Degree 437 example in Gr(3,9) solved numerically via lifting a primal-dual formulation.

There are 437 3-planes which nontrivially meet 4 general fixed 4-planes and 6 general fixed 6-planes in complex 9-space.

[We wrote a Maple script to generate these equations for a random instance of this problem.]

1e-2*(-21*x1-56*x2-56*b12*x4-8*x3-8*b12*x5-50-50*b12*x6+30*b12*x7+62*b12*x8+62*b13*x10-79*b12*x9-79*b13*x11-71*b12-71*b13*x12+28*b13);
1e-2*(66*x1-36*x2-36*b12*x4-41*x3-41*b12*x5+20+20*b12*x6-7*b12*x7+16*b12*x8+16*b13*x10-34*b12*x9-34*b13*x11-62*b12-62*b13*x12-90*b13);
1e-2*(97*x1-82*x2-82*b12*x4-66*x3-66*b12*x5+55+55*b12*x6+68*b12*x7+26*b12*x8+26*b13*x10+13*b12*x9+13*b13*x11-65*b12-65*b13*x12+5*b13);
1e-2*(-42*x1-82*x2-82*b12*x4+59*x3+59*b12*x5-69-69*b12*x6+23*b12*x7+25*b12*x8+25*b13*x10+5*b12*x9+5*b13*x11-75*b12-75*b13*x12+38*b13);
1e-2*(22*x1-81*x2-81*b12*x4-99*x3-99*b12*x5-2-2*b12*x6-98*b12*x7-41*b12*x8-41*b13*x10+24*b12*x9+24*b13*x11+65*b12+65*b13*x12+b13);
1e-2*(-90*x1-66*x2-66*c12*x4-11*x3-11*c12*x5+35+35*c12*x6+61*c12*x7+96*c12*x8+96*c13*x10+53*c12*x9+53*c13*x11+61*c12+61*c13*x12-70*c13);
1e-2*(22*x1-34*x2-34*c12*x4+69*x3+69*c12*x5+55+55*c12*x6-12*c12*x7+98*c12*x8+98*c13*x10+56*c12*x9+56*c13*x11-54*c12-54*c13*x12-88*c13);
1e-2*(89*x1-70*x2-70*c12*x4-44*x3-44*c12*x5-48-48*c12*x6-52*c12*x7+37*c12*x8+37*c13*x10+58*c12*x9+58*c13*x11-45*c12-45*c13*x12+72*c13);
1e-2*(-99*x1+14*x2+14*c12*x4+42*x3+42*c12*x5-68-68*c12*x6-19*c12*x7+51*c12*x8+51*c13*x10-35*c12*x9-35*c13*x11-52*c12-52*c13*x12+71*c13);
1e-2*(-30*x1-52*x2-52*c12*x4+9*x3+9*c12*x5-43-43*c12*x6+78*c12*x7+72*c12*x8+72*c13*x10+50*c12*x9+50*c13*x11+32*c12+32*c13*x12-23*c13);
1e-6*(16734.92485*x3*x4*x12-34783.80776*x3*x6*x10-34821.62384*x3*x6*x11+3876.960730*x3*x6*x12+6814.496906*x3*x11-63156.91232*x3*x7*x10-56665.71956*x3*x7*x11+5537.616164*x3*x7*x12+57885.14324*x3*x8*x11-13251.89729*x3*x8*x12-57885.14324*x3*x9*x10-6814.496906*x3*x9*x12-43046.50819*x1+39602.25373*x2+3981.296662*x3+27882.56563*x4+25519.56523*x5-37310.67304*x7-22541.27160*x8-8363.759803*x9+24505.62895*x10+26689.85718*x11-51818.23844*x1*x5+9219.690107*x1*x6+90391.73035*x1*x7+27161.62477*x1*x8+20491.27640-40755.22449*x1*x4+18982.25271*x2*x6*x10-16901.42166*x2*x6*x11-47568.31862*x2*x6*x12+130319.2589*x2*x7*x10-4449.569272*x2*x9-56815.28103*x2*x10+31596.55135*x2*x11-108436.2342*x2*x7+17499.37416*x2*x8-27882.56563*x2*x6+43902.75697*x2*x5+67408.01203*x1*x9*x12-70379.78724*x2*x5*x10+36907.97327*x2*x5*x11-16734.92485*x2*x5*x12-7345.977006*x1*x9*x10+257732.4599*x1*x7*x10+154409.7784*x1*x7*x11-40424.87358*x1*x7*x12+110590.6166*x1*x8*x12+2900.272673*x1*x5*x12+40717.16287*x1*x6*x10+23025.34822*x1*x6*x11-26991.03876*x1*x6*x12-15427.94469*x1*x4*x11-152091.8419*x1*x4*x12-134956.0582*x1*x5*x10-82066.76640*x1*x5*x11+13696.39229*x1*x9-110590.6166*x1*x10-67408.01203*x1*x11-80095.60833*x2*x7*x11-57139.11684*x2*x7*x12+7578.264486*x2*x8*x11+56815.28103*x2*x8*x12+7345.977006*x1*x8*x11-7578.264486*x2*x9*x10-31596.55135*x2*x9*x12+70379.78724*x3*x4*x10-36907.97327*x3*x4*x11-8736.687327*x1*x4*x10-43902.75697*x3*x4-18982.25271*x4*x10+16901.42166*x4*x11+47568.31862*x4*x12+34783.80776*x5*x10+34821.62384*x5*x11-3876.960730*x5*x12-47373.53878*x7*x10-49618.76137*x7*x11+2503.087241*x7*x12-19357.70720*x8*x11-24505.62895*x8*x12+19357.70720*x9*x10-26689.85718*x9*x12-25519.56523*x3*x6+51508.90397*x3*x8+9096.761187*x3*x9+13251.89729*x3*x10-40498.57406*x3*x7);
