On Partial Opimality by Auxiliary Submodular Problems

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1 On Partal Opmalty by Auxlary Submodular Problem Alexander Shekhovtov Mc., engneer Václav Hlaváč Prof., head of the Center for Machne Percepton Center for Machne Percepton, Department of Cybernetc Faculty of Electrcal Engneerng, Czech Techncal Unverty n Prague Techncka 2, Prague 6, Czech Republc Abtract In th work, we prove everal relaton between three dfferent energy mnmzaton technque. A recently propoed method for determnng a provably optmal partal agnment of varable by Ivan Kovtun (IK), the lnear programmng relaxaton approach (LP) and the popular expanon move algorthm by Yur Boykov. We propoe a novel uffcent condton of optmal partal agnment, whch baed on LP relaxaton and called LP-autarky. We how that method of Kovtun, whch buld auxlary ubmodular problem, fulfll th uffcent condton. The followng lnk thu etablhed: LP relaxaton cannot be tghtened by IK. For non-ubmodular problem th a non-trval reult. In the cae of two label, LP relaxaton provde optmal partal agnment, known a pertency, whch, a we how, domnate IK. Relatng IK wth expanon move, we how that the et of fxed pont of expanon move wth any truncaton rule for the ntal problem and the problem retrcted by one-v-all method of IK would concde.e. expanon move cannot be mproved by th method. In the cae of two label, expanon move wth a partcular truncaton rule concde wth one-v-all method. Keyword: energy mnmzaton, partal optmalty, pertency, max-um, WCSP, MRF, autarky, LP-relaxaton, expanon move. 1 Introducton 1.1 Energy Mnmzaton In th work 1 we conder mnmzaton problem of the followng form: [ ] mn x L f 0 + V f (x ) + t E f t (x t ) = mn f(x). (1) x L 1 The work wa upported bu EU project FP7-ICT NIFT and FP7-ICT HUMAVIPS and the Czech project 1M0567 CAK. Here, V a fnte et and E V V. A concatenated vector of all varable x = (x V) called a labelng. Varable x take t value n a dcrete doman L, called label. Labelng x take value n L, the Cartean product of all doman L. In th paper all L wll have the ame number of label, but may have dfferent aocated orderng, etc. Notaton t denote the ordered par (, t) and x t denote the par of correpondng varable, (x, x t ). The objectve compoed of term f 0 R and functon f : L R and f t : L L t R. The problem (1) condered n everal feld. It 1

2 alo known a the labelng problem, the Weghted Contrant Satfacton (WCSP) and for the cae of two label ( L = 2, ) a the peudo-boolean 2 optmzaton [1]. Our termnology come from conderng probabltc model n the form of Gbb dtrbuton. There certan dfference between problem wth two label and more than two label, the later wll be referred to a mult-label problem. 1.2 Partal Optmalty Energy mnmzaton (1) an NP-hard problem n general. Technque whch allow u to fnd a part of the optmal labelng are of our central nteret here. The dea that t may be poble to fx a part of varable to take certan label uch that any optmal labelng wll provably have the ame partal agnment. More precely, we conder a ubet of varable A V and the agnment of label over th ubet y = (y A). The par (A, y) called a trong optmal partal agnment (trong pertency [3]), f for any mnmzer x t hold x A = y, where notaton x A the retrcton of x to A,.e. (x A). Lkewe, f there ext at leat one mnmzer x, for whch x A = y hold we ay that (A, y) a weak optmal partal agnment. Two or more trong optmal partal agnment can be combned together, becaue each of them preerve all optmal oluton. Th not true for weak agnment, even f they agn dfferent varable, they may not hare any globally optmal oluton n common. However, f we want to fnd a mnmzer of (1) (or at leat localze t a much a poble), a weak optmal partal agnment could be more helpful the bet one agn all varable. 