Triangle-based Consistencies for Cost Function Networks

Transcription

1 Nonme mnuscrpt No. (wll e nserted y the edtor) Trngle-sed Consstences for Cost Functon Networks Hep Nguyen Chrstn Bessere Smon de Gvry Thoms Schex Receved: dte / Accepted: dte Astrct Cost Functon Networks (k Weghted CSP) llow to model vrety of prolems, such s optmzton of determnstc nd stochstc grphcl models ncludng Mrkov rndom Felds nd Byesn Networks. Solvng cost functon networks s thus n mportnt prolem for determnstc nd prolstc resonng. Ths pper focuses on locl consstences whch defne essentl tools to smplfy Cost Functon Networks, nd provde lower ounds on ther optml soluton cost. To strengthen rc consstency ounds, we follow the de of trngle-sed domn consstences for hrd constrnt networks (pth nverse consstency, restrcted or mx-restrcted pth consstences), descre ther systemtc extenson to cost functon networks, study ther reltve strengths, defne enforcng lgorthms, nd experment wth them on lrge set of enchmrk prolems. On some of these prolems, our mproved lower ounds seem necessry to solve them. Keywords Cost functon networks Weghted CSP Constrnt optmzton prolems Hgh order consstences Restrcted pth consstency Pth nverse consstency Mx-restrcted pth consstency Hep Nguyen MIAT, UR 875, Unversté de Toulouse, INRA, Cstnet-Tolosn, Frnce E-ml: f.nthhep@gml.com Chrstn Bessere Unversty of Montpeller, Montpeller, Frnce E-ml: essere@lrmm.fr Smon de Gvry MIAT, UR 875, Unversté de Toulouse, INRA, Cstnet-Tolosn, Frnce E-ml: degvry@toulouse.nr.fr Thoms Schex MIAT, UR 875, Unversté de Toulouse, INRA, Cstnet-Tolosn, Frnce E-ml: tschex@toulouse.nr.fr

2 2 Hep Nguyen et l. 1 Introducton Grphcl model processng s centrl prolem n AI. The Cost Functon Network frmework (CFN [25] s n nstnce of the vlued CSP frmework.), where the gol s to optmze the comned cost of locl cost functons, cptures prolems such s weghted MxSAT, Weghted CSP or Mxmum Prolty Explnton n prolstc networks. 1 CFNs hve pplctons n resource llocton [4], comntorl uctons, onformtcs [27, 26]... Dynmc progrmmng pproches such s ucket or cluster tree elmnton cn e used to tckle such prolems ut re nherently lmted y ther exponentl tme nd spce ehvor on grphcl models wth hgh tree-wdth. Insted, Depth Frst Brnch nd Bound llows to keep polynoml spce complexty ut requres good (strong nd chep) lower ounds on the mnmum cost to e effcent. In the lst yers, ncresngly etter lower ounds hve een desgned y enforcng soft locl consstences on CFNs. Arc consstences such s AC*, DAC*, FDAC*, EDAC* [17] or VAC [6] re nspred from rc consstency n hrd constrnt networks. They hve smll order polynoml enforcng tme ut do not lwys provde tght enough lower-ounds. The lner progrmmng sed OSAC consstency [9] provenly gves the strongest lower ound tht cn e otned y rc consstency ut s usully too expensve to compute. It now ecomes useful to look eyond rc consstences. Up to now, few hgher order consstences hve een proposed for CFNs [8, 12]. In ths pper, we show tht strong soft consstences cn e defned for CFNs y extendng hrd hgh order consstences defned for CSPs. Among hrd hgh order consstences, the fmly of trngle-sed consstences (Restrcted Pth Consstency or RPC, Pth Inverse Consstency or PIC, nd mxrestrcted Pth Consstency or mxrpc) re specfclly nterestng ecuse they hve stronger prunng power thn rc consstency, nd cheper computtonl cost thn other hgh order consstences. Ther extenson to CFNs s however non trvl, nd enforcng lgorthms crete ternry cost functons. The rest of the pper s orgnzed s follows. Secton 2 s ckground secton on constrnt nd cost functon networks, ssocted locl consstences nd enforcng opertons. The next sectons focus on our contrutons. Secton 3 gves defnton of new locl consstences. Secton 4 ntroduces dfferent wys to compre the strength of soft consstences n generl. Ths s then used to compre the proposed consstences to ech other nd to exstng consstences. Secton 5 focuses on the lgorthms for enforcng the proposed consstences. The lst secton gves expermentl results when these consstences re used s pre-processng or mntned durng serch on lrge set of enchmrks. We oserve tht the strengthened ound provded y trngle consstences (TRICs) re necessry to solve some prolems tht could not e solved otherwse. 1 We use the termnology of Cost Functon Networks y smlrty to Constrnt Networks. The Weghted Constrnt Stsfcton Prolem (WCSP) s the prolem of solvng CFN. For outsders, guessng wht Cost Functon Network could e, s lso much eser.

3 Trngle-sed Consstences for Cost Functon Networks 3 2 Bckground A constrnt stsfcton prolem (CSP) s trple (X, D, C) where X s set of n vrles, D s set of n domns (vrle X tkes vlues from D D), nd C s set of constrnts. Ech constrnt c S C defned over set S of vrles specfes the llowed ssgnments τ S of vlues for vrles n S, denoted y τ S c S. S nd S re the scope nd the rty of the constrnt c S. For smplcty, c {}, c {,j} re denoted s c, c j. The constrnts c, c j, c S wth S > 2 re respectvely clled unry, nry nd non-nry. A vlue for vrle s denoted y (, ) or y. Gven set of vrles S, l(s) denotes the set of ssgnments (tuples) of vlues for vrles n S, tht s, l(s) = D S = S D. Gven tuple τ S, vrle S nd suset S S, τ S [] nd τ S [S ] denote the projecton of tuple τ S on nd S respectvely. A tuple τ S s consstent f t stsfes ll the constrnts whose scope s ncluded n S. A soluton s consstent complete ssgnment. The prolem s consstent f t hs t lest one soluton. Defnton 1 (Locl consstences) Gven CSP P = (X, D, C), P s rc consstent (AC) f X, D, c S C such tht S, there exsts tuple τ l(s) such tht τ[] = nd τ c S. Such tuple τ s clled the support of vlue (, ) n the constrnt c S. P s restrcted pth consstent (RPC, [3]) ff t s AC nd X, D, c j C on whch hs only one support D j, k lnked to nd j y c k, c jk, there exsts vlue c D k such tht (, c) c k nd (, c) c jk. P s pth nverse consstent (PIC, [14]) ff t s AC nd X, D, j, k such tht, j, k re lnked one-y-one y nry constrnts, there exsts vlue D j, c D k such tht (, ) c j, (, c) c k nd (, c) c jk. P s mx-restrcted pth consstent (mxrpc, [11]) ff t s AC nd X, D, c j C, hs support D j such tht k lnked to oth, j y c k, c jk, there exsts vlue c D k such tht (, c) c k nd (, c) c jk. Let Φ e hrd consstency nd P CSP. The Φ closure of P s the CSP Φ(P ) = (X, D Φ, C) such tht DΦ X DX, Φ(P ) s Φ-consstent, nd there does not exst D such tht DΦ X D X nd (X, D, C) s Φ-consstent. CFNs extend CSPs y ssoctng costs to tuples [25,24]. A CFN s tuple (X, D, C, m) where X nd D re respectvely sets of vrles nd domns, s n clsscl CSPs. C s set of cost functons. Ech cost functon c S C ssgns non negtve nteger costs to tuples τ S l(s).e. c S l(s) [0..m] where m {1,..., + }. The ddton nd sutrcton of costs re ounded opertons, defned s + m = mn( +, m), m = f < m nd m otherwse. The comned cost of tuple τ S n CFN P s the sum of costs Vl P (τ S ) = (S S) (c S C) c S (τ S [S ]), where summton s done usng + m. τ S s nconsstent f Vl P (τ S ) = m, nd consstent otherwse. A soluton of P s complete consstent tuple τ X. An optml soluton hs mnmum Vl P (τ X ). It s mportnt to note tht CSPs re just CFNs usng m = 1. In such prolems, tuple whch receves cost of 1 s fordden. A cost of 0 s used for llowed tuples.