1e-6*(12196.80184*x3*x4*x12+91818.67008*x3*x6*x10-72755.71720*x3*x6*x11-23542.92060*x3*x6*x12+7606.793440*x3*x11-46855.76427*x3*x7*x10+4696.383720*x3*x7*x11-4555.833400*x3*x7*x12+51244.49295*x3*x8*x11+26182.00700*x3*x8*x12-51244.49295*x3*x9*x10-7606.793440*x3*x9*x12-12145.35906*x1+41579.43240*x2+3962.432664*x3+169079.5408*x4+87071.34908*x5-81164.18045*x7-93687.34986*x8+25940.95484*x9+168487.8228*x10-120448.7406*x11+3524.532035*x1*x5+146132.9773*x1*x6+8578.672752*x1*x7-140370.2065*x1*x8-29631.41241*x1*x4-135754.3968+327736.3153*x2*x6*x10-182658.4477*x2*x6*x11+170821.3766*x2*x6*x12+5173.083100*x2*x7*x10-37326.63635*x2*x9+102681.9873*x2*x10+7207.537740*x2*x11+6531.935910*x2*x7+181408.8718*x2*x8-169079.5408*x2*x6-21733.44696*x2*x5+16857.12353*x1*x9*x12-106885.5761*x2*x5*x10+7181.939030*x2*x5*x11-12196.80184*x2*x5*x12+88303.76032*x1*x9*x10+81856.55286*x1*x7*x10-10765.09857*x1*x7*x11+9935.447087*x1*x7*x12-46681.29060*x1*x8*x12+1084.851497*x1*x5*x12-157218.2812*x1*x6*x10+122970.1529*x1*x6*x11+34994.63060*x1*x6*x12+18205.12187*x1*x4*x11-26000.95415*x1*x4*x12-1008.523530*x1*x5*x10+2416.327159*x1*x5*x11+27714.68002*x1*x9+46681.29060*x1*x10-16857.12353*x1*x11+3386.940450*x2*x7*x11-2511.554650*x2*x7*x12+123625.6532*x2*x8*x11-102681.9873*x2*x8*x12-88303.76032*x1*x8*x11-123625.6532*x2*x9*x10-7207.537740*x2*x9*x12+106885.5761*x3*x4*x10-7181.939030*x3*x4*x11-186616.6902*x1*x4*x10+21733.44696*x3*x4-327736.3153*x4*x10+182658.4477*x4*x11-170821.3766*x4*x12-91818.67008*x5*x10+72755.71720*x5*x11+23542.92060*x5*x12+148114.6634*x7*x10-85132.28349*x7*x11+68654.31390*x7*x12-50928.65112*x8*x11-168487.8228*x8*x12+50928.65112*x9*x10+120448.7406*x9*x12-87071.34908*x3*x6+81378.06668*x3*x8-15887.75442*x3*x9-26182.00700*x3*x10-7069.057086*x3*x7);
1e-6*(-136174.3475*x3*x4*x12+7425.218235*x3*x6*x10+5005.792607*x3*x6*x11+60801.78302*x3*x6*x12-67372.35809*x3*x11+51020.60723*x3*x7*x10+42987.31410*x3*x7*x11-188.2650810*x3*x7*x12-1643.736244*x3*x8*x11+79969.82703*x3*x8*x12+1643.736244*x3*x9*x10+67372.35809*x3*x9*x12+68886.77773*x1+174886.1004*x2+116132.8654*x3+30840.06890*x4+19898.21120*x5-44730.05307*x7-32292.83184*x8-27720.56890*x9-118626.1755*x10-98154.20664*x11-10265.31393*x1*x5-16528.95915*x1*x6-38447.59544*x1*x7+6700.728960*x1*x8-14331.18520*x1*x4-24742.39031*x2*x6*x10-23359.59348*x2*x6*x11+154253.5349*x2*x6*x12-33212.01453*x2*x7*x10-16705.99531*x2*x9+62797.72363*x2*x10+48907.45847*x2*x11-110518.8250*x2*x7-15496.65640*x2*x8-30840.06890*x2*x6-1900.787810*x2*x5-11285.44744*x1*x9*x12-12748.93877*x2*x5*x10-13539.61630*x2*x5*x11+136174.3475*x2*x5*x12-241.7227310*x1*x9*x10-10802.43550*x1*x7*x10-7719.198570*x1*x7*x11-24912.30907*x1*x7*x12-15013.01525*x1*x