A Markov Chain Monte Carlo approach to the Steiner Tree Problem in water network optimization

Minimal water networks serving a farm districtA Metropolis algorithm for the Steiner Tree Problem

Carlo Lancia, Alessandro Checco

University of Rome TorVergata

Cortina D’Ampezzo, January 28, 2009

Carlo Lancia, Alessandro Checco Minimal Water Networks

Outline

1 A Model for the Shortest Network ProblemFarm Districts and Water NetworksThe Minimal Steiner Tree Problem

2 Metropolis AlgorithmThe Statistical Mechanics approach to OptimizationMarkov Chain Monte Carlo for the Steiner Tree Problem

3 Numerical Data and ConclusionsNumerical Comparison with Primal-DualConclusions


A Model for the Shortest Network Problem Farm Districts and Water Networks

Introduction

DefinitionA farm district is a set of neighboring farmlands

DefinitionA water network is a set of pipes bringing water to the lands

ConstraintsWe look for the minimal water network such that

Its pipes lie on land boundaries only, so no land bridgingIt does reach at least a corner of each land boundary



Modeling the District

It is very difficult to model the shortest network problem on thegraph induced by the district topology.

Graph induced by the district



Modeling the District

We do prefer to work on a slightly different graph. =Steiner Nodes; =Distribution Node; ◾ =Terminal Nodes

Creation of a new graph G


A Model for the Shortest Network Problem The Minimal Steiner Tree Problem

Water Networks and Steiner Trees

Water networks are equivalent to Steiner Trees on G:

Example



Water Networks and Steiner Trees

Water networks are equivalent to Steiner Trees on G:

Example



The Shortest Network Problem

Some NotationWe call VS the set of Steiner nodes, VT the set of TerminalsA Steiner Tree is any tree T ⊂ G spanning VTThe cost vector c ∶ E → R+ is such that

c(u) = c(v) for fictitious edges u, v situated in the same landfictitious-edge cost is comparable with regular-edge cost

Formulation as Minimal Steiner Tree ProblemOn graph G a water network serving the district is a SteinerTree, so what we are looking for is the shortest Steiner Treehaving root in , with the additional requirement of terminalnodes being leaf nodes (no land crossings allowed)



The Shortest Network Problem

Some NotationWe call VS the set of Steiner nodes, VT the set of TerminalsA Steiner Tree is any tree T ⊂ G spanning VTThe cost vector c ∶ E → R+ is such that

c(u) = c(v) for fictitious edges u, v situated in the same landfictitious-edge cost is comparable with regular-edge cost

Minimal Steiner Tree Problem – NP-hard

minT ⊂G

∑x ∈E(T )

c(x)

s.t. VT ⊂ V (T )deg(v) = 1 ∀ v ∈ VT


Metropolis Algorithm The Statistical Mechanics Approach

The Statistical Mechanics Approach

Optimization Problem

Problem instance

Cost function, f(ξ)

Optimal solutions

Many Particles System

Particles configuration

Hamiltonian, H(ξ)

Ground states

All we need is a sampling algorithm

When β →∞ Gibbs distributione−βH(ξ)

Zis concentrated on

ground states, that is to say optimal configurations





Problem instance


Optimal solutions



Hamiltonian, H(ξ)

Ground states



Zis concentrated on






Problem instance


Optimal solutions



Hamiltonian, H(ξ)

Ground states



Zis concentrated on






Problem instance


Optimal solutions



Hamiltonian, H(ξ)

Ground states



Zis concentrated on






Problem instance


Optimal solutions



Hamiltonian, H(ξ)

Ground states



Zis concentrated on




MCMC Sampling

Let Ω be the set of all Steiner trees over GDefine a Markov chain (Xt) on Ω that is ergodic with limitmeasure equal to Gibbs distribution

Sampling Algorithm1 Follow the evolution of the

chain for a very long time τ2 Return min

t∈[0,τ]Xt



MCMC Sampling

Let Ω be the set of all Steiner trees over GDefine a Markov chain (Xt) on Ω that is ergodic with limitmeasure equal to Gibbs distribution

Sampling Algorithm1 Follow the evolution of the

chain for a very long time τ2 Return min

t∈[0,τ]Xt


Metropolis Algorithm MCMC for the Steiner Tree Problem

Cooking Up An Appropriate Markov Chain

Target

We want a chain which limit measure is π(ξ) = e−βH(ξ)

Z

Transition ProbabilitiesMetropolis rule allow us to design a chain that is reversiblewith respect to Gibbs distribution

P (ξ, η) = minπ(η)π(ξ) ,1 = exp−β [H(η) −H(ξ)]+Such design strategy is very general and independent ofthe functional form of Hamiltonian



Which Hamiltonian?