4 4 Hep Nguyen et l. We ssume the exstence of unry cost functon c for every vrle, nd nullry cost functon, noted c. Ths constnt non-negtve cost defnes lower ound on the cost of every soluton. A CFN P cn e trnsformed nto n equvlent CFN P (.e., Vl P (τ) = Vl P (τ) τ) y pplyng so-clled equvlence-preservng trnsformtons (EPTs) tht shft costs etween cost functons. The EPT Shft(τ S, c S, α) (Algorthm 1) moves n mount of cost α etween cost functon c S nd tuple τ S such tht S S. The condtons (2) nd (3) gurntee tht the operton wll not crete ny negtve cost n the prolem. Shft llows to defne the three usul EPTs [10] Project (from c S to τ S, α > 0), Extend (from τ S to c S, α < 0) nd UnryProject (from to c, α > 0, S = {}, S = ). Algorthm 1: Operton for shftng costs n CFNs 1 Procedure Shft(τ S, c S, α) // condton:(1) S S, (2)c S (τ S )+ mα 0, (3)c S (τ S ) α τ S l(s ), τ S [S] = τ S 2 c S (τ S ) c S (τ S ) + m α; 3 forech τ S l(s ) s.t τ S [S] = τ S do 4 c S (τ S ) c S (τ S ) m α; By pplyng EPTs to n orgnl CFN, t s possle to trnsform t n n equvlent CFN tht stsfes gven locl consstency property. Ths my ncrese the lower ound c. The smplest locl consstency, node consstency (NC [16]), requres tht X, D c ()+c < m nd there exsts vlue D such tht c () = 0. Defnton 2 (Soft rc consstences) Gven nry CFN P = (X, D, C, m) nd n order < on vrles, P s rc consstent (AC [24]) ff X, D nd c j C, there exsts D j such tht c j (, ) = 0. s clled (smple) support for (, ) n c j. 2 P s drectonl rc consstent (DAC [7]) w.r.t < ff, D, c j such tht < j, there exsts vlue D j such tht c j (, ) + c j () = 0. s clled full support for (, ) n c j. P s full drectonl rc consstent (FDAC [19]) w.r.t < ff t s AC nd DAC. P s exstentl rc consstent (EAC [17]) ff X, there exsts vlue D such tht c () = 0 nd c j C, there exsts D j such tht c j (, ) + c j () = 0. Vlue s clled the exstentl rc consstent support of. P s exstentl drectonl rc consstent (EDAC [17]) ff t s EAC nd FDAC. 2 There exsts tny vrtons on the defnton of AC for CFNs. Ths pper uses the defnton n [20] whch smplfes the defnton n [10] y not consderng the propgton of nconsstent tuples.

5 Trngle-sed Consstences for Cost Functon Networks 5 Bool(P ) s CSP defned s CFN (X, D, C, 1) such tht c S C ff c S C, S nd τ c S c S (τ) > 0. P s vrtul rc consstent (VAC [6]) ff the AC closure of Bool(P ) s non-empty. For smplcty, we restrct ourselves to nry CFNs. A nry CFN s AC, DAC, FDAC, EAC, EDAC f t s NC nd respectvely AC, DAC, FDAC, EAC, EDAC [16]. Defntons of soft rc consstences for non-nry CFNs hve een gven n [10,5,21,22]. 3 Soft trngle-sed consstences (TRICs) In ths secton, we extend the hrd locl consstences RPC, PIC nd mxrpc, defned on trngles of vrles to CFNs. For ech hrd consstency, we defne sx soft vrnts, lso clled softenng levels: smple, drectonl, full drectonl, exstentl, exstentl drectonl, nd vrtul. Ths gves rse to eghteen new soft locl consstences. In ddton to soft ACs, ll these soft versons gurntee the extenslty of rc supports on extr thrd vrles on so-clled wtness. Defnton 3 (Wtness) Gven vlue (, ), pr of vlues (, j ) nd vrle k lnked oth to nd j, A smple wtness of (, j ) on k s vlue c D k such tht c k (, c) + c jk (, c) + c jk (,, c) = 0. A full wtness of (, j ) on k s vlue c D k such tht c k (c) + c k (, c) + c jk (, c) + c jk (,, c) = 0. Defnton 4 (Extenslty of pr of vlues on vrle) Gven pr of vlues (, j ) nd vrle k lnked oth to nd j, (, j ) s smply extensle on k f there exsts smple wtness on k for t. (, j ) s fully extensle on k f there exsts full wtness on k for t. Defnton 5 (Extenslty of vlue on trngle) A trngle s trple of vrles (, j, k) tht re lnked one-y-one y nry cost functons. It s noted s jk. Gven vlue (, ) nd trngle jk. (, ) s smply extensle on trngle jk f there exsts smple rc support for (, ) n c j tht s smply extensle on k. (, ) s fully extensle on trngle jk f there exsts full rc support for (, ) n c j tht s fully extensle on k. Defnton 6 (Extenslty of pr of vlues) For pr of vlues (, j ) nd n order < on the vrles, (, j ) s: smply extensle f t s smply extensle on every k lnked to oth nd j. fully extensle f t s fully extensle on every k lnked to oth nd j. drectonlly-fully extensle f t s fully extensle on every k > lnked to oth nd j. sem-fully extensle f t s smply extensle on every k < lnked oth to nd j nd s fully extensle on every k > lnked oth to nd j.

6 6 Hep Nguyen et l. Notce tht full extenslty mples sem-full extenslty. Sem-full extenslty mples drectonl-full nd smple extenslty. Conversely, oth drectonl-full nd smple extenslty do not mply ny other extenslty. Exmples n Fgure 1 llustrte the dfferent extensltes of prs of vlues. k 2 c 1 k 2 c 1 c k 2 c k 2 c k 2 k 1 c () j k 1 1 c () j k 1 c (c) j k 1 1 c (d) j k 1 c (e) j Fg. 1 Exmple of dfferent extensltes of the pr of vlues (, j ). k 1 < < j < k 2. An edge ppers etween prs of vlues wth non zero cost. In CFN(), (, j ) s not smply extensle on k 1. In CFN(), (, j ) s smply extensle (on oth k 1, k 2 ) ut s not drectonlly-fully extensle (ecuse t s not fully extensle on k 2 ). In CFN(c), (, j ) s drectonlly-fully extensle w.r.t k 2 ut s not sem-fully extensle (ecuse t s not smply extensle on k 1 ). In CFN(d), (, j ) s sem-fully extensle (fully extensle on k 2 nd smply extensle on k 1 ) ut s not fully extensle (ecuse t s not fully extensle on k 1 ). In CFN(e), (, j ) s fully extensle (on oth k 1, k 2 ). 3.1 Soft restrcted pth consstences The de of soft RPC consstences s to only check the extenslty of prs of vlues (, j ) tht wll mke vlue soft rc nconsstent f ther nry cost ecomes postve. If vlue (, ) hs only one smple support (j, ) on c j nd ths support (, j ) s not extensle on some thrd vrle k, every 3-vlues tuple over {, j, k}, nvolvng (, j ), hs postve comned cost. Becuse (j, ) s the unque rc support of (, ), every complete tuple nvolvng (, ) hs postve cost evluton. Thus, the unry cost c () cn e ncresed y equvlence preservng trnsformtons. Defnton 7 (Soft restrcted pth consstences (Soft RPCs)) Gven CFN P = (X, C, D, m) nd n order < on vrles, P s RPC f t s AC nd X, D, c j C on whch (, ) hs only one smple rc support D j, (, j ) s smply extensle. P s drectonl RPC (DRPC) f t s DAC nd X, D, c j C such tht < j nd (, ) hs only one full rc support D j, (, j ) s drectonlly-fully extensle. P s full drectonl RPC (FDRPC) f t s FDAC nd X, D, c j C such tht (1) f > j nd (, ) hs only one smple rc support D j then (, j ) s smply extensle, or (2) f < j nd (, ) hs only one full rc support D j then (, j ) s sem-fully extensle.