8*x12+46412.43226*x1*x5*x12+1984.325540*x1*x6*x10+566.9166000*x1*x6*x11+57433.05149*x1*x6*x12-2968.287613*x1*x4*x11-10881.69800*x1*x4*x12+2277.415514*x1*x5*x10+964.6769570*x1*x5*x11+3927.873260*x1*x9+15013.01525*x1*x10+11285.44744*x1*x11-23928.96451*x2*x7*x11-73048.72096*x2*x7*x12+1665.063050*x2*x8*x11-62797.72363*x2*x8*x12+241.7227310*x1*x8*x11-1665.063050*x2*x9*x10-48907.45847*x2*x9*x12+12748.93877*x3*x4*x10+13539.61630*x3*x4*x11-4181.783643*x1*x4*x10+1900.787810*x3*x4+24742.39031*x4*x10+23359.59348*x4*x11-154253.5349*x4*x12-7425.218235*x5*x10-5005.792607*x5*x11-60801.78302*x5*x12+79674.49884*x7*x10+65318.15444*x7*x11+32402.95958*x7*x12-2220.302740*x8*x11+118626.1755*x8*x12+2220.302740*x9*x10+98154.20664*x9*x12-19898.21120*x3*x6-11988.86939*x3*x8-12487.33718*x3*x9-79969.82703*x3*x10-74064.33711*x3*x7+53464.68265);
1e-6*(108711.8356*x1*x6*x12-121795.2869*x1*x6*x11-48262.31139*x1*x6*x10-39752.53950*x1*x5*x12+88583.56920*x1*x5*x11+549.8109800*x1*x5*x10-1350.441020*x1*x4*x12+41616.37150*x1*x4*x11-145958.5172*x3*x9*x12+17116.72443*x3*x9*x10+6775.338491*x3*x8*x12-17116.72443*x3*x8*x11-24778.65024*x3*x7*x12-14812.03756*x3*x7*x11+3593.390135*x3*x7*x10+137558.9614*x3*x6*x12-78253.17101*x3*x6*x11-12499.21656*x3*x6*x10-7023.031720*x3*x4*x12+88815.72103*x3*x4*x11-3299.192677*x3*x4*x10+8648.171890*x2*x9*x12-59822.48868*x2*x9*x10-5051.664520*x2*x8*x12+59822.48868*x2*x8*x11+4939.480370*x2*x7*x12-43021.94438*x2*x7*x11-9037.699380*x2*x7*x10-14532.94345*x2*x6*x12+85351.30456*x2*x6*x11+50673.21741*x2*x6*x10+7023.031720*x2*x5*x12-88815.72103*x2*x5*x11+3299.192677*x2*x5*x10-77017.33121*x1*x9*x12+67686.50250*x1*x9*x10+29896.80150*x1*x8*x12-67686.50250*x1*x8*x11-32723.61563*x1*x7*x12+35969.01644*x1*x7*x11+14796.52090*x1*x7*x10+77017.33121*x1*x11-29896.80150*x1*x10+123242.4321*x1*x9+28931.05244*x1*x8+11567.11729*x1*x7-35816.57684*x1*x6-36861.94076*x1*x5-45041.31545*x1*x4+5051.664520*x2*x10-70046.98000*x2*x9-39552.35551*x2*x8+17862.10701*x2*x7+2902.933571*x2*x6+62584.64916*x2*x5-47391.15857*x3*x6-62584.64916*x3*x4-8648.171890*x2*x11+145958.5172*x3*x11-6775.338491*x3*x10+113918.0038*x3*x9+7829.770891*x3*x8+30690.84240*x3*x7-87354.88976*x1+11632.90646*x2-111858.7140*x3-2902.933571*x4+47391.15857*x5+14949.34354*x7-29587.16989*x8-47757.89569*x9-84925.73117*x10-132645.4689*x11-50673.21741*x4*x10-85351.30456*x4*x11+14532.94345*x4*x12+12499.21656*x5*x10+78253.17101*x5*x11-137558.9614*x5*x12+20951.93723*x7*x10+52139.75913*x7*x11-45473.67459*x7*x12-36258.94873*x8*x11+84925.73117*x8*x12+36258.94873*x9*x10+132645.4689*x9*x12+3620.352640-14967.92666*x1*x4*x10);
1e-6*(-51417.06373*x1*x6*x12-3679.363411*x1*x6*x11+24617.35055*x1*x6*x10-97952.85298*x1*x5*x12+12023.67940*x1*x5*x11-13643.35528*x1*x5*x10+129728.4084*x1*x4*x12-31028.26198*x1*x4*x11-4028.067160*x3*x9*x12-15325.22768*x3*x9*x10-120278.7743*x3*x8*x12+15325.22768*x3*x8*x11-100190.0274*x3*x7*x12+8723.638060*x3*x7*x11-120694.7335*x3*x7*x10+151249.3212*x3*x6*x12-20398.36601*x3*x6*x11-33653.90190*x3*x6*x10-58705.55174*x3*x4*x12+4435.605164*x3*x4*x11-90903.84518*x