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Dagstuhl Seminar Proceedings 04401 Algorithms and Complexity for Continuous Problems http://drops.dagstuhl.de/opus/volltexte/2005/152
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1Dagstuhl Seminar Proceedings 04401Algorithms and Complexity for Continuous Problemshttp://drops.dagstuhl.de/opus/volltexte/2005/152

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º öé �¶¼ �lk ü 2 ê é Ø Ñ é Ø#e � öO �UThd Y é ] ¯B � é Ø 4 ÕSR î é=Û 4 ÕSR î Ý Ø � Y�� ] ø � � � � Y2 ] ¼

h�kKJeW]U � 1h� ��yöO �UT�d Y é ] ¯B � é Ø 4 Õ R î é�Û 4 Õ R î Ý Ø � Y�� ] ø � � � � Y

2 ] T38 b éjb� öO �UThd Y é ] » î b O b [ b éb ¼GIHKJn`tR[PXWIPMk�c N_PI�VN�oVJ$mTN_m®� » Î�y.{�J!cdcdR » � R[m�fTWMHKJ$L�J!ZlUVL�J�iJnH�RhoVJ

Ø Ñ Û 4 2 ê ë Ñ Ø#�L� � � � Y2� ] T öé �D¼ ��k ü 2 ê é Ø Ñ é Ø#e � 8 b éjb� » î6Y z Y é ] î b éb ] b éb î éö× C �nm x � ê�� Û ÿ ê�ÿý o ¼ � 1�� �

¶jJnJ!P�WMN_cdR'W]J�WMHKJ�PMkKc U'Z«WMHKJ$P�JSbKNQmKUVcFNtR[`�^$UzJ!çF^!N_J$mxWMP)NQm@WMHKJeZlUV`_`QU�iDN_mK�Y`QJ$cFcrR�upq�r�����síçÊ��©gæ!±[¨�¨�±¤��Â�²z³dÂ�²�©utÕ"�¤V¨µÂ�¡¦´���±[¨ �'°b éb î éö× C �nm x Û ÿ êaÿý o T e 3 4 m x Û ÿ ê�ÿ

ÿ ê�ÿ o b éjb _ ÿ êaÿLT Í á ¹ _ x ' » ÿ ê�ÿ �~»[¸ ���Â�¡�� e 3 4 | º è 3 4 à á Í á è éæåié Ýjv î é à cdR��xw 1 ¹�� `_m » � 3 4 k � èUyz¢�±�±�æ%{9vwm@iDH�R'WIZlUV`Q`_U�iDP$yzi�J�iDN_`Q`�L�J$\KL�J$P�J$mxW`x R[P8x º Ú ÿ êaÿ[iDNQW]H Ú ' » u|Ð�UVmKP�N_fKJ!LD�KLMP�WIWMHKJ�^hR[P�JÚ · � »�¹ �'� u�GIHKJ�`tR'LM�VJ!P�Wao'R[`_kKJ�U[Z m x Û ÿ ê�ÿ

ý o N_PaZlUVL«ý�º}| � x Û ÿ êaÿú� R »~ R[mKfdW]HKJ)PMkKc N_m �~»[¸ �a^$R[mb�JnJ!P�WMN_cdR'W]J!fTbxq

ÿ êaÿ m � Ú Û 1 ��ÿ ê�ÿ| � Ú Û 1 ��ÿ ê�ÿ R »~ o ¼))P�N_mK��ázWMN_L�`_N_m��Kó�P�ZlUVLMcgkK`tR�y�� ·�º � � R è � â k » v � èU��� å Y é Ý â ] ZlUVLePMU[cdJ:� â · �£¸�¹ 1 �&y�NQW�N_P�JhR[PXqBW]U@PMHKU�iW]HKR'W m � Í o T � � Í � B � �� Û Í � â î B����� � è éæå Y�� â ]» v Í � � Û ÍK� ¼ �~» 1 �

1h�

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¶jJekKPMJ�WMHKJe`tR[PXWIJ$PXW]NQcrR'WMJ�ZlUVLp�Tº � Ú Û 1 ��ÿ ê�ÿVR[mKf¹ÍFº3| � Ú Û 1 ��ÿ ê�ÿ R »6~ u�væZ«�Tº 1 W]HKJ!mIÍdº 1 R'mKfm � Í o º 1 uDvæZ��jN_P�U�fKf�R[mKfBR�W�`QJhR[PXW�J!{zkKR[`�WMU � y�N~uÉJ[u�y���º » J ä 1 iDNQWMH}JÇ' 1 y�WMHKJ$mÇÍ;ºùJ ä 1R[mKf@iJnJ!P�WMN_cdR'W]J

� � Í � B � �� Û Í � â î B ���� � è éæå Y�� â ]» v Í � � Û Í�� T m » J ä 1J ä 1 o ï\[ é m » J ä 1J o ï ���� �¦» J ä 1 � è éæå Y�� â ]» v � J ä 1 ��JT » â m 1 ä 1» J o ï ���� �~» ä 13R J�� è éæå Y�� â ]» v � J ä 1 � T » â

0� è é [ éæå Y�� â ]v � ¼

GIHKJn`tR[PXWIJ$PXW]NQcrR'WMJeN_PDR[`QPMU�W]L�kKJ�NQZ���N_PIJ�oVJ$m�uGIHKJ!LMJ�ZlUVLMJ�ZlUVL � Ú Û 1 ��ÿ ê�ÿë' » i�J�H�RhoVJ

ÿ ê�ÿ m � Ú Û 1 ��ÿ ê�ÿ| � Ú Û 1 ��ÿ êaÿ R »6~ o T ÿ ê�ÿ » Y��gî é^] b éjb ���� � è é ß åié Ýv � Ú Û 1 ��ÿ ê�ÿ T » Y��gî é^] b éb á � Í á è é ß åié Ý R v ¼á�NQmK^$J » �gî é R � Ú Û 1 ��T è `Qm » y�iJSU[b�W%R[NQm �~»[¸ ��uÐ�UVmKP�N_fKJ!LmKU�i WMHKJ�LMJ$cdR[NQmKN_mK�e^$R[PMJ Ú ' � u«GIHKJ)WMJ$L�cdPaU[Z�W]HKJ�PMkKcÏNQm �¦»[¸ �=R[L�J)m�U�i N_mK^!LMJhR'PMN_m��PMNQmK^$JYý T � x Û ÿ ê�ÿ � R » Û 1 u�GIHKJ$L�J!ZlUVL�JOW]HKJrPMkKc N_m �~»'¸ �nNQPYPMcdR[`_`QJ$L�W]H�R'm�ÿ ê�ÿ m � Ú Û 1 ��ÿ ê�ÿÿ êaÿ o u¾�\�\K`QqzN_m�� �¦» 1 �ZlUVLI��º � Ú Û 1 ��ÿ êaÿL' » R[m�fUͽº ÿ êaÿ[i�JnJhR[P�N_`�qO^$U[mK^$`QkKfKJeW]H�R�W

ÿ ê�ÿ m � Ú Û 1 ��ÿ ê�ÿÿ êaÿ o T á ÿ ê�ÿ è éæåié Ý R v � Ú Û 1 � b éb è b éb T á Í á è éæåié Ýuv î é � Ú Û 1 � b éb è 3 4iDHKNQ^%H�NQcd\K`QN_J!P �~»'¸ �pR[mKfT^$UVcF\K`QJ!W]J!PpW]HKJ�\KL�U�U'Z�U[Z�W]H�Jn`QJ$cFcrR�u �

¶�JS^$UVmxWMN_mskKJSWMUdJ!P�WMN_cdR'W]JeW]H�JgJ$L�LMUVL Ø Ñ Û 4 2 ê ë Ñ Ø#� � � � � Y2� ] u�á�NQmK^$J¿x � ê��X' » ÿ êaÿsiJg^$R[m�R[\�\K`Qq{�J$cFcrR 1 WMU � 1�� ��yKR[mKf@UVb�W]R[N_m