It is quite natural to choose the Hamiltonian as the cost functionof the problem, that is the length of tree ξ

H(ξ) =∑x∈ξ

c(x)

Upgrading the Hamiltonian

We rather prefer to work with the following Hamiltonian function

H(ξ) =∑x∈ξ

c(x) + h ∑v∈VT

(deg(v) − 1)

where the second sum is extended to all terminal vertices.



Which Hamiltonian?

Land Bridging Penalties

Employing this Hamiltonian, Steiner trees crossing one or morelands have energy proportional to h. When temperature is lowand h is large such configurations are nearly never sampled.

Upgrading the Hamiltonian

We rather prefer to work with the following Hamiltonian function

H(ξ) =∑x∈ξ

c(x) + h ∑v∈VT

(deg(v) − 1)

where the second sum is extended to all terminal vertices.



Chain Evolution

At time t denote current tree by Xt. Select an edge x ∈ E(G)u.a.r., then set Xt+1 according to the following rule:

Current tree



Chain Evolution

Selected edge is the red one. Removing body edges from thetree would result in two separate componentsÔ⇒Xt+1 =Xt

Body edges are not removed from current tree



Chain Evolution

Adding edges non-adjacent to current tree would result in twoseparate components Ô⇒ Xt+1 =Xt

Separate edges are not added to current tree



Chain Evolution

When edge x connects a terminal leaf to current tree we setXt+1 =Xt not to compromise solution feasibility

Branch edges connecting terminal leaves are never removed



Chain Evolution

If x connects a steiner leaf to current tree we cut it out and setXt+1 =Xt ∖ x

Branch edges connecting steiner leaves are always removed



Chain Evolution

If edge x is adjacent to current tree we may add it. We setXt+1 =Xt ∪ x with probability exp−β c(x)

Edges adjacent to current tree may be added to it



Chain Evolution

In case adding x to the tree results in a land crossing we setXt+1 =Xt ∪ x with probability exp−β(c(x) + h)

Attention must be paid to land crossing and penalty applied



Chain Evolution

When adding x to the tree would result in a loop we observethe following procedure which removes the cycle:

Edges introducing a loop may be added by swap procedure



Chain Evolution

We select uniformly at random an edge y of the loop adjacentto x, then swap x and y with probability exp−β [c(x) − c(y)]+

Edges introducing a loop may be added by swap procedure



Chain Evolution

We select uniformly at random an edge y of the loop adjacentto x, then swap x and y with probability exp−β [c(x) − c(y)]+

Swap procedure: edge (+) is added and (–) is removed



Chain Evolution

In case adding x to the tree results in a land crossing loop swapprocedure must be performed more carefully:

Attention must be paid to possible land crossing



Chain Evolution

Interchanging edges may lead to a land crossing, in such caseswe swap x and y with probability exp−β [c(x) − c(y) + h]+

Swapping must be penalised if it leads to a land bridging



Chain Evolution

Interchanging edges may lead to a land crossing, in such caseswe swap x and y with probability exp−β [c(x) − c(y) + h]+

Swapping (+) and (–) leads to bridging Ô⇒ penalty imposed



Chain Ergodicity

The chain we have just described is found to beirreducible and aperiodicIt has a unique stationary distribution, namely Gibbs one

limt→∞

P t(ξ, η) = e−βH(η)

Z

Proving irreducibility

Swap procedure grants that the chain can move from a Steinertree to another, stepping through feasible solutions only


Numerical Data and Conclusions Numerical Comparison with Primal-Dual

Numerical Comparison with Primal-Dual

Both Metropolis and Primal-Dual procedure were implementedin Fortran 90 and run on a 1.6 GHz Intel Dual Core Processor

Shortest Network Length (km)

Metropolis Primal-Dual

Test District #1 4,288 5,867Test District #2 6,052 7,123Test District #3 6,588 9,353Test District #4 16,359 20,378Test District #5 5,131 6,660Test District #6 8,123 10,176Test District #7 16,044 22,222