7 Trngle-sed Consstences for Cost Functon Networks 7 P s exstentl RPC (ERPC) f X, there exsts vlue D such tht (1) c () = 0, (2) hs full rc support n every cost functon (.e., P s EAC), nd (3) c j C on whch (, ) hs only one full rc support D j, (, j ) s fully extensle. Such vlue (, ) s the ERPC support for. P s exstentl drectonl RPC (EDRPC) f t s ERPC nd FDRPC. P s vrtul RPC (VRPC) f the RPC closure of Bool(P ) s non-empty. VRPC s defned sed on the hrd CSP Bool(P ) nd hrd RPC. The other softenng levels of RPC dffer from ech other y (1) the strength of supports (smple or full) (2) the strength of wtnesses (smple, full, drectonl-full, semfull) nd (3) the scope of pplcton of these propertes (every domn vlue or one vlue per domn, every cost functon or some specfc cost functons). Exmple 1 Consder the CFNs n Fgure 1. CFN() s VRPC ecuse the RPC closure of Bool(P ) s not empty, contnng vlues ( ), (j, ), (k 1, ), (k 1, c), (k 2, ), (k 2, c). However, t s not RPC ecuse the unque support (, j ) of (, ) on c j s not smply extensle on k 1 CFN() s RPC: oth (, j ) nd (, j ) (respectvely the unque smple rc support of (, ), (j, ) on c j nd of (, ), (j, ) on c j ) re smply extensle on k 1 nd k 2 t smple wtnesses (k 1, ) nd (k 2, ) respectvely. However, t s not DRPC ecuse the unque full rc support (, j ) of (, ) n c j s not fully extensle on k 2 >. CFN(c) s DRPC ecuse oth (, j ) nd (, j ) (respectvely the unque full rc support of (, ) n c j nd of (, ) n c j ) re fully extensle on k 2 > t (k 2, ). Vrle k 1 < s not nvolved n DRPC for. However, t s not FDRPC ecuse the unque full support (, j ) of vlue (, ) on c j s not smply extensle on k 1. CFN(d) s FDRPC where the supports (, j ) nd (, j ) re fully extensle on k 2 t (k 2, ) nd smply extensle on k 1 t (k 1, ). At the sme tme, t s ERPC where (, ), (j, ), (k 1, ), (k 2, ) re ERPC supports for vrles, j, k 1 nd k Soft pth nverse consstences We now consder soft pth nverse consstences. They gurntee the extenslty of domn vlues on trngles of vrles. For ll trngles jk shrng two vrles, j of cost functon c j, PICs requre tht one of the rc supports of (, ) n c j s extensle on k. The rc supports of (, ) tht re extensle on dfferent k cn e dfferent. Defnton 8 (Soft pth nverse consstences (Soft PICs)) Gven CFN P = (X, C, D, m) nd n order < on vrles, P s PIC f t s AC nd X, D, jk, (, ) s smply extensle on jk.

8 8 Hep Nguyen et l. P s drectonl PIC (DPIC) f t s DAC nd X, D, jk such tht < j, < k, (, ) s fully extensle on jk. P s full drectonl PIC (FDPIC) f t s FDAC nd X, D, jk, (, ) s fully extensle on jk f < j, < k nd smply extensle on jk otherwse. P s exstentl PIC (EPIC) f X, there exsts vlue D such tht (1) c () = 0, (2) hs full rc support n every cost functon (.e., P s EAC) nd (3) (, ) s fully extensle on every trngle. P s exstentl drectonl PIC (EDPIC) f t s EPIC nd FDPIC. P s vrtul PIC (VPIC) f the PIC closure of Bool(P ) s non-empty. See exmples n Fgure 2. As n the cse of RPC, VPIC s defned sed on the hrd CSP Bool(P ) nd hrd PIC. The other softenng levels dffer from ech other y the strength of supports, the strength of wtnesses, nd the scope of pplcton of these propertes. k 2 c k 2 c 1 k 2 c k 2 c c j c j c j c j k 1 c () not PIC k 1 () PIC not DPIC c k 1 c (c) DPIC not FDPIC k 1 c (d) EDPIC Fg. 2 Exmple of soft PIC consstences. k 1 < < k 2 < j nd jk1, jk2. The CFN() s not PIC ecuse vlue (, ) s not smply extensle to trngle jk1. The CFN() s PIC ut s not DPIC ecuse vlue (, ) s not fully extensle to trngle jk2 wth < j, < k 2. The CFN(c) s DPIC (ecuse every vlue n D cn e fully extended to jk2 (the only trngle concerned y DPIC for ) ut t s not FDPIC (ecuse vlue (, ) s not smply extensle to trngle jk1 ). The CFN(d) s FDPIC where every vrle s smply extensle to 2 trngles nd s fully extensle to (, j, k 2 ). The CFN(d) s lso EPIC where (, ), (j, ), (k 1, ), (k 2, ) re respectvely EPIC supports of, j, k 1, k Soft mx-restrcted pth consstences Stronger thn PICs, soft mx-restrcted pth consstences (soft mxrpcs) check the exstence of n extensle rc support for ech vlue on ech nry cost functon whtever the numer of rc supports the vlue hs. In contrst to soft PICs, mxrpcs requre the extenslty of the sme rc support t the sme tme on ll thrd vrles. If vlue (, ) hs no such extensle rc support n some nry cost functon c j, ech support (, j ) of (, ) n c j s not extensle n some extr vrle k,.e. the comned cost of every tuple (, j, k c ) s postve. Thus, the nry cost of every rc support of (, ) n c j cn e ncresed y equvlence preservng trnsformtons nd then (, ) wll no longer e rc consstent.

9 Trngle-sed Consstences for Cost Functon Networks 9 Defnton 9 (Soft mx-restrcted pth consstences (Soft mxr- PCs)) Gven CFN P = (X, D, C, m) nd n order < on the vrles, P s mxrpc f t s AC nd X, D, c j C there exsts smple rc support D j such tht (, j ) s smply extensle. P s drectonl mxrpc (DmxRPC) f t s DAC nd X, D, c j C such tht < j, there exsts full rc support D j such tht (, j ) s drectonlly-fully extensle. P s full drectonl mxrpc (FDmxRPC) f t s FDAC nd for D, D, c j C (1) f > j, there exsts smple rc support D j such tht (, j ) s smply extensle. (2) otherwse, f < j, there exsts full rc support D j such tht (, j ) s sem-fully extensle. P s exstentl mxrpc (EmxRPC) f X, there exsts vlue D such tht (1) c () = 0, (2) hs full rc support n every cost functon (.e., P s EAC) nd (3) c j C, there exsts full rc support D j such tht (, j ) s fully extensle. P s exstentl drectonl mxrpc (EDmxRPC) f t s EmxRPC nd FDmxRPC. P s vrtul mxrpc (VmxRPC) f the mxrpc closure of Bool(P ) s non-empty See exmples n Fgure 3. Here gn, VmxRPC s defned sed on the hrd CSP Bool(P ) nd hrd mxrpc. The other softenng levels dffer from ech other y the strength of supports, the strength of wtnesses, nd the scope of pplcton of these propertes. k 2 c k 2 c 1 k 2 c k 2 c c j c j c j c j k 1 c () EmxRPC not mxrpc k 1 c () mxrpc not DmxRPC k 1 c (c) DmxRPC not FDmxRPC k 1 c (d) FDmxRPC EDmxRPC Fg. 3 Exmple of soft mxrpcs. k 1 < < k 2 < j nd jk1, jk2. The CFN() s not mxrpc ecuse vlue (, ) hs no rc support n c j (etween (, j ) nd (, j c)) tht s smply extensle on oth k 1, k 2. The CFN() s mxrpc ut s not DmxRPC ecuse vlue (, ) hs no full rc support n c j (etween (, j ) nd (, j c)) tht s fully extensle to k 2. The CFN(c) s DmxRPC (ecuse every vlue n D hs full rc support n c j, c k2 tht s respectvely fully extensle on k 2 nd j. Trngle jk1 s not nvolved n DmxRPC for ). The CFN(c) s not FDmxRPC ecuse vlue (, ) hs no full support n c j (etween (, j ) nd (, j c)) tht s smply extensle on k 1. The CFN(d) s oth FDmxRPC nd EmxRPC where (, ), (k 1, ), (j, ), (k 2, ) re respectvely EmxRPC supports of vrles, k 1, j, k 2.