1.3 Doman Contrant The dea of optmal partal agnment naturally extend to contranng a varable to a ubet of label K L. Let A V, let K L, A. Let K be the Cartean product of K, A. We ay that a par (A, K) a trong (rep. weak) optmal contrant f 2 Varable x {0, 1} are regarded a Boolean n th cae and peudo emphaze that a real-valued rather than Boolean functon of thee varable condered. x A K for all (rep. at leat one) mnmzer x. Th type of contrant called doman contrant. Obvouly, t nclude partal agnment a a pecal cae. 1.4 Autarke Some doman contrant follow from more pecfc properte called autarke. Th term occur n [3] for two-label problem and we conder t extenon [15] to mult-label problem. Let L = {0, 1... L} V, L N. Let x, y L. Defne component-we mnmum and maxmum of two labellng: (x y) = mn(x, y ), (2a) (x y) = max(x, y ). (2b) A par (x mn L, x max L) uch that x mn x max (component-we) called a weak autarky for problem (1), f x L f((x x mn ) x max ) f(x). (3) If addtonally for any x (x x mn ) x max trct nequalty f((x x mn ) x max ) < f(x) (4) hold, then the autarky called trong. The autarky provde doman contrant wth K = [x mn,..., x max ]. For any mnmzer x, we have that ˆx = (x x mn ) x max a mnmzer a well, and ˆx K. A trong autarky guarantee addtonally that x mut telf atfy x K. Indeed, f t wa not true then ˆx x and f(ˆx) < f(x ), whch a contradcton. Therefore a weak (rep. trong) autarky provde a weak (rep. trong) doman contrant. Determnng whether a gven par (x mn, x max ) a trong autarky an NP-hard decon problem [3]. Autarke can be combned together. A jon of two autarke (x 1, x 2 ), (y 1, y 2 ) the par (x 1 y 1, x 2 y 2 ). For trong autarke, the reult a trong autarky and th operaton commutatve, aocatve and dempotent, o that t defne a em-lattce. Proof. From defnton of autarke, we have f((((x x 1 ) x 2 ) y 1 ) y 2 ) f((x x 1 ) x 2 ) f(x) (5) 2

3 Note, that for x 1 x 2 we have (x x 1 ) x 2 = (x x 2 ) x 1. We can rewrte the labelng n the left hand de (LHS) a follow ((x x 1 ) x 2 ) y 1 ) y 2 = ((x x 2 ) (x 1 y 1 )) y 2. = (x (x 2 y 2 )) (x 1 y 1 ), (6) where doted equalty hold f y 2 x 1. Th atfed for trong autarke, becaue t would be a contradcton that all optmal labellng are below y 2 and above x 1. Thu there ext an autarky, whch provde the maxmal amount of doman contrant among trong autarke. It the jon of all trong autarke. It alo poble to jon non-contradctve weak autarke together, but let u leave t ade for now. We wll conder a pecal cae of autarke wth one-de contrant, of the form (x mn, L) or (0, x max ), where L and 0 repreent the labelng wth all component equal to L (rep. 0). For uch autarke nequalty (3) mplfe, becaue x 0 = x and x L = x. Method [10, 11] compute trong autarke of th form. By takng the jon of two trong autarke (x mn, L) and (0, x max ) we can obtan a trong autarky (x mn, x max ). However, the revere not true: f (x mn, x max ) a trong autarky, t doe not mply that (x mn, L) or (0, x max ) an autarky. And t the cae that other method (roof-dual [1] n the cae of two-label problem and