Numerical Comparison with Primal-Dual

Algorithm Execution Time (sec)

lands edges Metropolis Primal-Dual

T-D #1 110 909 8 58T-D #2 132 1091 11 62T-D #3 201 1670 16 71T-D #4 458 3858 23 380T-D #5 169 1413 15 66T-D #6 274 2450 20 75T-D #7 605 5117 38 744

Total runtime 131 1456



Test District #1 - Primal Dual



Test District #1 - Metropolis




1

2

3 4

6

5

17

25

7

13

8

9

18

10

11

20

12

21

24

14

40

15

16

28

19

29

22

85

23

26

32

34

27

41

58

3130

59

35

38

33

3637

46

39

53

4442

43

56

45

51

47

60

48

5049

55

78

52

63

54

62

57

61

104

99

76

72

6465 66

67

71

6869

70

87

100

73

74

106

75

102

82

77

91

79 80

90

81

120

86

8384

97

101

115

128

88

108

89

111

98

92

124

93

9495

123

96

157

118

105

103

107

130

137

110

144

117

138

113

109

141

126

148

180

112

114

143

116

146

119

135

195

121122

129

125

152

127

189

131

139

132

134133

158

136

162

140

156

147

142

173

154

145

149

168

150

181

151

153

170

207

280

155

166

159

193

160

161

179

251

163

205

164

165

167

169

216

171

191

200

172

223

174 176175

209

177178

214

182183 185

225

184187

226

186188

190

198

252

192196

206

194197

199

203

227

201 202

212

204

208

244

210

256

211

218

232

213

215

281

217

219

220221

222

265

224

246

236

228229

233

230

255

286

231

234

248

243

235

247

237

250

238

259

239240

260

241 242

245

273

249

275

282

271

253

254

277

257

258

287

272

261262

264263

283

266

279

267

278

268

269

274270

285

313

317

276

291

284

289

297

349

290

315

322

288

301

296

321

292

293

309

294

306

331

295

298

329

299

302300

307

303

304305

308

343

372

310311

314

312

323

316

335

319318

326

367

320

351

325

328

324

333

368

373

364

327

330

334

332

346

342

337

345

336

338

350

339340

341

352

371

344

380

347

348

354

353

355

356

365

357

358

362

359360

363

361

376

384

366

370

369

374

387

375

382

377

386

378

379

381

383

385




1

2

3

13

4

6

5

7

8

15

9

18

10

19

11

20

12

24

14

16

17

28

2122

23

42

27

25

26

34

30

58

29

59

70

31

35

32

36

33

43

3738

39

44

40

41

49

64

45

50

46

47

60

48

75

51

78

5253

56

54

63

55

83

57

68

104

61

69

62

72

6665

77

85

67

71

87

73

97

7476

91

79 80

90

81

120

82

112

84

86

115

88

108

89

111

96

92

124

93

94

130

95

123

118

98

137

99100

101

102 103

105

106107

116

113

109

141

110

114

128

134

143

117

119

122121

127

159

129

125126

189

131

199

135

132

139

162

133

153

145

158

136

138

140

156

142

173

154

144

146

147148

149

168

150151

170

152

280

171

155

157

193

160

236

161

251

163

205

164

180

165

166

169

167

216

200

172

223

174 176178 175 177 179

181

214

221

182

190

183 185

225

184

231

187

226

186

243

228

188

202

234

191

252

192196

206

195194197

198

232

262

201

204

241

203

208

242

210

207

209

244

211

215

256

213

239

212

220

217

230

218

255

219

253

237

222

235

224

261

246

227

229

233

286

287

250

238

245

240

260

254

263

247

330

248249

275

282

271

279

277

257

258259

266

283

264

267 268

276

318

265

270

288

269

284

281

272 273274

291

303

278

297

331

349

310

294

290

285

329

301

289

295

321

327

292

293

306

296

298299

302300

342

336

304305

313315

308311307

372

312

314

309

320

316317

366

324

335

344

319

326

351

322323

328

333

325

364

334

332

346

370

337

345

339

338

347

341340

352

371

343

359

380

350

348

362

353

361

355

354

356357

365

358

378

379

360

381

375376

363

383

367368 369

373

374

387

382

377

385

386

384




1

2

3

4

5

6

97

21

44

8

47

10

11

38

1213

15

79

14

19

16

94

17

18

20

23

25

22

56

26

211

24

111

31

27

48

28

29

53

30

32

3534

