202:
variable is selected by some pricing strategy, based on the dual multipliers (node potentials), and forms a cycle with the arcs of the tree. The leaving variable is the arc of the cycle with the least augmenting flow. The substitution of entering for leaving arc, and the reconstruction of the tree is called a pivot. When no non-basic arc remains eligible to enter, the optimal solution has been reached.
201:
The network simplex method is an adaptation of the bounded variable primal simplex algorithm. The basis is represented as a rooted spanning tree of the underlying network, in which variables are represented by arcs, and the simplex multipliers by node potentials. At each iteration, an entering
44:
For a long time, the existence of a provably efficient network simplex algorithm was one of the major open problems in complexity theory, even though efficient-in-practice versions were available. In 1995
193:
in 1997. Strongly polynomial dual network simplex algorithms for the same problem, but with a higher dependence on the numbers of edges and vertices in the graph, have been known for longer.
187:
104:
36:. The network simplex method works very well in practice, typically 200 to 300 times faster than the simplex method applied to general linear program of same dimensions.
502:
124:
467:
219:
414:
235:
492:
432:
482:
317:
Tarjan, Robert E. (1997-08-01). "Dynamic trees as search trees via euler tours, applied to the network simplex algorithm".
477:
472:
497:
33:
17:
369:
487:
264:
Orlin, James B. (1997-08-01). "A polynomial time primal network simplex algorithm for minimum cost flows".
378:
133:
230:
214:
52:
383:
396:
342:
299:
224:
428:
334:
291:
29:
424:
418:
388:
326:
281:
273:
241:
360:
109:
46:
210:
The network simplex algorithm can be used to solve many practical problems including,
461:
364:
190:
127:
346:
451:
400:
303:
25:
338:
295:
392:
330:
286:
277:
367:(June 1993), "Polynomial dual network simplex algorithms",
49:
provided the first polynomial algorithm with runtime of
136:
112:
55:
32:. The algorithm is usually formulated in terms of a
181:
118:
98:
454:Section 14, p B-113 shows an example execution
8:
382:
285:
135:
111:
66:
54:
256:
503:Computational problems in graph theory
7:
126:is maximum cost of any edges. Later
468:Optimization algorithms and methods
182:{\displaystyle O(VE\log V\log(VC))}
236:System of distinct representatives
14:
99:{\displaystyle O(V^{2}E\log(VC))}
220:Hitchcock transportation problem
176:
173:
164:
140:
93:
90:
81:
59:
1:
519:
229:Chains and antichains in
34:minimum-cost flow problem
22:network simplex algorithm
18:mathematical optimization
493:Polynomial-time problems
452:Solving Network Problems
370:Mathematical Programming
319:Mathematical Programming
266:Mathematical Programming
413:Chvatal, Vasek (1983).
240:Covers and matching in
423:. Macmillan. pp.
231:partially ordered sets
183:
120:
100:
28:specialization of the
483:Mathematical problems
363:; Plotkin, Serge A.;
215:Transshipment problem
184:
121:
101:
478:Network flow problem
134:
110:
53:
473:Linear programming
420:Linear Programming
393:10.1007/bf01580615
331:10.1007/BF02614369
278:10.1007/BF02614365
225:Assignment problem
179:
116:
96:
130:improved this to
119:{\displaystyle C}
30:simplex algorithm
510:
498:Graph algorithms
439:
438:
410:
404:
403:
386:
377:(1–3): 255–276,
357:
351:
350:
314:
308:
307:
289:
261:
242:bipartite graphs
188:
186:
185:
180:
125:
123:
122:
117:
105:
103:
102:
97:
71:
70:
518:
517:
513:
512:
511:
509:
508:
507:
458:
457:
448:
443:
442:
435:
412:
411:
407:
384:10.1.1.297.5730
361:Orlin, James B.
359:
358:
354:
316:
315:
311:
263:
262:
258:
253:
246:Caterer problem
208:
199:
132:
131:
108:
107:
62:
51:
50:
42:
26:graph theoretic
12:
11:
5:
516:
514:
506:
505:
500:
495:
490:
488:Network theory
485:
480:
475:
470:
460:
459:
456:
455:
447:
446:External links
444:
441:
440:
433:
405:
352:
325:(2): 169–177.
309:
272:(2): 109–129.
255:
254:
252:
249:
248:
247:
244:
238:
233:
227:
222:
217:
207:
204:
198:
195:
178:
175:
172:
169:
166:
163:
160:
157:
154:
151:
148:
145:
142:
139:
115:
95:
92:
89:
86:
83:
80:
77:
74:
69:
65:
61:
58:
41:
38:
13:
10:
9:
6:
4:
3:
2:
515:
504:
501:
499:
496:
494:
491:
489:
486:
484:
481:
479:
476:
474:
471:
469:
466:
465:
463:
453:
450:
449:
445:
436:
434:9780716715870
430:
426:
422:
421:
416:
409:
406:
402:
398:
394:
390:
385:
380:
376:
372:
371:
366:
362:
356:
353:
348:
344:
340:
336:
332:
328:
324:
320:
313:
310:
305:
301:
297:
293:
288:
283:
279:
275:
271:
267:
260:
257:
250:
245:
243:
239:
237:
234:
232:
228:
226:
223:
221:
218:
216:
213:
212:
211:
205:
203:
196:
194:
192:
191:dynamic trees
170:
167:
161:
158:
155:
152:
149:
146:
143:
137:
129:
113:
87:
84:
78:
75:
72:
67:
63:
56:
48:
39:
37:
35:
31:
27:
23:
19:
419:
408:
374:
368:
355:
322:
318:
312:
269:
265:
259:
209:
206:Applications
200:
43:
21:
15:
365:Tardos, Éva
287:1721.1/2584
462:Categories
251:References
379:CiteSeerX
339:0025-5610
296:0025-5610
162:
153:
79:
347:18977577
197:Overview
425:320–351
401:5838223
304:3107792
40:History
431:
399:
381:
345:
337:
302:
294:
189:using
128:Tarjan
106:where
20:, the
397:S2CID
343:S2CID
300:S2CID
47:Orlin
24:is a
429:ISBN
415:"20"
335:ISSN
292:ISSN
389:doi
327:doi
282:hdl
274:doi
159:log
150:log
76:log
16:In
464::
427:.
417:.
395:,
387:,
375:60
373:,
341:.
333:.
323:78
321:.
298:.
290:.
280:.
270:78
268:.
437:.
391::
349:.
329::
306:.
284::
276::
177:)
174:)
171:C
168:V
165:(
156:V
147:E
144:V
141:(
138:O
114:C
94:)
91:)
88:C
85:V
82:(
73:E
68:2
64:V
60:(
57:O
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.