Ø Ñ Û 4 2 ê ë Ñ Ø#� � � � � Y2� ] T öé �¶¼ �jk ü 2 ê é Ø Ñ é Ø#e � e 3 4 8 b éb� » î6Y z Y é ] î b éjb ] m x � ê�� Û ÿ ê�ÿ

ÿ ê�ÿ o b éjb ¼á�N_m�^$J Ø Ñ Ø Ýe � º Ó é �¶¼ � Ø Ñ é Ø Ýe � T 1 y�i�Jn^$U[mK^$`QkKfKJeW]H�R�W

Ø Ñ Û 4 2 ê ë Ñ Ø � � � � � Y2� ] T e 3 4��� öé �D¼ � ü 2 ê é 8 Ý b éb� » î Ý Y z Y é ] î b éb ] m x � ê�� Û ÿ ê�ÿ

ÿ êaÿ o Ý b éb �� éæå Ý ¼ �~»V» �

1�È

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Ð�UVmKP�N_fKJ!L�WMHKJËZlkKmK^�W]NQUVm Ñ � Ú � º � Ú Û 1 � » î6Y��gî é^] ZlUVL Ú ' » u væW�NQPBJhR[PXq�W]U ^%HKJ!^ Æ W]H�R�WcdR�� �U� Ý Ñ � Ú � º Ñ � 1 ä 1SR `Qm » � º 1SR � è `Qm » �&yYR[m�f WMH�R'W Ñ NQP�fKJ!^$LMJ$R[PMNQmK� W]U ª!J$LMU ZlUVL Ú '1 ä 1SR `Qm » º »z¼ �V� ¼h¼$¼ u¶�JnR[L�J�LMJhR'f�qOW]U½fKJ���mKJ¿x � ê��pR[P x � ê��dº | Ú é ÿ êaÿ ~�¹ �~» � �iDHKJ!LMJ Ú é N_PpWMHKJ�PMcdR[`_`QJ$PXWImskKcYb�J$L Ú ' 1 ä 13R `Qm » ZlU[LpiDHKN_^%Hw � Ú Û 1 � » î6Y��gî é^] y b éb T k ½'é �~» � �iDNQWMH ½'é | º } Ý Ø ¾)íaí 2 Ø Ýe Ý3 4 �� 7 b éb�

ü 2 êé 8 Ý b éb� �� éæå Y é [xï å Ý ] 1

Ó Ä �¶¼ � w ü 2 ê Ä 8 Ý b Ä b� y ï å Y Ý [xï ] 7 b Ä b å Y é [xï å Ý ]� ¼

�)J!LMJPJrR[mKfp7 � R[L�JpW]HKJDmskKcYb�J$L�P|ZlL�UVc � 1�È �&u�GIH�R'W|NQP$yxNQZ ½�é ' � è `_m » � î Ý b éjb WMHKJ$m Ú é º 1 ä 1SR `_m » yR[mKfTNQZ ½�é � � è `Qm » � î Ý b éb W]HKJ!m Ú é NQPpW]HKJ�kKm�N_{skKJ�PMUV`Qk�W]NQUVm;U[Z�W]HKJ�mKU[m�¿w`QN_mKJ$R[L�J!{zkKR'W]NQUVm� Ú Û 1 � » î6Y��gî é^] º ½ éæå Y Ý b éjb ]é ¼

Ð�`_J$R[LM`�qVy�| Ú é ÿ ê�ÿ ~ ^hR'm�b�J�^$UVcF\Kk�WMJ$f@J���R[^�W]`Qq@bxq;k�PMN_m�����k�P�W)RYZlJ!i PXW]J$\�P)U[Z�î)J�iIW]UVm�ó�P�NQWMJ$L]R�W]N_U[mKP$uGIHKN_P�^$UVmK^!`_kKf�J$P«WMHKJpfKJ���mKN�W]N_U[mgU[Z�WMHKJpR[`Q�VUVL�NQW]H�cdP 4 2 ê ë ZlUVL�W]HKJ�R[\K\KL�U���NQcrR'WMN_UVmn\KL�UVbK`_J!c ¾�íaí 2 u¶jJ�P�WMLMJ!PMPpW]HKR'W�x � ê���fKJ!\�J!mKfKPIUVm;R'`_`�\�R[LMR[cdJ�W]J!LMP�� ¹ }zyzR[mKf ü ua��L�UVcÜW]HKJ�fKJ!��m�NQW]NQUVmrU'Z�x � ê��R'mKf�~»[» �i�JnUVb�W]R[N_mOWMH�R'W