3*x4*x10+18496.08445*x2*x9*x12+37513.50641*x2*x9*x10+117432.8805*x2*x8*x12-37513.50641*x2*x8*x11+112424.8275*x2*x7*x12-13791.55400*x2*x7*x11+140455.0024*x2*x7*x10-169498.5504*x2*x6*x12+53997.28517*x2*x6*x11-942.2649787*x2*x6*x10+58705.55174*x2*x5*x12-4435.605164*x2*x5*x11+90903.84518*x2*x5*x10-21960.26542*x1*x9*x12+5515.550500*x1*x9*x10+69851.79660*x1*x8*x12-5515.550500*x1*x8*x11+33815.54360*x1*x7*x12-23639.18762*x1*x7*x11+66699.02870*x1*x7*x10+21960.26542*x1*x11-69851.79660*x1*x10-8098.234240*x1*x9+19912.40168*x1*x8-12588.29080*x1*x7-22861.20060*x1*x6-23376.31550*x1*x5+14948.58680*x1*x4-117432.8805*x2*x10+856.8580600*x2*x9+16247.27040*x2*x8+9181.363000*x2*x7-23407.99270*x2*x6+3997.444670*x2*x5+42718.35250*x3*x6-3997.444670*x3*x4-18496.08445*x2*x11+4028.067160*x3*x11+120278.7743*x3*x10+825.6338000*x3*x9-30732.00480*x3*x8-10991.36900*x3*x7+23278.65110*x1+5328.416140*x2-14557.48694*x3+23407.99270*x4-42718.35250*x5-33753.26338*x7+6333.524520*x8-13991.43662*x9-44722.08095*x10+36023.31659*x11+28303.17680+942.2649787*x4*x10-53997.28517*x4*x11+169498.5504*x4*x12+33653.90190*x5*x10+20398.36601*x5*x11-151249.3212*x5*x12+53249.51087*x7*x10-42773.78124*x7*x11-376.7013500*x7*x12-14046.88975*x8*x11+44722.08095*x8*x12+14046.88975*x9*x10-36023.31659*x9*x12+66112.87373*x1*x4*x10);
1e-5*(16543.60921*x1*x6*x12-34021.18990*x1*x6*x11+7855.453784*x1*x6*x10-2978.309175*x1*x5*x12+1493.592624*x1*x5*x11+11669.34525*x1*x5*x10+2148.562673*x1*x4*x12-4510.845099*x1*x4*x11+10342.75148*x3*x9*x12-64579.17789*x3*x9*x10-47967.31106*x3*x8*x12+64579.17789*x3*x8*x11-16331.43584*x3*x7*x12+28417.81363*x3*x7*x11-29823.34063*x3*x7*x10+42939.04197*x3*x6*x12-63021.22678*x3*x6*x11+24170.62347*x3*x6*x10+2557.415944*x3*x4*x12-6819.694681*x3*x4*x11+15659.91361*x3*x4*x10-1954.501235*x2*x9*x12+38821.97994*x2*x9*x10+19046.49377*x2*x8*x12-38821.97994*x2*x8*x11+7166.748200*x2*x7*x12-16251.52995*x2*x7*x11+16018.05487*x2*x7*x10-16770.70085*x2*x6*x12+34992.71640*x2*x6*x11-7887.813707*x2*x6*x10-2557.415944*x2*x5*x12+6819.694681*x2*x5*x11-15659.91361*x2*x5*x10+6413.090798*x1*x9*x12-34213.03377*x1*x9*x10-18117.67315*x1*x8*x12+34213.03377*x1*x8*x11-5374.300311*x1*x7*x12+15237.51347*x1*x7*x11-14376.40195*x1*x7*x10-6413.090798*x1*x11+18117.67315*x1*x10+11246.18766*x1*x9-3164.665771*x1*x8+3316.223776*x1*x7+564.7493719*x1*x6-3973.993353*x1*x5-3.932902800*x1*x4-19046.49377*x2*x10-14055.09568*x2*x9+3958.521696*x2*x8-4142.067455*x2*x7-712.3658101*x2*x6+4974.132573*x2*x5-5542.604221*x3*x6-4974.132573*x3*x4+1954.501235*x2*x11-10342.75148*x3*x11+47967.31106*x3*x10+21656.28643*x3*x9-2626.243968*x3*x8+8845.431454*x3*x7-5362.268210*x1+6696.293099*x2-15664.97551*x3+712.3658101*x4+5542.604221*x5-5882.236330*x7+4787.038683*x8-18762.87548*x9-5236.837097*x10-35008.29780*x11+7887.813707*x4*x10-34992.71640*x4*x11+16770.70085*x4*x12-24170.62347*x5*x10+63021.22678*x5*x11-42939.04197*x5*x12+9701.548927*x7*x10-4366.011291*x7*x11+13233.54423*x7*x12-27392.42016*x8*x11+5236.837097*x8*x12+27392.42016*x9*x10+35008.29780*x9*x12+1281.311884*x1*x4*x10+9705.027812);