58

33

37

59

45

36

39

60

88

40

62

77

41

42 43

50

46

117

49

52

96

51

54

57

55

148

67

61

65

86

63

66

64

163

69

128

70

68

113

75

7371

76

81

72

116

121

74

90

104

78

185

80

8382

84

85

97

131

139

95

87

112

89

143

106

91

100

92

93

103

98

191

142

165

193

99

101

102

196

108

105

107

166

109

122

110

144

123

114

170

115

126

120

118

132

135

119

127

236

134

154

129

124 125

171

156

179

187

130

136

147

149

133

145

141

137

138140

224

151

146

162

150

155

216

218

158

152

204

153

235

159

157

214

161

215

160

219

223

178

177

164

173

167

168

230

169

174

182

172

175

176

252

181184

180

209

183186188

200

190 189

296

192194

285

195

199

197198

249

309

205201202

311

206

203

212

316

208210207

228

362

275

220

241

322

213

237

217

222

225

221

227

341

229

232

226

234

269

367

231

248

233

342

240

243

282

244

261

238

343

239

246

294

242

301

245

254

291

247

256

477

361

250

332

351

251

278

253

266

276

255

355

262

494

257

356

258

264

259

347

260

283

263

271

453

268

288

265

452

272

267

398

273

270

290

287

279

274

280 281

277

353

286

390

299

284

302

289

292

297

293

300

389

295

306303

298

310

331

379

372

305

312

304

317

323

307308

368

324

314313

321

475

315

337

318 319320

330

336

327 325326328

350

329

333

335334

340 338

418

344

339

357

345

482

348

378

346

412

359 358

349

352

354

427

360

373

363365

431

366

401

364

428

382

376

446449

369

399

370

371

380

374

439

375

407

385

404

377

386

381

388

383

448

384

442

391

392

387

400

485

405

394

396

393

480

395

464

421

397

414

406

410

402 403

433

413

430

408409

435

411

432

498

468

415

445

416417

526

419420423422

496

424

512

425

426

429

447

472

444

450

434

461

436437438

440441 443

551

467

462 463

451

459

454

517

510

456 455

460

521

457458

465

488

500

481

486

532

466

540

487

530

469

502

509

470

528

471

483

473474

476

478

543

479

537

519

484

505

567

513

489490

545

491

529

492 493

507

495 497

550

499

520

525

501 504503506

555

514

556 557

508

534

511

563

515516

598

561

518

542

522

573

523

524

590

569

581

636

527

538

587

547

531

588

535 533

595

582

536

553

591

539

541

544

549

546

568

609

548

613

574

552

554

559 558

594

560

580

562

577

564

575

565566

596

674

584

570571572

615

585

610

576578

606

579

597

593

583

618

627

586

612

589

643

592

600 599

608

632

677

601 602

623

603

604

605

607

626

638

611

625

614

631

616

676

617

622

619 620

621

624

633

630

628

635

629

663

691

634

646

642

660

637639640

641

647

666

648

644

662

645

667

678

655

649650654 651

723

652653657

734

656

665

733

658 659

694

661

664

683

689

669

679

668

715

670

758

671

701

672

673

675

712

721

749

698

680681

682

688

684

690

685

753

686 687

703

713

692

706 708

693

695 697

705

696

722

769

699

711

727

700

731

702

729

704

773

707

762

709

710

785

716

725

714

728

717718

815

719

743

720

741

724

810

764

726

730

746

804

735732

757

777

736

796

737 738

752

739740

806

747

742

744

755

745

772

759

748

750751

761

754

789

756

768

760

822

763

765

778776

766

767

801

770 771

856

774775

830

793

853

813

779780

839

781

782

849

783784

787786

791

877

788

878

790

871

792

820

794

795

819

797798 799

800

802

821

803

805

831

807

845

808814

827

809 811 812

828

816 817

826

818

865

835

829

857

823825824

861

854

832

909

833

837

834

842843

836

901

838

935

846

840841

844

889

847848

850

851852

867

858

855

937

859862

917

860

874

882

898

863864

866

875

868869870

872

873876

888

885

879

883

880 881

932

884

890891

886887