Ø Ñ Û 4 2 ê ë Ø#� � � � � Y2� ] T e 3 4 �� öé �D¼ � ü 2 ê é 8 Ý b éb� ½�é �� éæå Ý º } Ø ¾�íaí 2 Ø ¼

�)J!mK^$J'y�W]HKJ�J$L�LMUVLIU'Z 4 2 ê ë NQPDR'WDcFUVP�WI} Ø ¾�íaí 2 Ø u��UVLpW]HKJ��VJ!mKJ$LMR[`�`_NQmKJhR'Lp\KLMU[bK`_J!cdP�ë Ï 2 ìsyKiJ�fKJ!��m�JnWMHKJnR[`_�VU[LMNQWMHKcFPK�4 2 ê ë bxq�4 2 ê ë | º Ï2�� 4 2 ê ë 4 iDNQW]H } á | º } Ø

Ï2 ØÑ 2 Ø ¾�íaí 2 Ø �~» È �

iDNQWMH�W]HKJ�mskKcgb�J!LMPfÑ 2 ZlLMUVc � � �&u�væW�N_P�^!`_JhR'L�W]H�R�W�W]HKJ�J!LML�UVL�U[Z �4 2 ê ë N_P�b�UVkKmKf�J$f�bxq�} Ø Ï 2 Ø u�¶jJPMk�cdcdR[LMNQª$J�WMHKN_P�R[mKR[`QqzPMNQPpN_m;WMHKJeZlUV`_`QU�iDNQmK�Y`QJ$cFcrR�u1 Î

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pq�r������M���©$¡?�����xÃ�� �'¡F��±[¨ �£¢«çÊ��©O¤V¨É³x±[¢&Â�¡��zÁg° 4 2 ê ë ³[Â���©$² §$´¤����¥�¡§¦@���Y¨%¡§¦�¤V².�s�©�£ª%¡§¦e¤V².� �4 2 ê ë³[Â���©$²�§$´«�©���'¡���¤L��©Y©$¢&¢�±[¢]°e°!¤V¡¦Â_°+æ&´�Â�²z³è éëê�ì ��4 2 ê ë ¯DE 2 �¹T } Ø ¾)í|í 2 Ø ¤V².� è éLê�ì � �4 2 ê ë ¯¶E 2 �¹T } ØÏ2 Ø _ �r·�v E [ ¹ _ }Y· �¦¸�¹ 1 � ¼

GIHKJBfKJ���mKNQWMN_UVm�PrU[Z�WMHKJBR[`Q�VUVL�NQW]H�cdP 4 2 ê ë R[mKf¬�4 2 ê ë ZlUVL�crR[`Q`Qq�LMJ${skKNQLMJ�WMHKJ Æ mKU�iD`_J!fK�VJ�U'ZnR[`_`iJ$N_�[HsWMP ü 2 êé R'Pgi�J!`_`�R[PgW]H�JOmKUVL�c Ø ¾)í|í 2 Ø iDHKN_^%HþR'LMJOkKP�J$fþN_mjW]HKJ;fKJ���mKN�W]N_U[m�U[Zyx � ê��&u�î)U[WMJW]HKR'W<x � ê��)NQPeROfKJ!^$LMJ$R[PMNQmK�OZlkKmK^�W]NQUVmBU[Z Ø ¾�íaí 2 Ø y�R[mKf�WMHKJ$L�J!ZlUVL�JYi�J½^hR'mBkKPMJ½RO`QU�iJ$Leb�UVkKm�fBUVmØ ¾�íaí 2 Ø N_mFW]HKJ)fKJ!�KmKNQWMN_UVmFU[Z�x � ê��&u��)J!mK^$J'yKNQZ�W]HKNQPm�UVLMcÏNQPkKm Æ m�U�iDm�yKNQW|^hR'mOb�J)LMJ!\K`tR[^!J$fdbsqFW]H�J