t mult-label extenon [15]) can fnd an autarky of the form (x mn, x max ), whch not a jon of two one-de autarke. be t mnmzer. Then we have the followng properte: f(x x ) f(x), f(x x ) f(x). (9a) (9b) They ealy follow from ubmodularty, notng that f(x x ) f(x ) and f(x x ) f(x ). So, n fact, any par of optmal oluton (x 1, x 2 ) a weak autarky for th problem. Moreover, f we let x mn = arg mn x f(x), (10a) x max = arg mn x f(x), (10b) where arg mn the et of mnmzer, we ee that both x mn and x max are mnmzer of f and that (x mn, x max ) a trong autarky for f. In fact, t the jon of all trong autarke for f. Th trong autarky can be determned from a oluton of the correpondng maxflow problem. 2 Approach by Kovtun In th ecton, we revew technque [10, 11] for buldng autarke (and hence doman contran) by contructng auxlary problem. We take a omewhat dfferent perpectve on thee reult, however, our tatement and proof here are n a ene equvalent to one gven n [10, 11]. Theorem 1. Let f = g + h, let (x mn, x max ) be a trong autarky for g and a weak autarky for h. Then (x mn, x max ) a trong autarky for f. 1.5 Submodular Problem Functon f called ubmodular f x, y L f(x y) + f(x y) f(x) + f(y). (7) Proof. We have f((x x mn ) x max ) = g((x x mn ) x max ) + h((x x mn ) x max ) g(x) + h(x), (11) In the cae f defned by (1), t ubmodular ff (ee e.g. [18]) t E, x t, y t L t = L L t f t (x t ) + f t (y t ) f t (x t y t ) + f t (x t y t ). (8) Mnmzng a parwe ubmodular functon reduce to mncut problem [7], [13]. Let f be ubmodular and x and the nequalty trct f (x mn, x max ) trong for ether h or g. The dea of auxlary problem to contruct a ubmodular g, for whch, a we know, a trong autarky (x mn, L) can be found by choong x mn a the lowet mnmzer of g, gven by (10a). The trck to fnd 3

4 uch g that (x mn, L) at the ame tme an autarky for h = f g. The followng uffcent condton were propoed [11]: Statement 1. Let h atfy V, x L, ˆx K h (x ˆx ) h (ˆx ) t E, x t L t, ˆx t K t h t (x t ˆx t ) h t (x t ). (12a) (12b) Then for any x mn uch that x mn K, the par (x mn, L) a weak autarky for h. If addtonally V, ˆx K, x < ˆx h (x ˆx ) < h (ˆx ), then (x mn, L) a trong autarky. (12c) Proof. For any x L, ummng correpondng nequalte from (12a) and (12b), we obtan h (x x mn ) + t h t (x t x mn t ) h (x ) + t h t (x t ). If x x mn x, then V x < x mn mple trct nequalty. (13) and (12c) Two practcal method were propoed [11] to contruct g and (K V). We frt decrbe a more general approach. Algorthm 1: Sequental contructon of g, (K V), [11] 1. Start wth K =, V; 2. Fnd g uch that h = f g atfe (12) and g atfe ubmodularty contrant (8). 3. Fnd x mn = arg mn x g(x); 4. If x mn K for all V then top. 5. Set K K {x mn } V and go to tep 2. In tep 2 for each edge t E a ytem of lnear nequalte n g t ha to be olved. Whle [11] provde an explct oluton, for our conderaton t wll not be neceary. When the algorthm top, g ubmodular and (x mn, L) a trong autarky for g and a weak autarky for f g. By Theorem 1, t a trong autarky for f. It may top, however, wth x mn = 0 for all, o that effcently no contrant are derved. Beng a polynomal algorthm t cannot have a guarantee to mplfy the problem (1). A mpler non-teratve method propoed n [10] hown n Algorthm 2. It attempt to dentfy node where the label L better than any other label. The contructed auxlary problem g ha a property that t lowet mnmzer x mn = arg mn g(x) guar- Algorthm 2: One v