10 10 Hep Nguyen et l. 4 Comprng soft locl consstences In ths secton, we compre the strength of the dfferent soft consstences proposed n the prevous secton nd soft rc consstences. Soft consstences rse specfc dffcultes when comprson of strength s consdered. Most of the consstences we consdered re domn consstences n the sense tht they defne propertes tht vlues must stsfy nd enforcng them my ncrese unry costs tht NC my ultmtely use to ncrese the lower ound c. However, vrtul locl consstences re dfferent ecuse they drectly try to ncrese c nd do not try to ncrese unry costs for NC. Thus, the strength of vrtul consstences cn e drectly mesured y the qulty of the lower ound provded. For the other soft consstences, ths strength s etter mesured y the lty to move costs to lower rtes. We therefore need to ntroduce two dfferent order reltons etween locl consstences to cpture ths dfference etween vrtul nd other consstences. We denote y c [P ] the lower ound c n prolem P. Furthermore, soft locl consstences re not confluent. A sngle prolem P my hve dfferent equvlent prolems stsfyng gven locl consstency property A. For gven CFN P nd soft locl consstency A, A(P ) s therefore defned s the set of prolems tht cn e otned fter enforcng A n P. When P lredy stsfes A, we ssume tht A(P ) = {P }.e., tht enforcng A on prolem stsfyng A does not chnge P (whch s effectvely the cse for ll enforcng lgorthms). Smlrly, focusng on lower ounds, enforcng weker consstency wll not chnge the lower ound. Defnton 10 (Stronger relton) Gven two soft consstences A nd B, A s stronger thn B, noted y A B, ff for every CFN P tht stsfes A, P lso stsfes B,.e. B(P ) = {P }. A s stronger thn B n terms of lower ound, noted y A c B, ff for every CFN P tht stsfes A nd ny P B(P ), then c [P ] = c [P ]. A s strctly stronger thn B, noted A > B, ff A B nd CFN P such tht P stsfes B nd A(P ) {P }. A s strctly stronger thn B n terms of lower ound, noted A > c B, ff A c B nd CFN P such tht P stsfes B nd P A(P ), c [P ] > c [P ]. We frst show tht entls c. Proposton 1 Gven two soft consstences A nd B, f A B then A c B. Proof Becuse A B, B(P ) = {P } for every P tht stsfes A. So we hve P B(P ), c [P ] c [P ] nd thus A c B. Smlrly to the stronger nd strctly stronger reltons for hrd consstences, our reltons for soft consstences re trnstve. Proposton 2 (Trnstvty) Gven three soft consstences A, B, nd C,. If A B nd B C then A C.

11 Trngle-sed Consstences for Cost Functon Networks 11. If A > B nd B > C then A > C. c. If A > B nd B c C then A c C. Proof. Let P e CFN tht stsfes A. Becuse A B nd P stsfes A, B(P ) = {P },.e. P lso stsfes B. Becuse B C nd P stsfes B, C(P ) = {P }. Thus, f P stsfes A, C(P ) = {P },.e. A C.. (1) Becuse > mples, we hve A B nd B C. So A C from the property (). (2) Becuse A > B, there exsts CFN P stsfyng B nd A(P ) {P }. Becuse P stsfes B nd B C, P lso stsfes C. Thus there exsts P tht stsfes C nd A(P ) {P }. So A > C. c. Becuse > mples, we hve A B. Let P e CFN tht stsfes A, P lso stsfes B. Becuse B c C nd P stsfes B, P C(P ), c [P ] = c [P ]. Thus, for every CFN whch stsfes A, P C(P ), c [P ] = c [P ]..e. A c C. To show tht soft consstency A s not stronger (resp. not stronger n terms of lower ound thn B), t s enough to show tht there exsts CFN P n whch A holds nd B does etter thn A: B(P ) {P } (resp. P B(P ), c [P ] c [P ]). Two consstences A nd B re ncomprle ff A s not stronger thn B nd B s not stronger thn A. Defnton 11 (Incomprle relton) Gven two soft consstences A nd B, A nd B re ncomprle, noted A / B, ff A / B nd B / A A nd B re ncomprle n terms of lower ound, noted A / c B, f A / c B nd A / c B Fgure 4 s the Hsse dgrm tht summrzes the reltons mong soft ACs, RPCs, PICs nd mxrpcs. A row of the grph corresponds to sx soft consstences ssocted wth sme hrd consstency nd column corresponds to the soft consstences t sme softenng level. A drected pth from consstency A to B, wthout or wth dshed rrow, respectvely mens tht A > B or A > c B. If there does not exst ny drected pth etween A nd B, they re ncomprle. Frst, we consder the relton etween vrtul consstences nd domn consstences. Then, domn consstences re consdered ccordng to the rows nd the columns of the grph. Theorem 1 Gven two hrd locl consstences A, B {AC,RPC,PIC,mxRPC}, f we denote y V A, V B ther correspondng vrtul consstences nd A, B ny other softenng level of A nd B,. VA > c A.. If A > B then 1 VA > VB 2 VA > c B. Proof. We frst prove tht VA c A y contrdcton. Suppose tht there exsts CFN P stsfyng VA nd enforcng A cn stll ncrese c from

12 12 Hep Nguyen et l. AC DAC FDAC EAC RPC DRPC EDAC VAC FDRPC ERPC PIC DPIC EDRPC VRPC FDPIC EPIC mxrpc DmxRPC EDPIC VPIC FDmxRPC EmxRPC EDmxRPC VmxRPC Fg. 4 Hsse dgrm of reltons etween soft consstences A B A > B A B C mples A C A B A > c B A B C mples A C vrle x. All vlues nd tuples whose costs hve een necessry for ncresng c y A re lso fordden when enforcng A n the clssc CSP Bool(P ). So, f we elmnte these vlues nd tuples n the sme order tht costs re moved y A n P, x wll e wped-out n Bool(P ). Thus P s not VA nd the ssumpton s flse. Ths mens tht VA c A. Secondly, Fgure 12 shows prolem tht stsfes every non-vrtul vrnt of AC, RPC, PIC, mxrpc ut not the vrtul ones. Enforcng the vrtul one wll led to strctly stronger c.. Consder frst the cse of V B. Frst, we prove tht VA V B. Let P e CFN whch s VA. The A closure of Bool(P ) s not empty. Becuse A B, the B closure of Bool(P ) wll e not empty. Thus, P lso stsfes VB,.e. VB(P ) = {P }. Now we prove tht VA V B,.e. VmxRPC > VPIC > VRPC > VAC. Fgures 5, 6, 7 respectvely show CFN whch s VAC ut not VRPC, VRPC ut not VPIC, VPIC ut not VmxRPC nd c cn e ncresed y the unstsfed consstences. We now consder the cse of ny other soft consstency B ssocted wth B. From VA > VB (just ove) nd from V B > c B (Theorem 1()), y Proposton 2(c), we deduce tht VA c B. Now, we wll prove tht VA > c B. Becuse V B > c B, there exsts CFN P such tht P s B nd V B cn stll ncrese the lower ound c [P ]. Ths mens tht the B closure of Bool(P ) s empty. Becuse A > B, the A closure of Bool(P ) s lso empty. Thus, enforcng VA on P wll ncrese c whle P stsfes B. The followng theorem shows tht gven softenng level, the correspondng soft mxrpc s strctly stronger thn the correspondng soft PIC, whch