ZlUV`Q`_U�iDN_mK�g`_U�i�J!LDb�UVk�mKf@UVb�W%R'N_mKJ!f@WMHKLMU[kK�VH@W]HKJ�N_mxWMJ$�VLMR'W]NQUVm;\KLMU[bK`_J!c�y�PMJ$Jr� � » yKGIHKJ!UVLMJ!c 1$� yØ ¾�íaí 2 Ø Ý ' Ø v5î�G 2 Ø Ý º ü 2 ê » ä öé �¶¼ � ü 2 ê é­e b éjb ¹

iDNQWMH e | º 1 2 � 5 �+� � 5 � Ú � �þ�+�&¹ Ú ��f �+�&¹ Ú � ¼{�J$cFcrR » R[P�i�J!`_`�R[P½W]HKJ@mKJ&�zWrW]H�J$UVL�J$c HKUV`Qf ZlUVL�W]HKJ�R[`_�[UVLMN�W]HKcFPSiDN�W]H Ø ¾�íaí 2 Ø LMJ$\�`tR[^!J$f�bsqØ v5î)G 2 Ø R[P�R[b�U�o[J[u¶�JrmKU�ièJ!P�W]NQcrR�W]JdWMHKJO^$R[LMfKNQm�R[`QNQW]NQJ$P�U[ZIWMHKJ@R[`_�[UVLMN�W]HKcFP®�4 2 ê ë R'mKf 4 2 ê ë uBGIHKJO^hR'LMfKNQm�R[`_N�Wwq�U[Z�4 2 ê ë fKUzJ$P�m�U[W�J&��^$J$J!fBW]HKJS^hR[L�fKN_m�R'`_NQWwqOU[Z 4 2 ê ë y�R[m�f�W]HKJS`tR'W�W]J!L)^$R[LMfKNQm�R[`QNQWwq;^$R[m�b�JgJ$PXW]NQcrR'WMJ$f�bsqW]H�JnP�kKc U[Z�^hR[L�fKN_mKR[`_N�W]N_J!PU'Z�W]H�Jn^!UVLML�J$P�\�UVm�fKN_mK�FR[`Q�VUVL�NQW]H�cdP N é ê O uaGIH�R'W�N_P!y^$R[LMf �+4 2 ê ë �ÃT 1 ä öé �¶¼ � ^$R[LMf � ê�� iDNQWMH ^hR'LMf � ê�� | º öO �UT Y é ] ^$R[LMf m 2PBDC é Ð B � ê ¹ B¦� o ¼

¾�P)N_mË� � Ê � y�^hR[L�f � ê��kT Ó O �§T Y é ] ^hR[L�f ��¯ B � é 4 ÕgR î é �kT Ó O �§T Y é ] @ B � é ^hR[L�f � 4 ÕgR î é ��uDGIHKJ!LMJ�ZlUVLMJYf�kKJW]U � 1�È ��y^$R[LMf � ê��ÃT öO �§T Y é ] 7 b éjb� » ïÕY b O b î b éb ] º 7 b éb� z Y é ] î b éjböF C b éb » ï F m L Û 1ÿ êaÿ Û 1 o ¼

GIHKJFbKN_m�UVcdN_R[`_P�N_mBW]HKJF`tR'P�WnPMkKcÌR'LMJ½N_mK^!LMJ$R[PMNQmK�Ky�PMU@i�Jd^hR[m®J$PXW]N_cdR'WMJ½WMHKJ$c bxqBWMHKJd`_R[LM�[J$P�WnUVm�JUVb�W]R[N_m�J$f;ZlUVL�L)º«x � ê�� Û ÿ ê�ÿòu«¶�JeW]H�J$mT�VJ!W

^hR[L�f � ê��¬T 7 b éb�» ï Û 1 » ïÕY z Y é ] î b éb [ é^] m x � ê�� Û ÿ ê�ÿ Û 1ÿ ê�ÿ Û 1 o ¼1 Ê