all method, [10, 11] 1. For each choe uch orderng of L that 0 arg mn f (). L 2. Set g = f, V. 3. Set K = {0, L}. a t, = L, j = L, b t, = L, j L, 4. Set g t (, j) = c t, L, j = L, d t, L, j L, where a t, b t, c t, d t are uch that f t g t atfy (12b) and ubmodularty contrant. One of the oluton a follow: a t = f t (L, L), b t = mn j L f t(l, j), c t = mn f t(, L), L ( d t = mn b t + c t a t, mn L,j L [ f t (, j) + mn { b t f t (L, j), c t f t (, L) }]). (14) anteed to atfy x mn K V. (becaue all cot (g t (, j) < L, j < L) are equal and g (0) g () V, < L, ee proof n [10]). Therefore (x mn, L) a weak autarky for f g and Theorem (1) apple. Both method allow u to chooe varou orderng of et L. Strong doman contrant derved from varou orderng can be then combned. 4

5 3 LP-autarke In th ecton we ntroduce a pecal ubcla of autarke, whch preerve optmal oluton of the LPrelaxaton. Unlke wth general autarke, the memberhp to th ubcla polynomally verfable. We how that autarke contructed by algorthm 1, 2 belong to th ubcla. Th ha ueful mplcaton for LP relaxaton. 3.1 LP Relaxaton Let φ(x) be a vector wth component φ(x) 0 = 1, φ(x), = [[x =]] and φ(x) t, = [[x t =]], where [[ ]] 1 f the expreon nde true and 0 otherwe. Let f denote a vector wth component f 0, f, = f () and f t, = f t (). Wth repect to component of energy functon we wll be ung th ndex and parenthe notaton completely nterchangeably. Let, denote a calar product. Then we can wrte energy mnmzaton a mn f, φ(x). (15) x L It relaxaton to a lnear program wrtten a mn f, µ, (16) µ Λ where Λ the local polytope. It approxmate conv{φ(x) x L} from the outde, ee e.g. [18] for more detal. It gven by the lnear contrant µ 0 = 1, µ, 0, µ t, 0, L t µ t, = 1 t E, j L t µ t, = µ, L, t E, L µ t, = µ t,j j L t, t E. Vector µ Λ called a relaxed labelng. 3.2 LP-autarky (17) We now extend the noton of autarky to relaxed labellng. Defnton 1. A bnary operaton : Λ L Λ, defned a follow. Let y L and µ Λ. Then ν = µ y Λ contructed a: ν, = ν t, = µ,, < y, y µ,, = y, 0, > y ; µ t,, < y, j < y t, y µ t, j, = y, j < y t, j y t µ t,, < y, j = y t, µ t, j, = y, j = y t, y j y t 0, > y or j > y t. (18a) (18b) By contructon, the relaxed labelng ν ha non-zero weght only for label below y: ν, = 0 for > y and the ame for par t,. Let u check that ν Λ. Proof. Normalzaton contrant: ν, = µ, + µ, = µ, = 1. (19) <y y Margnalzaton contrant: µ t, + µ t, j, j < y t, <y y ν t, = µ t, + µ t, j, j = y t, <y y j y t j y t 0, j > y t, = ν t,j. (20) Operaton ν = µ y defned completely mlarly, havng ngleton component µ,, > y, (µ y), = µ,, = y, (21) y 0, < y. Defnton 2. We ay that a par (x mn, x max ) a weak LP-autarky for f, f µ Λ f, (µ x mn ) x max f, µ. (22) If addtonally for all µ uch that (µ x mn ) x max µ the trct nequalty hold then we ay that t a trong LP-autarky. 5

6 3.3 Properte of LP-autarke Statement 2. Any weak (rep. trong) LP-autarky a weak (rep. trong) autarky. Proof. By ubttutng µ = φ(x). Statement 3. Checkng whether (x mn, x max ) an LP-autarky for f can be olved n a polynomal tme. Proof. By contructon, (µ x mn ) x max a lnear map n µ, let u denote t Aµ. Inequalty (22) hold ff mn f, µ Aµ 0, (23) µ Λ whch a lnear program. To verfy whether A a trong LP-autarky we need to olve mn f, µ Aµ > 0 µ Λ,.t. µ, < V. x mn x max Statement 4. If f ubmodular, then µ Λ, y L µ, f + φ(y), f µ y, f + µ y, f. (24) (25) Proof. Scalar product n (25) are compoed of um of ngleton