COMPLEXITY OF COUNTING PLANAR TILINGS BY TWO BARS

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1 OMPLXITY O OUNTING PLNR TILINGS Y TWO RS KYL MYR strt. W show tht th prolm o trmining th numr o wys o tiling plnr igur with horizontl n vrtil r is #P-omplt. W uil o o th rsults o uquir, Nivt, Rmil, n Roson in [] in whih thy show tht th prolm trmining xistn o suh tilings ws shown to NP-omplt.. Introution or this prolm w onsir th pln s gri o unit squrs, n w in igurs to susts o ths squrs. W onsir th omplxity o ounting how mny wys igur n til y l rtngl n n m rtngl whr l n m r oth t lst 2. In th s tht l = m = 2 this prolm n ormult in trms o ounting prt mthings o plnr grphs whih Kstlyn show oul on in polynomil tim [3]. Thus w look t th s ithr l or m is t lst 3; in prtiulr w will show: Thorm.. ounting th numr o wys to til plnr igur with l n m rtngls is #P-omplt i t lst on o l n m is t lst 3 n th othr is t lst 2. Our proo is moiition o tht givn in [] tht ws us to show tht th ssoit ision prolm is NP-omplt. W n ssoit iprtit grph to onjuntion o ooln luss y ltting th luss n vrils nos n hving gs onnting luss to th vrils thy ontin. Th proo in [] rus rom plnr 3-N St y onvrting th ssoit plnr grph into igur whih is till i n only i th plnr 3-N xprssion is stisil. W will o similr rution rom plnr-x3monost. inition.2. Plnr -x3monost ooln xprssion is -x3mono i it th onjuntion o sris o luss suh tht h lus ontins xtly 3 non-ngt vrils n th lus is tru i xtly on o thos vrils is tru. Suh n xprssion is plnr i th ssoit grph is plnr. inlly, plnr -x3monost is th qustion: Givn plnr -x3mono xprssion, is thr n ssignmnt o ooln vlus to h vril suh tht ll luss in th xprssion r tru? ounting solutions o plnr -x3monost xprssions ws shown to #Pomplt y Hunt, Mrth, Rhkrishnn, n Strns in [2]. 2. Proo o th s l = 2, m = 3 W will irst prov our lim or th s l = 2,m = 3, n thn xtn ths rsults to l 2, m 3. In orr to o this w introu vril igurs, lus

2 2 KYL MYR xmpl o Nottion or Tiling o igur igur xmpl 2 o Nottion or Tiling o igur igur 2 igurs, n g igurs tht orrspon to th vril nos, lus nos, n gs rsptivly o th plnr grph ssoit to plnr -x3mono xprssion. ollowing th onvntion o [], w will pit our igurs y th lttrs -, whr th lttr w us is trmin y th position o th squr mo 2 horizontlly n mo 3 vrtilly, n whr th lttr is pitliz i it is ovr y th vrtil r n lowr-s i it is ovr y th horizontl r(s igurs n 2). Th irst igur w onstut is wir. wir will onnt vril igur to lus igur n will trnsmit th vril vlus rom th vril igur to th lus igur. Wirs n til in two wys orrsponing to tru n ls. Th tru tiling lvs squr on h n until, whil th ls tiling ovrs ll th squrs. W hv two typs o wirs; thos tht strts on n thos tht strt on, n h o ths will n on th othr lttr, n mor spiilly w n xtn th wir so tht it ns on ny o tht lttr, w show wir o th irst typ with oth tru n ls tilings in igurs 3 n 4. Two wirs, on o h typ, run long si h othr orm n g igur. W nxt onstrut th vril igur. This igur will onnt to on g igur or h lus in whih th orrsponing vril pprs, thus th siz vris pning on how mny luss vril is in. Thr r only two wys to til this igur: on wy will or th onnting g igurs to hv tru tiling n th othr will or thm to hv ls tiling. Ths two wys orrspon to