13 Trngle-sed Consstences for Cost Functon Networks 13 s strctly stronger thn the correspondng soft RPC, whch tself s strctly stronger thn the correspondng soft AC. Theorem 2 (Vertcl comprson). mxrpc > PIC > RPC > AC.. DmxRPC > DPIC > DRPC > DAC. c. FDmxRPC > FDPIC > FDRPC > FDAC. d. EmxRPC > EPIC > ERPC > EAC. e. EDmxRPC > EDPIC > EDRPC > EDAC. Proof Frst, we cn note tht the stronger thn relton holds etween the consdered prs of consstences, sed on ther defnton: t ech softenng level, the soft vrnt of mxrpc mples the soft vrnt of PIC. The sme pples for PIC nd RPC, s well s RPC nd AC. Second, we prove the strctly stronger thn relton etween them y showng CFNs n whch the weker consstences hold whle the stronger ones do not.. Fgure 5 shows CFN tht stsfes AC ut does not stsfy RPC. Fgure 6 shows CFN tht stsfes RPC ut does not stsfy PIC. Fgure 7 shows CFN tht stsfes PIC ut does not stsfy mxrpc. Thus mxrpc > PIC > RPC > AC. -e. The proof s smlr to tht for () y usng Fgure 5, 6 nd 7. The followng theorem wll show tht for ny hrd consstency: (1) the ssocted exstentl drectonl consstency s strctly stronger thn oth the exstentl nd the full drectonl ones, (2) the ssocted full drectonl consstency s strctly stronger thn oth the non-drectonl nd the drectonl ones, (3) other prs of soft consstences re ncomprle. Theorem 3 (Horzontl comprson) Gven two dfferent hrd consstences A nd B n {AC, RP C, P IC, mxrp C}, gven A, DA, F DA, EA, EDA the smple, drectonl, full drectonl, exstentl, exstentl drectonl vrnt of A nd B, DB, F DB the smple, drectonl, full drectonl vrnt of B,. (column 2-1): A / DB. (column 3-1,2): F DA > A, DA c. (column 4-1,2,3): EA / B, DB, F DB d. (column 5-3,4): EDA > F DA, EA Proof. A / DB: usng Fgures 8 nd 9.. F DA > A, DA. The stronger thn relton s mpled y the defnton of the consstences. F DA > A: Fgure 8 shows prolem whch s mxrpc, PIC, RPC, AC ut s not FDmxRPC, FDPIC, FDRPC, FDAC. F DA > DA: Fgure 9 shows prolem whch s DmxRPC, DPIC, DRPC, DAC ut s not FDmxRPC, FDPIC, FDRPC, FDAC. c. EA / B, DB, F DB: usng Fgures 10 nd 11. d. EDA > F DA, EA. The proof drectly follows from the defntons.

14 14 Hep Nguyen et l. Theorem 4 (Incomprlty) For ny pr of consstences whch s not covered y the three prevous theorems, the consstences re ncomprle. Proof FDAC / RPC,PIC,mxRPC, DRPC,DPIC,DmxRPC: usng Fgures 5, 8 nd 9. FDRPC / PIC,mxRPC, DPIC,DmxRPC: usng Fgures 6, 8 nd 9. FDPIC / mxrpc, DmxRPC: usng Fgures 7, 8 nd 9. EDAC / (E/FD/D/-)(RPC/PIC/mxRPC): usng Fgures 5, 10 nd 11. EDRPC / EPIC, EmxRPC, FDPIC, FDmxRPC, DPIC, DmxRPC, PIC, mxrpc: usng Fgures 6, 10 nd 11. EDPIC / EmxRPC, FDmxRPC, DmxRPC, mxrpc: usng Fgures 7, 10 nd 11. VAC / c (ED/E/FD/D/-)(RPC/PIC/mxRPC): usng Fgures 5 nd 12. VRPC / c (ED/E/FD/D/-)(PIC/mxRPC): usng Fgures 6 nd 12. VPIC / c (ED/E/FD/D/-)mxRPC: usng Fgures 7 nd 12. j l k AC DAC FDAC EAC EDAC VAC RPC DRPC FDRPC ERPC EDRPC VRPC PIC DPIC FDPIC EPIC EDPIC VPIC mx Dmx FDmx Emx EDmx Vmx Fg. 5 A CFN tht stsfes ll rc consstences ut does not stsfy ny soft RPC (hence does not stsfy ny soft PIC, mxrpc). j < k < < l. The tle on the rght ndctes whch consstences re stsfed or not (strkethrough). mxrpc s refly wrtten s mx, nd the sme for other vrnts of mxrpcs. The prolem does not stsfy ny soft RPC ecuse of vrle j (the unque support (j, k ) of (j, ) n c jk s not smply extensle on nd the unque support (j, k ) of (j, ) s not smply extensle on l). k c j l c m RPC DRPC FDRPC ERPC EDRPC VRPC PIC DPIC FDPIC EPIC EDPIC VPIC mx Dmx FDmx Emx EDmx Vmx c c Fg. 6 A CFN tht stsfes ll RPC consstences ut does not stsfy ny PIC consstency. < j < k < l < m. Every vlue of stsfes RPC consstences ecuse t hs more thn 2 full (hence smple) rc supports n c k, c j, c l, c m. The prolem does not stsfy ny PIC consstency ecuse of vrle (vlue (, ) s not smply (hence not fully) extensle to trngle lm whle (, ) s not smply (hence not fully) extensle to trngle jk ).

15 Trngle-sed Consstences for Cost Functon Networks 15 j 3 j 6 c c RPC DRPC FDRPC ERPC EDRPC VRPC PIC DPIC FDPIC EPIC EDPIC VPIC mx Dmx FDmx Emx EDmx Vmx j 1 j 2 j 5 j 4 Fg. 7 A CFN tht stsfes ll PIC consstences ut does not stsfy ny mxrpc consstency. < j 1 < j 2 < j 3 < j 4 < j 5 < j 6. There re only zero unry costs n ths prolem, thus smple nd full supports (or wtnesses) re dentcl. The prolem s EDPIC snce oth (, ), (, ) cn e fully extended to ll 4 trngles. However, the prolem does not stsfy ny mxrpc consstency ecuse of vrle (no rc support of vlue (, ) n c j1 cn smultneously e extended on j1 j 2 nd j1 j 3, the sme for vlue (, ) n c j4 ). k 1 j AC DAC FDAC EDAC RPC DRPC FDRPC EDRPC PIC DPIC FDPIC EDPIC mxrpc DmxRPC FDmxRPC EDmxRPC Fg. 8 A CFN whch s non-drectonl consstent ut s drectonl nconsstent for order < j < k. The prolem s not DAC ecuse vlue (, ) hs no full rc support n c k. Therefore, t does not stsfy FDAC, EDAC, FDRPC, EDRPC, FDPIC, EDPIC, FDmxRPC, EDmxRPC. However, the prolem s mxrpc (hence PIC, RPC) ecuse t s AC nd every domn vlue s smply extensle to the trngle. k j AC DAC FDAC EDAC RPC DRPC FDRPC EDRPC PIC DPIC FDPIC EDPIC mxrpc DmxRPC FDmxRPC EDmxRPC Fg. 9 A CFN whch s drectonl consstent ( > j > k) ut s non-drectonl nconsstent. The prolem s not AC ecuse (, ) hs no rc support n c j. However, the prolem s DAC ecuse every vlue of j nd k hs full rc support n c j, c k. Moreover, the prolem s DmxRPC (hence DPIC, DRPC) ecuse every vlue of j nd k cn e fully extended on the trngle (n the trngle jk, only the smllest vrle k nd c k, c kj re concerned y trngle-sed drectonl consstences). 5 Algorthms In ths secton, we present lgorthms for enforcng soft PIC, DPIC, FD- PIC, EPIC, EDPIC, mxrpc, DmxRPC, FDmxRPC, EmxRPC, nd EDmxRPC. Soft RPCs hve not een mplemented ecuse they re weker thn ther PIC nd mxrpc counterprts nd ecuse t s costly to mntn the unqueness of rc supports per vlue n ech cost functon rc supports cn e tertvely creted nd roken when EPTs re ppled. For vlue (, ) tht does not stsfy gven TRIC (trngle consstency), the common de s to crete support for vlue (, ) on c j tht s

