Added metabolic polytope simplification

[DimitriosPavlouGR]

Metabolic polytope simplification

Adds redundancy removal and dimension fixing for metabolic polytopes, with two methods: an exhaustive one that tests every bound against the full constraint set, and a Clarkson like output-sensitive method that keeps each LP as small as the essential set found so far.

Additions

• metabolic_polytope.hpp - a representation for {x : A_eq x = b_eq, b_l <= x <= b_u}
• bigg_parser.hpp - reads BiGG models from JSON
• exhaustive_simplification.hpp - tests each bound by relaxing it and solving up to four LPs
• clarkson_simplification.hpp - Clarkson's method, starts with every bound relaxed and iteratively adds constraints to the essential set
• transformation.hpp - projects onto the affine hull, giving a full dimensional equivalent H-Polytope representation
• test/metabolic/* - unit tests and benchmarks for both methods
• examples/metabolic_simplification - example of parsing a BiGG model, simplifying and transforming it to quickly approximate its volume

Results

19 BiGG models, dimension fixing enabled. Compared against PolyRound (Gurobi) with matching settings (1e-7 tolerance)

Bounds relaxed: agree closely across all three implementations - identical on 11 of 19 models.

Dimension fixing: counts differ substantially because the two implementations test different things compared to PolyRound. PolyRound looks for inequality rows of zero width, while these implementations solve a max/min LP pair per variable against the equality system.

Runtime: Clarkson implementation is 2.7x faster than the exhaustive method and 3.1x faster than PolyRound on the set of tested instances.

Bounds relaxed

Model	n	m	bounds	PolyRound	exhaustive	Clarkson
e_coli_core	95	72	190	154	154	154
iNF517	754	650	1508	1309	1309	1309
iAT_PLT_636	1008	738	2016	1423	1422	1423
iSynCJ816	1044	928	2088	1864	1944	1865
iJN746	1054	907	2108	1884	1633	1863
iAM_Pc455	1074	907	2148	1758	1744	1748
iEK1008	1226	998	2452	2161	2160	2152
iCN900	1229	885	2458	2317	2317	2317
iYO844	1250	990	2500	2278	2278	2278
iND750	1266	1059	2532	2259	2259	2244
iPC815	1961	1552	3922	3481	3481	3481
iRC1080	2191	1706	4382	3457	3457	3455
iECED1_1282	2706	1929	5412	4622	4618	4530
iEC042_1314	2714	1926	5428	4640	4640	4639
iECIAI39_1322	2721	1953	5442	4668	4668	4668
iECSF_1327	2742	1951	5484	4658	4658	4657
iECBD_1354	2748	1952	5496	4708	4708	4704
iEcolC_1368	2768	1969	5536	4740	4740	4736
RECON1	3741	2766	7482	5362	5004	5111

Dimensions fixed

Model	PolyRound	exhaustive	Clarkson
e_coli_core	4	8	8
iNF517	50	241	241
iAT_PLT_636	0	0	0
iSynCJ816	37	460	263
iJN746	66	347	402
iAM_Pc455	13	223	223
iEK1008	53	333	333
iCN900	266	926	926
iYO844	125	594	582
iND750	104	635	635
iPC815	132	896	896
iRC1080	72	622	622
iECED1_1282	236	1062	1062
iEC042_1314	243	1070	1070
iECIAI39_1322	257	1152	1152
iECSF_1327	234	999	999
iECBD_1354	258	1097	1097
iEcolC_1368	257	1115	1115
RECON1	136	1272	1191

Runtime (seconds)

Model	PolyRound	exhaustive	Clarkson	vs exhaustive	vs PolyRound
e_coli_core	0.358	0.054	0.030	1.8×	11.9×
iNF517	6.605	3.417	1.285	2.7×	5.1×
iAT_PLT_636	12.944	6.096	3.584	1.7×	3.6×
iSynCJ816	12.622	6.466	3.582	1.8×	3.5×
iJN746	9.680	8.883	5.865	1.5×	1.7×
iAM_Pc455	12.040	38.696	6.451	6.0×	1.9×
iEK1008	14.587	11.392	4.199	2.7×	3.5×
iCN900	14.498	5.101	2.240	2.3×	6.5×
iYO844	12.817	14.591	17.695	0.8×	0.7×
iND750	15.917	30.207	5.143	5.9×	3.1×
iPC815	28.685	17.418	9.021	1.9×	3.2×
iRC1080	48.612	64.885	28.750	2.3×	1.7×
iECED1_1282	62.152	201.296	20.105	10.0×	3.1×
iEC042_1314	58.337	62.210	18.685	3.3×	3.1×
iECIAI39_1322	61.671	76.595	18.818	4.1×	3.3×
iECSF_1327	63.685	72.860	23.426	3.1×	2.7×
iECBD_1354	70.345	61.653	20.663	3.0×	3.4×
iEcolC_1368	63.408	73.278	19.958	3.7×	3.2×
RECON1	159.775	207.606	214.521	1.0×	0.7×

Example

examples/metabolic_simplification runs the full pipeline on a BiGG model:

./metabolic_simplification e_coli_core.json

input polytope
 reactions     : 95
 metabolites   : 72
 finite bounds : 190

clarkson: starting simplification on polytope with 95 reactions, 72 metabolites, and 190 finite bounds
clarkson: fixed 8 dimensions
clarkson: interior LP optimal, slack objective = 0.58199
clarkson: simplification finished, fixed 8 dimensions, relaxed 154 bounds

simplified polytope
 finite bounds    : 36
 bounds relaxed   : 154
 dimensions fixed : 8

transformed polytope
 dimension                   : 24
 constraints (finite bounds) : 36

volume
estimate : 8.28757e+27

Limitations

• Clarkson often requires dimension fixing - degenerate dimensions make it difficult for Clarkson to find an interior point, so the run fails, falling back to the exhaustive method instead
• Clarkson misses some bounds - due to numerical error Clarkson might not be able to classify some of the constraints as redundant, marking them as essential instead and returning a not minimal representation of the polytope as a result