[We used these parameters to solve the random instance in Bertini using regeneration.]

USEREGENERATION: 1;
TRACKTOLBEFOREEG: 1e-7;
TRACKTOLDURINGEG: 1e-7;
FINALTOL: 1e-11;
SLICETOLBEFOREEG: 1e-7;
SLICETOLDURINGEG: 1e-7;
SLICEFINALTOL: 1e-11;
SECURITYMAXNORM: 1e8;
MAXNORM: 1e8;
PRINTPATHMODULUS: 1000;
SAMPLEFACTOR: 0.1;
NUMSAMPLEPOINTS: 3;
ODEPREDICTOR: 2;
RANDOMSEED: 1000;

[This is how long it took to solve using 7 processors on a 3.6 GHz machine.]

real 11m20.025s
user 78m57.601s
sys 0m13.459s

[This is how long it took to certify the solutions (including a reality check) using one processor on a 3.6 GHz machine.]

real 9m22.862s
user 9m18.907s
sys 0m0.164s

We previously formulated and solved a random instance of this problem using a primal-dual formulation. The square primal-dual formulation with similar coordinates involves 24 variables and equations, while the newer square lifted formulation involves 16 variables and equations. The primal-dual computation and certification consumed 20.37 gigaHertz-hours and 2.00 gigaHz-hours respectively. The lifted computation and certification appear to be improvements as they only consumed 4.75 gigaHertz-hours and 33.54 gigaHertz-minutes respectively.