906

894

892

908

893

923

897

933

895

896

900

931

913

899

903

928

902

904905

925

907

910

911

919920

912

918914 915 916

940

929

936

921922

926

967

924

927

930

934

941

938

939

942

979

943

944

945946

947948

950

955

949

954953

951

952

956 957

966

958959

965

960 961

1018

962

1019

964 963

968

1017

970

983

973

971

1029

969

978977

972

974975

980

976

984

1028

986

1037

988

981982985

1031

987

1054

1003

989990

1056

991

1057

992993 994

998

995

1058

996997

1034

999

1061

1000

1001 1002

1042

1004

1064

1005

100610071008100910101011

1079

10121013

1077

101410151016

1036

1045

10201021 10221027 1023 1025

1101

1024

1041

1105

1026

11001098

1032 1030

1074

10331035

1071

1038 1039 1040

1053

1103

1046

1117

1043

1047

1044

1055

1171

1119

1048

1050

1121

1049

1065

1051

1052

1059

1060

10631062

1102

1085

1069

10661067

1068

1092

1070

1086

1072

1115

10731075

1076 1078 1080 1081 1082

1149

1083

1148

1084

1146

1087

1132

1088

10901089

1116

1095

1091

1093

1094 1096 1097

1120

1099

1165

1129

110411061108 11071109

1113

1110111111121114

1118

1131

1122

1123

11241125

1130

1128

1145

11261127

1157

1133 1134 1135

1155

11361137

1172

1138 1141 1139 1140

1167

1169

1142

1143

1144

1147

1150

1152

1151

1153

1154

1176

1156

1158

1159

1160

1161

1162

1163

1164

1166

1168

1170

1173

1177

1174

1175




1

2

3

43

4

5

9

6

7 8

49

10

24

11

38

1213

79

15 14

17

19

102

16

18

22

20

23

21

33

26

111

25

99

27

48

28

52

29

53

30

54

32

57

31

92

34

58

3637

59

35

87

80

39

60

40

62

77

41

71

4244

45

117

46 475051

76

5556

67

61

86

63

64

68

65

163

66

74

113

6970

81

72

108

73

84

123

75

88

83

78

82

91

89

85

97

131

95

112

105

101

90

106

93

100

103

94

9896

193

125124

104

107

166

109

110

115

144

114

126

182

116

120

118

119

127

142

121 122

154

129

171

156

128

130

136

148

132

147

133134135

141

137

138

200

139 140

143

151

229

145 146

152

149

162

150

155

158

197

153

159

157

160161

227

164

173

165

168

230

167

170169

174

172

176

175

181

177

178 179

180

204

183184

198

185186

250

187 188 189

296

190191

326

192

285

194195 196

223

207

249

199

201202

311

206

203

208

205

211

228

362

214

209210 213 212

215

219

237

216

217

329

218

225

220

221

233

222

341

224

232

226

242

367

245

231

236

248

342

234 235

282

244

238

239

243

240

246

241

301

259

252

254

247

256

251

277

267

253255

262

257

260258

261

265

360

263

271

453

264

288

270

452

266

275

398

268

269

416

290

397

272273

292

274

304

276

280

300

278

279

297

291

281

331

325

283284

286

302

287

289

371

293

389

294295

305303

298 299

312

380

440

317

306 307308309

368

310

314313

316

321

315

323

438

322

337

318 319320

336

476

327 324328

350

333330

332

334

465

335

338

418

344

357

339340

411

343

345

348346

412

347

425

358

349351

352353

354355356

363

359361

366

401

364

428

365

441

376

386

446449

369

370

372 373374

439

375

385

404

377 378 379

422

381

382383

448

384

393

391

392

387

447

388

485

390

396

480

394395

464

403

409

399

406

400

402

413

405

410

430

407

415

408

424

414

445

417419

420421 423

427

437

431

426

436

429 432

433434435

442443

471

466

444

467

463

451 450

454456 455

521

457458

459

488

460461

501

462

486

532

491

487

468

530

469

500

509

470

472 473474

541

475

477 478

543

479

537

481482483 484

505

499

513

527

489490

492 493

536

494

495 496497

550

498

520

525

508

502503

540

504

529

566

506

514

507

557

510

511

563

512

515516

598

517518

524

519

522523

590

526

581

538

587

528

547

531

533

588

535534

582

553

544

539

542

545546

584

609

551548 549

552

570

554