16 16 Hep Nguyen et l. 1 1 j k l AC DAC FDAC EAC EDAC RPC DRPC FDRPC ERPC EDRPC PIC DPIC FDPIC EPIC EDPIC mxrpc DmxRPC FDmxRPC EmxRPC EDmxRPC Fg. 10 A CFN whch s full drectonl consstent ut s exstentl nconsstent for l < j < k <. The prolem s not EAC (hence not ERPC, EPIC, EmxRPC) ecuse of vlue ( hs no full support n c j whle hs no full support n c l ). However, the prolem s FDmxRPC (hence FDPIC, FDRPC) ecuse t s FDAC nd every vlue of, k cn e smply extended to oth trngles nd every vlue of j, l cn e fully extended to jk nd lk respectvely. k j AC DAC FDAC EAC EDAC RPC DRPC FDRPC ERPC EDRPC PIC DPIC FDPIC EPIC EDPIC mxrpc DmxRPC FDmxRPC EmxRPC EDmxRPC Fg. 11 A CFN whch s exstentl consstent ut not full drectonl consstent for > j > k. The prolem s not AC (hence s not RPC, PIC, mxrpc) ecuse of vlue (, ) (no rc support n c j ) nd s not DAC (hence s not DRPC, DPIC, DmxRPC) ecuse of vlue (j, ) (no full rc support n c j ). However, the prolem s EmxRPC (hence EPIC, ERPC, EAC) where (, ), (j, ), (k, ) re respectvely EmxRPC supports of, j, k. 1 1 j k l m AC DAC FDAC EAC EDAC VAC RPC DRPC FDRPC ERPC EDRPC VRPC PIC DPIC FDPIC EPIC EDPIC VPIC mx Dmx FDmx Emx EDmx Vmx Fg. 12 A CFN whch s exstentl drectonl ut s not vrtul consstent l < < j < k < m. The prolem s not VAC (hence not VRPC, VPIC, VmxRPC) ecuse AC mkes Bool(P) wped-out t j or k. Conversely, the prolem s EDmxRPC where vrles j, m, l re FDmxRPC n jm nd, j, k, l, m re EmxRPC supports of vrles. lso extensle on vrles k y shftng costs n trngles jk (consstng of nry, ternry nd possly unry costs) to c. We denote y jk (,, c) = c j (, )+c jk (, c)+c k (, c)+c jk (,, c) the comned cost defned y the sum of nry nd ternry costs nvolved n tuple (, j, k c ), where c jk (,, c) = 0 f c jk does not exst. Algorthm 2 presents ll the sc opertons for shftng costs nsde trngles nd prunng vlues. Extend2To3(,, j,, c jk, α) extends cost of α from pr of vlues (, j ) to ternry cost functon c jk. Project3To2(c jk,,, j,, α) projects cost of α from c jk on (, j ). Project3To1(c jk,,, α) projects cost of α from c jk on vlue (, ). Extend1To2(,, c j, α) extends cost of α from vlue (, ) to c j. Project2To1(c j,,, α) projects cost of α from c j on vlue (, )

17 Trngle-sed Consstences for Cost Functon Networks 17 PruneVrs() removes ll nconsstent vlues hvng unry cost equl to m. The queues Q, P, S, T store vrles or cost functons whch hd some chnge n domn or n cost. They wll e used for the propgton of chnges n our lgorthm. Q stores vrles such tht some vlue of D hs een deleted (Procedure PruneVrs(), lne 24). P stores vrles such tht some vlue of D hs ncresed ts cost from 0 (Procedure Project3To1 t lne 13 nd Project2To1 t lne 17 S s n uxlry queue wth the sme contents s P (Procedure Project3To1 t lne 13 nd Procedure Project2To1 t lne 17). It s used to effcently uld the propgton queue R whch contns vrles tht need to e checked for the exstentl consstency. These re ll vrles of S (those tht hve vlues whch cost ncresed from 0) nd ther neghors ecuse: (1) for S, the vlue n D tht hs ncresed ts unry cost my e the exstentl support of nd (2) the exstentl support of neghor vrles j my e fully supported y ths vlue. T contns nry cost functons c j tht hve een modfed (ecuse of unry cost extenson n Procedure Extend1To2, lne 4) for whch, j nd ther common neghors my hve lost smple support/wtness nd need to e revsed. 5.1 Enforcng PICs Enforcng PIC supports Smple PIC supports re enforced y Procedure fndpicsupport n Algorthm 3. To crete smple PIC support for vlue on jk, nry nd ternry costs nvolved n jk re moved to n such wy tht there s tuple (, j, k c ) whose ternry nd nry costs decrese to 0. The order for movng costs s presented n Fgure 13. Frst, nry costs c j, c k, c jk re extended on ternry cost functon c jk y Procedure Extend2To3 (lnes 10 12). Then, ternry costs c jk re projected on y Procedure Project3To1 (lne 13). The mxmum cost tht cn e projected on ech vlue D s stored n P []. It s computed sed on the vlle nry nd ternry costs (lne 3). Bnry cost extensons E j, E k, E jk re then computed sed on P [] nd the ternry nd nry costs (on the two other sdes of the trngle, see lnes 4 9). The extensons should e suffcently lrge so tht lter projectons of P [] wll not crete negtve costs nd suffcently smll so tht zero trngle cost remns fter projecton: there should exst vlues k c, j nd such tht the fnl resultng ternry cost c jk (,, c)+e j (, )+E k (, c)+e jk (, c) P [] = 0. Ech computed prwse extended cost E (, ) s suffcent to stsfy the mxmum cost requrements on the thrd vrle. Snce these extensons re supposed to e done sequentlly, lne 7 sutrcts E j (, ), whch wll e ncluded n the ternry cost, nd does not requre c j (, ). The sme resonng pples for lne 9, for oth prevous extensons.

18 18 Hep Nguyen et l. Algorthm 2: Elementry opertons 1 Procedure Extend1To2(,, c j, α) 2 // precondton: c () α > 0 3 Shft((, ), c j, α); 4 T T {c j }; 5 Procedure Extend2To3(,, j,, c jk, α) // precondton: c j (, ) α > 0 6 Shft((, j ), c jk, α); 7 Procedure Project3To2(c jk,,, j,, α) 8 // precondton: c D k, c jk (,, c) α 9 Shft((, j ), c jk, α); 10 Procedure Project3To1(c jk,,, α) 11 // precondton: D j, c D k, c jk (,, c) α 12 f c () = 0 α > 0 then 13 P P {}; S S {}; 14 Shft(, c jk ); 15 Procedure Project2To1(c j,,, α) // precondton: D j, c j (, ) α 16 f c () = 0 α > 0 then 17 P P {}; S S {}; 18 Shft(, c j ); 19 Procedure PruneVrs() 20 forech X do 21 forech D do 22 f c () + c m then 23 D D {}; 24 Q Q {}; 25 Procedure ssmllest(, jk ) 26 return (( < j) ( < k)); In the end, nry cost extensons on ternry functons do not led to the loss of ternry AC supports. Moreover, nry cost extensons do not led to the loss of PIC supports ecuse PIC supports nvolve only zero nry costs whch cnnot e used for extenson. Full PIC supports re smlrly enforced y Procedure fndfullpicsupport n Algorthm 3. The dfference s tht unry costs on j, k re extended on nry functons c j nd c k y Procedure Extend1To2, n order to crete full PIC supports wth zero unry costs (lnes 23, 24 respectvely). Then nry nd ternry costs re moved to s for smple PIC supports (lne 25). The order n whch costs re moved to enforce full PIC supports s lso vsle n Fgure 13. The unry costs of j, k re tken nto ccount for the computton of P [] s well s for the computton of unry cost extensons E j, E k (lnes 18,20,22). As for nry extensons, unry cost extensons should e suffcently lrge to vod the creton of negtve costs y lter projectons of P [] nd