term and parwe term. We frt how that um of ngleton term are equal, expandng ngleton term n the rght hand de (RHS): [ ] (µ y), + (µ y), f() = µ,f () + µ, f (y )+ <y y µ,f () + µ, f (y ) = >y y µ,f () + ( ) µ, f (y ) = µ,f () + [[= y ]]f (y ). Now conder ubmodularty contrant: t E, L t, y t L t f t () + f t (y t ) f t ( y t ) + f t ( y t ). (26) (27) Multplyng th nequalty by µ t, and ummng over, we obtan on the LHS: µ t, f t () + f t (y t ) = µ t, f t () + [[=y t ]]f t (y t ) and on the RHS: [ µ t, ft ( y t ) + f t ( y t ) ] = [ ] (µ y)t, + (µ y) t, ft (), where the equalty verfed a follow: µ t, f t ( y t ) = µ t,f t() + µ t,f t(y, j)+ <y y j<y t j<y t µ t,f t(, y t ) + µ t,f t(y t) = <y y j y t j y t (µ y) t, f t,. (28) (29) (30) The term wth rewrtten mlarly. By ummng nequalte (28) (29) over t E and addng equalte (26) of the ngleton term, we get the reult. Statement 5. Let f be ubmodular and x arg mn f(x). Then µ Λ x µ x, f µ, f, (31a) µ x, f µ, f. (31b) Proof. Let u how (31a). For ubmodular problem LP-relaxaton (16) tght. Thu for any ν Λ there hold ν, f f(x ) = φ(x ), f. In partcular, for ν = µ y we have µ y, f φ(x ), f, whch when combned wth (25) mple the tatement. Statement 6. Let (x mn, L) be a trong LP-autarky for f, then: V, < x mn, µ arg mn µ, f µ Λ µ, = 0. (32) Proof. Let µ arg mn µ Λ µ, f and µ, > 0. Then µ x mn µ and f(µ x mn ) < f(µ ), whch contradct optmalty of µ. 6

7 3.4 Implcaton for Algorthm 1, 2 We have already een n tatement 5 that for a ubmodular problem g, takng y a a mnmzer (rep. the lowet mnmzer) of g gve a weak (rep. trong) LPautarky (y, L). Let u how that tatement 1 extend to LP-autarke too. Th would mply that autarke derved by algorthm 1, 2 are n fact LP-autarke for f = g + h. Statement 7. Let h atfy nequalte (12). Then for any y L uch that y K, the par (y, L) a weak LP-autarky for h. Proof. Let µ Λ. From nequalty (12a) we have ((µ y), µ, )h, 0. (33) Multplyng (12b) by µ t, and ummng over L t and over t E we obtan [ ] (µ y)t, µ t, ht, 0. (34) t Addng (33) and (34), we get: whch equvalent to (22). µ y µ, h 0, (35) We have hown that algorthm 1, 2 derve doman contrant n the form of trong LP-autarke. We know too that optmal oluton of LP-relaxaton wll obey doman contrant derved va trong LPautarke. Note, whle algorthm 1, 2 depend on the orderng of the label, oluton of the LP-relaxaton doe not. Hence, Corollary 1. Let (K L V) be a trong doman contrant derved by Algorthm 1, 2 w.r.t. any orderng of et L. Then the et of optmal oluton of LP relaxaton wth and wthout thee doman contrant would concde. We proved that LP relaxaton cannot be tghtened by algorthm 1, 2. It may only be mplfed by elmnatng all varable whch are guaranteed to be 0 n every optmal oluton. Th may be ueful n practcal method olvng LP relaxaton. For problem wth two label, the followng relaton alo hold. Let Λ = arg mn µ Λ f, µ. Let x mn = mn{ µ Λ µ, > 0}, x max = max{ µ Λ µ, > 0}, (36) then (x mn, x max ) a trong autarky for f. Th the roof-dual autarky [1]. Becaue for any other autarky derved va algorthm 1 and 2 tatement 6 hold, we conclude that roof-dual autarky domnate algorthm 1 and 2. 