution	18-09-2015	Mond/MP	minute	click here (large picture)
add32	4960	4960	23884	4	62	solution	02-09-2015	Mond/MP	day	click here (large picture)
bcsstk21	3600	3600	26600	0	0	solution	31-03-2016	Mond/MP	minute	click here (large picture)
zenios	2873	2873	27191	0	1	solution	30-03-2016	Mond/MP	minute	click here (large picture)
mhd4800b	4800	4800	27520	2	810	solution	18-09-2015	Mond/MP	hour	click here (large picture)
G62	7000	7000	28000	0	0	solution	01-04-2016	Mond/MP	minute	click here (large picture)
plbuckle	1282	1282	30644	0	802	solution	31-03-2016	Mond/MP	minute	click here (large picture)
G65	8000	8000	32000	0	0	solution	01-04-2016	Mond/MP	minute	click here (large picture)
G66	9000	9000	36000	0	0	solution	01-04-2016	Mond/MP	minute	click here (large picture)
Chebyshev3	4101	4101	36879	4	5	solution	04-09-2015	Mond/MP	minute	click here (large picture)
G67	10000	10000	40000	0	0	solution	31-03-2016	Mond/MP	minute	click here (large picture)
TS	2142	2142	45262	0	48	solution	31-03-2016	Mond/MP	minute	click here (large picture)
rosen2	1032	3080	47536	8	1396	solution	18-09-2015	Mond/MP	minute	click here (large picture)
lp_fit2p	3000	13525	50284	21	340	solution	18-09-2015	Mond/MP	minute	click here (large picture)
rosen10	2056	6152	64192	8	456	solution	27-08-2015	Mond/MP	minute	click here (large picture)
c-30	5321	5321	65693	30	1	solution	03-09-2015	Mond/MP	minute	click here (large picture)
conf5_0-4x4-10	3072	3072	119808	0	0	solution	06-04-2016	Mond	minute	click here (very large picture)
conf5_0-4x4-14	3072	3072	119808	0	0	solution	06-04-2016	Mond	minute	click here (very large picture)
conf5_0-4x4-18	3072	3072	119808	0	0	solution	10-04-2016	Mond	minute	click here (very large picture)
conf5_0-4x4-22	3072	3072	119808	0	0	solution	10-04-2016	Mond	minute	click here (very large picture)
conf5_0-4x4-26	3072	3072	119808	0	0	solution	10-04-2016	Mond	minute	click here (very large picture)
conf6_0-4x4-20	3072	3072	119808	0	0	solution	04-04-2016	Mond	minute	click here (very large picture)
conf6_0-4x4-30	3072	3072	119808	0	0	solution	04-04-2016	Mond	minute	click here (very large picture)
lp_fit2d	25	10524	129042	21	3244	solution	28-08-2015	Mond	minute	click here (very large picture)
Muu	7102	7102	170134	0	0	solution	02-03-2016	Mond	minute	click here (very large picture)

Last updated August 10, 2023.
to Home page Rob Bisseling.