19 Trngle-sed Consstences for Cost Functon Networks 19 Fg. 13 The order of cost movements for enforcng smple or full PIC supports on vrle, where unry cost extensons re not ncluded n the enforcement of smple PIC supports. The rcs ndcte the drecton of cost movements nd the numers under the rcs ndcte the order n whch the correspondng cost movements re performed. smll enough so tht the the fnl nry costs c j (, ) + E j () E j (, ) nd c k (, c) + E k (c) E k (, c) re equl to 0. Therefore, unry cost extensons on nry functons cnnot led to the loss of nry AC supports. However, unry cost extensons on nry functons cn led to the loss of smple PIC supports, thus modfed nry functons re stored n the lst T n order to lter enforce PIC supports for relted vlues. Exmple 2 Consder the CFN() n Fgure 14. It hs 4 vrles < j < k < l nd 5 nry cost functons c j, c k, c l, c jk, c jl. Bnry costs re represented y edges (contnuous lne) nd ternry costs re represented y hyper-edges (dshed lnes for c jk nd dotted lnes for c jl ). The sence of (hyper)edges ndctes zero cost. The ntl prolem s FDAC ut not FDPIC ecuse vlue (, ) s not fully extensle on jk. Now, consder enforcng full PIC supports for the vlues of. Procedure fndfullpicsupport(, j, k) computes the mounts of cost for projectons/extensons: P [] = E j [] = 1. Other shfted costs re zero. After extendng cost of 1 from (j, ) on c j, t wll cll Procedure fndpicsupport(, j, k), compute the mounts of shfted cost s follows: P [] = E j [, ] = E k [, ] = E jk [, ] = 1. nd perform the cost shfts. The resultng prolem, presented n Su-fgure 14(d) s stll not FDPIC ecuse vlue (, ) cnnot e fully extended on trngle jl. Then Procedure fndfullpicsupport(, j, l) computes nd performs the followng cost shftng: P [] = E j [, ] = E l [, ] = E jl [, ] = 1. The fnl prolem, presented n Su-fgure 14(g) s FDPIC. Contrrly to hrd PIC, enforcng smple nd full PIC supports cn crete new ternry functons, e.g., c jk, c jl. Whenever nry cost need to e extended to ternry cost functon tht does not exst, the ternry cost functon needs to e creted nd ntlzed wth n empty cost for every tuple Soft PIC lgorthms Enforcng EDPIC requres enforcng PIC, DPIC, nd EPIC smultneously. We thus only present n lgorthm for EDPIC. PIC, DPIC, FDPIC, nd EPIC lgorthms cn e derved y removng locks of code.

20 20 Hep Nguyen et l. Algorthm 3: Algorthms enforcng PIC supports 1 Procedure fndpicsupport(, jk ) 2 forech D do 3 P [] mn Dj,c D k jk (,, c); 4 forech D, D j do 5 E j [, ] mx c Dk {P [] c jk (,, c) c k (, c) c jk (, c)}; 6 forech D, c D k do 7 E k [, c] mx Dj {P [] c jk (,, c) c jk (, c) E j (, )}; 8 forech D j, c D k do 9 E jk [, c] mx D {P [] c jk (,, c) E j (, ) E k (, c)}; 10 forech D, D j do Extend2To3(,, j,, c jk, E j [, ]); 11 forech D, c D k do Extend2To3(,, k, c, c jk, E k [, c]); 12 forech D j, c D k do Extend2To3(j,, k, c, c jk, E jk [, c]); 13 forech D do Project3To1(c jk,,, P []); 14 α mn D {c ()}; 15 UnryProject(, α); 16 Procedure fndfullpicsupport(, jk ) 17 forech D do 18 P [] mn Dj,c D k jk (,, c) + c j () + c k (c); 19 forech D j do 20 E j [] mx D,c D k {P [] jk (,, c) c k (c); 21 forech c D k do 22 E k [c] mx D, D j {P [] c jk (,, c) E j []; 23 forech D j do Extend1To2(j,, c j, E j []); 24 forech c D k do Extend1To2(k, c, c k, E k [c]); 25 fndpicsupport(, jk ); 26 Procedure fndepicsupport() 27 α mn D {c () + jk,>j or >k mn Dj,c D k { jk (,, c) + c j () + c k (c)}}; 28 f α > 0 then 29 forech jk do 30 f ssmllest(, jk ) then fndfullpicsupport(, jk ); 31 R R jk {j, k}; 32 UnryProject(, α); EDPIC s enforced y Procedure enforceedpic n Algorthm 4. Ths procedure conssts of four nner-whle loops tht respectvely enforce EPIC, DPIC nd PIC. It lso enforces NC y cllng PruneVrs t lne 24. The frst whle-loop (lnes 5-7) enforces EPIC. It frst puts n R ll vrles tht need to e checked for EPIC sed on the uxlry queue S (lne 4). EPIC supports of vrles R re enforced y Procedure fndepicsupport (lne 7). When enforcng the exstentl support for, EPIC s only responsle for trngles on whch s not the smllest vrle ecuse DPIC wll tke cre of the remnng ones (Algorthm 3, lne 27). If hs no fully supported vlue (.e., α > 0) such vlue cn e creted y enforcng full PIC supports for every vlue of on every trngle n whch s not the smllest vrle

21 Trngle-sed Consstences for Cost Functon Networks 21 Fg. 14 Cost evoluton n CFN durng the enforcement of full PIC supports () orgnl prolem wth 5 nry cost functons c j, c k, c l, c jk, c jl, < j < k < l. It s FDAC ut not FDPIC ecuse of vrle where (, ) nd (, ) cnnot e fully extended on jk nd jl respectvely. () extendng cost of 1 from j on c j wth E j [] = 1. (c) extendng cost of 1 from (, j ), (, k ) nd (j, k ) on c jk wth E j [, ] = E k [, ] = E jk [, ] = 1. (d) projectng cost of 1 from c jk on wth P [] = 1. (e) extendng cost of 1 from (, j ), (, l ) nd (j, l ) on c jk wth E j [, ] = E l [, ] = E jl [, ] = 1. (f) projectng cost of 1 from c jk on wth P [] = 1 nd then enforcng NC y projectng cost of 1 from c on c. The resultng prolem s FDPIC. (Algorthm 3, lne 30). The EPIC supports of neghor vrles of cn lso e destroyed (due to new vlues of non-zero cost mde y the enforcement of full PIC supports on ) nd thus re pushed ck to R to e lter checked for EPIC (Algorthm 3, lne 31). DPIC s enforced y the second whle-loop t lne 8. For vrle j P, only vrles tht re lnked to j y trngle (lne 10) nd re the smllest vrle of the trngle (lnes 11, 12) re consdered for checkng for DPIC. PIC s enforced y two whle-loops t lnes 13 nd 18. For vrle j Q, every neghor vrle of s checked for PIC. For ech c j T,, j nd ll vrles connected to oth nd j re checked for PIC. Smple PIC supports re enforced n the reverse drecton of the DAC order,.e. n trngles n whch the consdered vrles re not the smllest (lnes 16-17, 21-23). From Algorthm 4, lgorthms for enforcng other levels of PICs cn e otned y pproprtely keepng the rght nner whle-loops: the frst loop (lnes 4-7) for EPIC, the second one t lne 8 for DPIC, the thrd one t lne 13 for PIC, nd three loops t lnes 8, 13, 18 for FDPIC.