Degree 128 example in Fl(2,4,5;8) solved numerically via lifting or a primal-dual formulation.

We formulated a certain Schubert problem in Fl(2,4,5;8) using the lifting formulation and the primal-dual formulation. The steps involved are similar to those of the previous Grassmannian example. Both formulations are square, but the lifted formulation appeared to be more efficient, using 33 equations and variables while the primal-dual formulation uses 41 equations and variables. We similarly chose and solved random instances. Solving and certifying improved from 2.95 gigaHertz-days of processing power to solve and 1.78 gigaHertz-hours to certify in the primal-dual instance to 1.13 gigaHertz-days of computing to solve and 1.68 gigaHertz-hours to certify.

Degree 28,490 example in Gr(3,10) solved numerically via lifting.

We formulated the Schubert problem asking for the 3-planes in complex 10-space that nontrivially meet 3 given 5-planes and 12 given 7-planes.

[Here are the 17 equations for the lifted formulation of a random instance. We include the parameters we used in Bertini to solve this system..]


USEREGENERATION: 1;
TRACKTOLBEFOREEG: 1e-8;
TRACKTOLDURINGEG: 1e-8;
FINALTOL: 1e-12;
SLICETOLBEFOREEG: 1e-8;
SLICETOLDURINGEG: 1e-8;
SLICEFINALTOL: 1e-12;
SECURITYMAXNORM: 1e9;
MAXNORM: 1e9;
PRINTPATHMODULUS: 2000;
SAMPLEFACTOR: 0.1;
NUMSAMPLEPOINTS: 3;
ODEPREDICTOR: 2;
RANDOMSEED: 1000;

1e-2*(-90*x1-21*x2-21*b12*x5-56*x3-56*b12*x6-8*x4-8*b12*x7-50-50*b12*x8+30*b12*x9+30*b13*x12+62*b12*x10+62*b13*x13-79*b12*x11-79*b13*x14-71*b12-71*b13*x15+28*b13);
1e-2*(-65*x1+5*x2+5*b12*x5+66*x3+66*b12*x6-36*x4-36*b12*x7-41-41*b12*x8+20*b12*x9+20*b13*x12-7*b12*x10-7*b13*x13+16*b12*x11+16*b13*x14-34*b12-34*b13*x15-62*b13);
1e-2*(5*x1-75*x2-75*b12*x5+38*x3+38*b12*x6+97*x4+97*b12*x7-82-82*b12*x8-66*b12*x9-66*b13*x12+55*b12*x10+55*b13*x13+68*b12*x11+68*b13*x14+26*b12+26*b13*x15+13*b13);
1e-2*(-41*x1+24*x2+24*b12*x5+65*x3+65*b12*x6+x4+b12*x7-42-42*b12*x8-82*b12*x9-82*b13*x12+59*b12*x10+59*b13*x13-69*b12*x11-69*b13*x14+23*b12+23*b13*x15+25*b13);
1e-2*(61*x1+96*x2+96*b12*x5+53*x3+53*b12*x6+61*x4+61*b12*x7-70-70*b12*x8+22*b12*x9+22*b13*x12-81*b12*x10-81*b13*x13-99*b12*x11-99*b13*x14-2*b12-2*b13*x15-98*b13);
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[This is how long it took to (1) solve using 8 processors, (2) refine the solutions using 2 iteration of Newton's method with 96-bit precision, (3) further refine using 2 iterations with 192-bit precision, and (4) certify on a 3.6 GHz machine.]

real 3015m38.627s
user 20435m55.377s
sys 34m48.568s

real 29m0.954s
user 28m55.404s
sys 0m3.977s

real 37m41.367s
user 37m35.714s
sys 0m3.921s

real 13m6.791s
user 13m4.313s
sys 0m1.515s

The main computation was by far the longest, taking 1.71 GHz-months to solve. A primal-dual formulation of this Schubert problem would require 33 equations and variable. For this reason (together with the high degree), we did not attempt to solve a random instance of this problem using a primal-daul formulation.