4 Expanon Move Expanon move algorthm [4] eek to mprove the current oluton x by conderng a move, whch for every V ether keep the current label x or change t to the label k. Algorthm 3: Expanon-Move [4] 1. Let x L, let k L. The move energy functon g(z) of bnary confguraton z {0, 1} V defned by: g 0 = f 0, g (0) = f (x ), g (1) = f (k), g t (1, 1) = f t (k, k), g t (1, 0) = f t (k, x t ), g t (0, 1) = f t (x, k), g t (0, 0) = f t (x, x t ). (37) 2. Let z arg mn g(z). z 3. If g(z ) < g(0), agn x { x, f z = 0, k, f z = 1. If the above procedure repeated for all label k L and no mprovement to x found then x ad to be a fxed pont of th method. In the cae f a metrc energy [4], the move energy g ubmodular for arbtrary x and tep 2 eay. Statement 8. Let f be metrc [4]. Let (x mn, L) be a trong autarky for f uch that x mn {0, L}, V. Then for any fxed pont x of the expanon-move algorthm there hold x x mn. (38) 7

8 Proof. Aume V uch that x < x mn. Then f(x x mn ) < f(x) and nce x {1, L}, t x x mn x, x mn = 1, = (39) L, x mn = L, whch a vald expanon move from x to label k = L, trctly mprovng the energy. In the cae when a move energy not ubmodular, t can be truncated to make t ubmodular whle tll preervng the property that the move doe not ncreae f(x) [12]. Let t = g t (1, 1) + g t (0, 0) g t (0, 1) g t (1, 0). Par t ubmodular ff t < 0. Defnton 3. Truncaton g of g dfferent from g only n non-ubmodular parwe component of g, whch are et a: g t,00 = g t,00 β t t, g t,01 = g t,01 + α t t, g t,10 = g t,10 + (1 α t β t ) t, g t,11 = g t,11, (40) where α t and β t are free parameter, atfyng α t 0, β t 0, α t + β t 1. It eay to verfy that g ubmodular, and g(z) g(0) g (z) g (0), (41) ayng that ncreae n g no more than ncreae n g when changng from 0 to z. Proof of (41). By contructon of g, for all t E uch that t > 0, enumeratng all z t, g t,00 g t,00 = 0, g t,01 g t,00 = g t,01 g t,00 + (α t + β t ) t, g t,10 g t,00 = g t,10 g t,00 + (1 α t ) t, g t,11 g t,00 = g t,11 g t,00 + β t t, (42) we ee that only potve value are added on RHS. It alo een that the added potve value do only ncreae wth β t. Th mean that the truncaton wth β t > 0 (let denote t g α,β ) never better than the truncaton wth β = 0 (let denote t g α ): z g(z) g(0) g α (z) g α (0) g α,β (z) g α,β (0). (43) Smlarly, the truncaton wth α = 0, β = 1 (g 0,1 ) not better than the truncaton g α,β : g α,β (z) g α,β (0) g 0,1 (z) g 0,1 (0). (44) Th verfed by examnng component: g 0,1 t (z t ) g 0,1 t (0) g α,β t (z t ) + g α,β t (0) = 0, z t = 00, t (1 (α + β)), z t = 01, t (1 (1 α)), z t = 10, t (1 β), z t = 11, 0. (45) If z an mprovng move for g 0,1 then t alo an mprovng move for any truncaton. We have the followng reult about Algorthm 2: Statement 9. Let (x mn, L) be a trong autarky for f obtaned by Algorthm 2. Let x be a fxed pont of the expanon-move algorthm wth any truncaton rule. Then x x mn. (46) Proof. We wll prove that the tatement hold for truncaton (α = 0, β = 1). We need to how that for a move from x to x x mn the truncated energy decreae at leat a much a doe auxlary problem bult by Alg. 2. Th can be verfed by npectng parwe component for the 4 cae z t = 00, 01, 10, Concluon We propoe a novel repreentaton of method [10, 11] a dervng doman contrant va LP-autarke. Th allow for comparon wth other method dervng doman contrant n the ame form [3, 15] and etablhng relaton wth common method of (approxmate) optmzaton. We alo beleve that label domnaton condton propoed by [5] can be nterpreted n the ame framework, allowng for the theoretcal comparon and or for the degn of combned method. Our reult open everal drecton for mprovement. A drect mprovement to Alg. 2 can be obtaned a follow. Alg. 2 contruct a mult-label auxlary problem, whch equvalent to a two-label problem (nce we know that there a mnmzer wth 8