22 22 Hep Nguyen et l. Algorthm 4: Algorthm enforcng EDPIC 1 Procedure enforceedpic() 2 S = P = Q = X; T ; 3 whle Q or P or S or T do 4 R S S, jk {j, k}; 5 whle R do 6 R.pop(); 7 fndepicsupport(); 8 whle P do 9 j P.pop(); 10 forech jk do 11 f ssmllest(, jk )) then fndfullpicsupport(, jk ); 12 f ssmllest(k, jk )) then fndfullpicsupport(k, jk ); 13 whle Q do 14 j Q.pop(); 15 forech jk do 16 f ssmllest(, jk ) then fndpicsupport(, jk ); 17 f ssmllest(k, jk ) then fndpicsupport(k, jk ); 18 whle T do 19 c j T.pop(); 20 forech jk do 21 f ssmllest(, jk ) then fndpicsupport(, jk ); 22 f ssmllest(j, jk ) then fndpicsupport(j, jk ); 23 f ssmllest(k, jk ) then fndpicsupport(k, jk ); 24 PruneVrs() 5.2 Enforcng mxrpcs In contrst to PICs tht re enforced on trngles shrng vrle, mxrpcs re enforced on trngles shrng two vrles of nry cost functon. The extensle rc support of vlue (, ) n nry cost functon c j s stored n mxrpcsupport[,, j] nd the wtness for ths support on vrle k s stored n mxrpcwtness[,, j, k]. In our lgorthms for enforcng soft mxrpcs, we wll use prmeter nmed fulllevel, where fulllevel = flse ndctes tht sem-fully extensle rc supports re used (FDmxRPC) nd fulllevel = true ndctes tht fully extensle supports re used (EmxRPC). We wll use the followng functons: jk (,, c) = c k (, c) + c jk (, c) + c jk (,, c): denotes the ncompletely comned cost of tuple (,, c) (excludng c j (, ) from jk (,, c)). k j(, ) =mn c Dk jk (,, c) denotes the mnmum ncompletely comned cost of tuples nvolvng two vlues (, j ). Ths s the mxmum cost tht cn e projected on the pr of vlues (, j ) from two sdes c k, c jk of the trngle jk wthout cretng negtve costs.

23 Trngle-sed Consstences for Cost Functon Networks 23 rgmn k j(, ) s used to denote vlue c D k for whch ths mnmum s reched. It s smple wtness for the pr (, ) on the vrle k. mn{ jk (,, c) + c k (c)} (fulllevel = true) ( < k) k j(,, fulllevel) = c D k mn{ jk (,, c)} (fulllevel = flse) c D k s smlr to k j(, ) ut t tkes nto ccount the unry cost c k of wtnesses n the cse of (1) fully extensle (fulllevel=true) or (2) sem-fully extensle rc supports on trngles w.r.t DAC order ( < k). rgmn k j(, ) s used to denote the vlue c D k for whch ths mnmum s reched. It s (full) wtness for the pr (, ) on the vrle k. j (, ) = k { k j(, )} s the mxmum sum of costs tht cn e projected on the pr of vlues (, j ) from ll trngles jk shrng, j. j (,, fulllevel) = k { k j(,, fulllevel)} s smlr to j (, ) ut tkes nto ccount the unry costs of wtnesses c k ccordng to fulllevel nd the order etween nd k s n the defnton of k j Enforcng mxrpc supports nd wtnesses Smple mxrpc support for vlue (, ) on c j s enforced y Procedure fndmxrpcsupport n Algorthm 5. The mn de s to move costs from two sdes c k, c jk of ll trngles jk to c j v c jk (lnes 18-20) nd fnlly project costs from c j to (, ) (lne 21) n such wy tht, for ech trngle jk, there exsts vlue D j nd vlue c D k such tht the nry nd ternry costs nvolved n the tuple (, j, k c ) decrese to 0. The mxmum cost P tht cn e projected to (, ) wthout cretng negtve costs s the mnmum over ll D j of the nry cost of (, j ) tht cn e otned y comnng the orgnl cost c j (, ) nd the cost tht cn e shfted to t from ll trngles (computed y functon j, lne 8).From ths, t ecomes possle to compute the ctul cost P j [, ] tht ech trngle jk provdes to (, j ) for ths mount of projecton to (, ). It s the mnmum of wht s needed for ths pr of vlues (P c j (, )) nd of wht cn e provded for t y jk (lne 13). Ths condton gurntees tht c j hs enough costs to mke unry cost projecton P on (, ) wthout cretng negtve costs. Moreover, f more cost s projected on c j, ths cnnot led to unry cost projecton greter thn P. In order to project cost of P j [, ] from c jk to (, j ) (lne 20), ech sde (, k c ) nd (j, k c ) hs to extend n mount of cost E k [, c] nd E jk [, c] to c jk (lnes 19 nd 18). These nry cost extensons E k [, c], E jk [, c] re lso the mnmum of the vlle cost c k (, c), c jk (, c) tht (, k c ), (j, k c ) hve nd the cost tht they need to provde to c jk (lnes 15, 17). Full mxrpc supports (coverng oth fully extensle supports for EmxRPC nd sem-fully extensle for FDmxRPC) re enforced y fndfullmxrpcsupport n Algorthm 5. The de for enforcng full mxrpc support for vlue (, ) on c j s to extend unry costs from j to c j (lne 28) nd from thrd vrles k to c k (lne 32). Then, costs re moved n the sme wy s for smple

24 24 Hep Nguyen et l. mxrpc support n Procedure fndmxrpcsupport (lne 33). The mxmum cost P tht cn e projected on (, ) s recomputed y tkng nto ccount the unry cost c j of supportng vlues nd the unry costs c k of wtnesses v (lne 25). In order to cheve ths unry projecton, ech vlue (j, ), (k, c) needs to extend respectvely on c j nd c k n mount of cost E j, E k (lnes 27 nd 31). The order n whch costs re moved when enforcng full mxrpc supports s descred n Fgure 15. An EmxRPC support for vrle s enforced thnks to Procedure fndemxrpcsupport() n Algorthm 5. It frst checks the EmxRPC property t lne 36. If there does not exst ny EmxRPC support (lne 37), the procedure wll serch for full mxrpc support for ny vlue of n ny cost functon c j y cllng fndfullmxrpcsupport wth the opton fulllevel=true. It only hs to tke cre of the trngles jk n whch s the smllest vrle, ecuse DmxRPC tkes cre of the remnng cses (the condton t lne 29 of Procedure fndfullmxrpcsupport). Fg. 15 The order of cost movements for enforcng smple full mxrpc supports where unry cost extensons re not ncluded n the enforcement of smple PIC supports. Exmple 3 Consder the CFN() n Fgure 16. It s FDPIC ut not FDmxRPC ecuse hs no full AC support n c j whch cn e cn e extended on oth jk nd jl : (, j ) cn e extended on jl ut not on jk whle (, j c ) cn e extended on jk ut jl. The postve projecton/extenson costs computed y Procedure fndfullmxrpcsupport(,, j, flse) re: P = 2, E j [] = 1. The procedure extends cost of 1 from j on c j nd then clls f ndmxrp C(,, j) whch computes the followng postve projectons/extenson costs: P = E jk [, ] = P j [, ] = E jl [c, ] = E jl [c, ] = P j [, ] = 2. The fnl prolem presented n Su-fgure 16(g) s FDmxRPC. Let j e vrle tht hd chnge n the domn D j or n unry cost c j (ncresng from 0). The former cse cn rek the wtness for smple or sem-full supports of vrle neghor to j n some c j, whle the lst cse cn rek the wtness for sem-full nd full supports. The check nd serch for new wtnesses s performed y Algorthm 6. Procedure fndwtness remove (, k, j) hndles the cse of domn reducton n D j. For ny vlue (, ), t checks the vllty of ts current (smple or sem-full) support n c k (lne 5, lgorthm 6), s well s the vllty of