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))P�N_mK� �¦»[¸ �iJn��mKR[`_`�qrUVb�W%R[NQm^hR[L�f � ê��¬T e 3 4 » ï» ï Û 1 7 b éb� » ïÕY z Y é ] î b éb ] m x � ê�� Û ÿ ê�ÿ

ÿ êaÿ o b éb ¼��LMUVc �~» � �i�JnH�RhoVJ¿x � ê�� Û ÿ êaÿ'º°| � Ú é Û 1 ��ÿ ê�ÿ ~ T 1 ä � Ú é Û 1 ��ÿ ê�ÿsR[m�f» ïÕY z Y é ] î b éb ] m x � ê�� Û ÿ êaÿ

ÿ ê�ÿ o b éb T » ïÕY�� É î é^] b éb � Ú é Û 1 � b éjb » ï è éæå Y�� É î é^] T » ï\[ é » ïÕY�� É î é^] b éb � Ú é Û 1 � b ébPMNQmK^$J 1SR � Ú é Û 1 �KT `_m » ua��L�UVc WMHKN_PIiJnU[b�W%R[NQm^hR'LMf �+4 2 ê ë �FT 1 ä e 3 4 » Ý ï�[ é» ï Û 1²± ��4 2 ê ë � iDNQWMH ± ��4 2 ê ë � | º öé �D¼ � 7 b éb� » ïÕY�� É î é^] b éb � Ú é Û 1 � b éb ¼

�~» Îx�G�UFJ!P�WMN_cdR'W]J ± �+4 2 ê ë �i�J�mKJ!J$f@W]HKJ�ZlUV`Q`_U�iDN_m��YN_mKJ!{sk�R[`_N�Wwq½iDHKN_^%HTN_PpJ$R[P�qrW]U½^%HKJ$^ Æ u��UVLIR[mxq�JIW ¸ yÚ ' 1 R[mKfÃã¬' 1

Ú T � e ï Ú îWï w�`_m�w » � � Ú î � y y ï\[ é iDNQWMH � e ï | º³wM`_m �~» � Û è î é y î6Y ï\[ é^] ¼ �~»VÊ �GIHKJ!m�`_J�WMWMN_mK� Ú º Ú é Û 1 R[m�f¹ã�º ÿ ê�ÿòyzi�JnUVb�W]R[N_m

± ��4 2 ê ë �¹T öé �¶¼ � � � e ï 7 � � b éb m » Y�� É î é^] b éb� Ú é Û 1 � b éjb o ï w�`Qm®w » Y�� É î é^] b éb � Ú é Û 1 � î b éb y y Y ï\[ é^] b éb ¼��LMUVc W]HKJ�fKJ���mKN�W]N_U[mTU'Z Ú é i�J�H�RhoVJ

m » Y�� É î é^]Ú é Û 1 o Ý b éb º 1´ é ¹iDHKJ!LMJ ´ é º cFN_m w ½'és¹�� è `_m » � î Ý b éb y ua�)J!mK^$J'y

± ��4 2 ê ë � T öé �D¼ � � � e ï 7 � � b éb ´ îWï å Ýé � 1» `_m ´ î éé � Y ï\[ é^] b ébT cdR��é �¶¼ � w�`Qm ´ îWï å Ýé y Y ï\[ é^] b éb öé �D¼ � w � e ï 7 � J î6Y ï�[ é^] y b éb ´ îWï å Ýé

T cdR�� �� � e ïJ ï\[ é ¹ m � e ïJ ï\[ é o 3 4 �� cdR��é �D¼ � w `Qm ´ îWï å Ýé y Y ï\[ é^] b éjb öé �D¼ � 7 b éb� ´ îWï å Ýé ¼

1

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¶�J�J!P�W]NQcrR�W]J�W]H�Jn`QUV�xR[L�NQWMHKcdNQ^�Z£R'^!W]U[LDR[PIZlUV`_`QU�iDP |cdR��é �D¼ � w `_m � ´ îWï å Ýé � y Y ï\[ é^] b éb º crR��é �¶¼ � w `_m � 7 b éb� ´ îWï å Ýé � Û `Qm � 7 b éjb� � y Y ï\[ é^] b éb

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