9 x {0, L}, V). For two label problem, we alo know that the autarky contructed by roof-dual domnate the autarky by truncaton, o t wll be better to et [ d t = mn ft (, j) L,j L + mn { b t f t (L, j), c t f t (, L) }] (47) and olve for roof-dual ung reducton to maxflow [2]. Th would be a non-ubmodular auxlary problem. We can alo attempt to contruct auxlary problem wth mxed ubmodular and upermodular term a n [15] or degn an algorthm whch wll propoe an autarky n ome greedy way and then verfy t va olvng lnear program (24). Reference [1] E. Boro and P.L. Hammer. Peudo-boolean optmzaton. Dcrete Appled Mathematc, (123(1-3)): , [2] E. Boro, P. L. Hammer, and X. Sun. Network flow and mnmzaton of quadratc peudo-boolean functon. Techncal Report RRR , RUTCOR, May [3] E. Boro, P. L. Hammer, and G. Tavare. Preproceng of uncontraned quadratc bnary optmzaton. Techncal Report RRR , RUTCOR, Apr [4] Y. Boykov, O. Vekler, and R. Zabh. Fat approxmate energy mnmzaton va graph cut. IEEE Tranacton on Pattern Analy and Machne Intellgence, 23(11): , Nov [5] J. Demet, M. D. Maeyer, B. Haze, and I. Later. The dead-end elmnaton theorem and t ue n proten de-chan potonng. Nature, 356: , [6] P.L. Hammer, P. Hanen, and B. Smeone. Roof dualty, complementaton and pertency n quadratc 0-1 optmzaton. Math. Programmng, page , [7] Hroh Ihkawa. Exact optmzaton for Markov random feld wth convex pror. IEEE Tranacton on Pattern Analy and Machne Intellgence, 25(10): , [8] Vladmr Kolmogorov. Convergent tree-reweghted meage pang for energy mnmzaton. In Robert G. Cowell and Zoubn Ghahraman, edtor, AI and Stattc, page Socety for Artfcal Intellgence and Stattc, [9] Vladmr Kolmogorov and Martn Wanwrght. On the optmalty of tree-reweghted max-product meage pang. In To appear n 21t Conference on Uncertanty n Artfcal Intellgence (UAI), July [10] I. Kovtun. Partal optmal labelng earch for a NPhard ubcla of (max, +) problem. In DAGM- Sympoum, page , [11] I. Kovtun. Image egmentaton baed on uffcent condton of optmalty n NP-complete clae of tructural labellng problem. PhD the, IRTC ITS Natonal Academy of Scence, Ukrane, In Ukranan. [12] Carten Rother, Sanjv Kumar, Vladmr Kolmogorov, and Andrew Blake. Dgtal tapetry. In CVPR 05: Proceedng of the 2005 IEEE Computer Socety Conference on Computer Von and Pattern Recognton (CVPR 05) - Volume 1, page , Wahngton, DC, USA, IEEE Computer Socety. [13] Dmtr Schlenger and Bor Flach. Tranformng an arbtrary mnum problem nto a bnary one. Reearch Report TUD-FI06-01, Dreden Unverty of Technology, Aprl [14] M.I. Schlenger. Syntactc analy of two-dmenonal vual gnal n the preence of noe. Cybernetc and Sytem Analy, 12: , [15] A. Shekhovtov, V. Kolmogorov, P. Kohl, V. Hlavac, C. Rother, and P. Torr. LP-relaxaton of bnarzed energy mnmzaton.. Reearch Report CTU CMP , Czech Techncal Unverty, [16] Martn Wanwrght, Tomm Jaakkola, and Alan Wllky. Exact MAP etmate by (hyper)tree agreement. In S. Thrun S. Becker and K. Obermayer, edtor, Advance n Neural Informaton Proceng Sytem 15, page MIT Pre, [17] Tomáš Werner. A lnear programmng approach to max-um problem: A revew. Reearch Report CTU CMP , Center for Machne Percepton, Czech Techncal Unverty, Dec [18] Tomáš Werner. A lnear programmng approach to max-um problem: A revew. IEEE Tranacton on Pattern Analy and Machne Intellgence, 29(7): , July