576

555556

559 558

562

594

561

560

578579

577

564

575

565

596

567 568

569

571

586

572

574573

592

585

606

624

580

593

671

583

620

591

595

589

604

672

613

616

597

623

602

599600

601603

607

605

622

621

631

608

645

610 611

625

612

615614

632

617618619

626

663

629

633

630

639

627

644

628

745

643

638

680

634 635636

660

637

690

661

670

640

641

647

642

662

646 648

649650

681

654

732

651

723

652653657

655

734

656

665

658 659

666

692

664

689

669

678

667 668

695

673

675

674

676677

698

679

742

736

744

682

694

683 684685686 687 688

691

713

708

693

696

705

722

697

700 699

727

704

731

701

702

703

729

714

706

773

707

762

758

709

710

794

711712

754

715 716

717

740

718

808

719

743

720

779

721

724

810

725

726

797

728

730739733 735737 738

757

748

741

756755

746747

801

749 750751753 752

761

796

769

759

807

760

822

763

766765

764

776

781

767

857

768

770 771

772774

830

775

833

793

853

777 778

816

780

798

866

782 783

852

784

867

785787

786

873

791788

878

789 790

871

792

880

820

795

819

870

799

800

802

803

804805 806

815 809

827824

811 812

823

813

825

814

828

817

826

818

821

909

829

861

831832

837

834

881

835

843

836

839 838

935

846

840

847

841 842

845 844

849

848

882

850

851

854

856855

937

859

858

863

860

874

862

864 865

868869

872

975

888

896

875876877

879

924

932

883 884885 886887

892

889 890

939

891

926

908

893

923

894895

922

897

898899

900

903901902

904905

925

906

907

910

920

911912

997

913

921

914 915 916

1007

917 918

940

919

936

927

928

930

929

931

982

933

934

938

951

942

941

943

944

945946

947948

955

949950

952

958

953

957

954

956

959

965

960

1018

961 962 964 963

1014

966

983

967

968 969

971970

972973974

979976

1028

977

986

978 980

988

981

1051

987

984985

993

1016

989

1054

1003

990

991

1056

992

1057

995

1001

994996

1058

1059

999998

1061

1000

10021004

1064

1005

100610081009

1010

1081

1011

1079

10121013

1077

1015

1076

1045

10171024

1019 1020

1094

1021 1022 10231026

1101

1031

1041

1025 10291027 10301032

10361033

1035 1034

1071

1037

1109

1038 1039

1053

1040

1103

1046

1042

1043

1044

1049

1047

1119

1048

1050

1123

1052

1126

1055

1060

1070

10631062

1065

1088

10661067

10681069

1086

1087

1115

1072

1073

1112

10741075

1154

1078 1080 1082

1083 1084

1085

1132

10901089

1095

10911092

1093

1122

1096 1097 1098 1099 1100

1165

11051102

1129

110411061108

1160

1107

1161

1113

1110

1162

1111

1163

1114

1156

1164

1127

1116

1117

1118

1120

1121

1143

11241125

1128

1157

1158

1152

1130

1131

1142

1133 1134 1135 1136

1175

1137 1138

1172

1141

1170

1139 1140

11681167

1169

1146

1144

1153

1145

11481147

1149

1151

1150

1155

1176

1159

1166

1171

1173

1174

1177


Numerical Data and Conclusions Conclusions

Conclusions

Final RemarksMetropolis algorithm appears to be more efficient in virtueof better solutions and shorter run-timeIt does not depend on fictitious-edge weightIt needs an initial Steiner tree to take its first move→ a brief high-temperature evolution seems to be a nice fix

Further developmentsMixing time of the chain, possibly via the coupling methodNumerical comparison on a larger data set


Numerical Data and Conclusions Conclusions

Conclusions

Final RemarksMetropolis algorithm appears to be more efficient in virtueof better solutions and shorter run-timeIt does not depend on fictitious-edge weightIt needs an initial Steiner tree to take its first move→ a brief high-temperature evolution seems to be a nice fix

Further developmentsMixing time of the chain, possibly via the coupling methodNumerical comparison on a larger data set


Documents

A Markov Chain Monte Carlo approach to the Steiner Tree Problem in water network optimization