3 OUNTING TILINGS Y TWO RS 3 Wir with Tru Tiling igur 3 Wir with ls Tiling igur 4 Vril igur with Tru Tiling igur 5 Vril igur with ls Tiling igur 6 vril ing tru or ls n r shown in igurs 5 n 6 or th s o 3 onnting g igurs. Our lst igur, shown in igur 7, is th lus igur. It tks in thr input g igurs n is till i n only i xtly on o thm hs tru vlu. This is th stp in whih our rution is not prsimonious, tht is, it os not prsrv th numr o solutions. Spiilly, thr r rossing o wirs in this igur suh tht i oth wirs hv ls vlus, thn th rossing is till in two wys, ithr y thr horizontl rs or two vrtil ons. ut w hv onstrut this igur in suh wy tht wirs orrsponing to h st o two o th thr vrils hs thr rossings. Thus h lus igur n til 2 3 = 8 wys i it n til. W hv thus onstrut th thr igurs tht w n, n w n onnt thm ll us o th plnrity o th xprssion. Thus w hv ru th ounting prolm or plnr -x3monost to plnr tiling prolm with rs o lngth 2 n 3 in whih th numr o tilings is 8 tims th numr o solutions or th -x3monost xprssion, whr is th numr o luss. Thus th prolm is #P-omplt.

4 4 KYL MYR lus igur igur 7 m 2 l igur 8 xtnsion rom l = 2,m = 3 to ny l 2, m 3

5 OUNTING TILINGS Y TWO RS 5 3. xtnsion or l 2, m 3 W now xtn ths rsults to longr rs; horizontl r o lngth l 2 n vrtil r o lngth m 3. W o this rpling h lttr - y rtngl s shown in igur 7. It is lr tht vry tiling o th originl igur will giv ris to tiling o th xpn igur, ut or gnrl igurs this xpnsion is not prsimonious. W lim tht this xpnsion is prsimonious or th igurs tht w r onstruting. To show this w just n to hk tht th tilings or h omponnt igur is prsimonious. W strt with th vril omponnt. To s tht this is prsimonious, irst onsir th ottom lt. This is singl squr, n thus it n ovr ithr y horizontl or vrtil r. On tht is hosn, thn w n work roun th igur n s tht thr is only on wy to til th rst o th igur. Similrly with wirs, on w hv ithr tru or ls vlu or th wir givn y th vril igur w work long th wir n thr is only on wy to til th wir. Th lst igur is th lus igur, ut th lus igur is only ompos o wirs with 3 irnt typs o smll igurs, n it is sy to s tht h o thm hv prsimonious xpnsion, n thus th whol igur os. This omplt th proo. 4. utur irtions On possil irtion or utur rsrh is to rstrit this prolm to simply onnt rgions. s mntion ov, in th gnrl s trmining th xistn o tiling is NP-omplt, ut in th simply-onnt s Knyon n Knyon show in [4], tht trmining th xistn o tiling n on in linr tim. Thus thr is unmntl irn twn ths two ss. itionlly, Pk n Yng show in [5] tht thr is lrg st o rtngls or whih tiling simply onnt rgions is NP-omplt n #P-omplt, so it woul intrsting to s i it is possil to ru th siz o tht st to 2. itionlly it woul intrsting to xtn ths rsults to mor gnrl pirs o rtngls in oth th simply n non-simply onnt ss. 5. knowlgmnts I woul lik to thnk J Yng n Kvin ilks or thir vi n guin in writing this ppr. I woul lso lik to thnk Univrsity o Minnsot, Minnpolis or hosting this RU. itionlly I thnk th RTG grnt NS/MS or proviing uning or this rsrh. Rrns. nil uquir, Muri Nivt, ri Rmil, n Mik Roson, Tiling igurs o th pln with two rs., omput. Gom. 5 (995), Hrry. Hunt III, Mhv V. Mrth, Vnktsh Rhkrishnn, n Rihr win Strns, Th omplxity o plnr ounting prolms, orr s./ (998). 3. Pitr W Kstlyn, imr sttistis n phs trnsitions, Journl o Mthmtil Physis 4 (963), no. 2, lir Knyon n Rihr Knyon, Tiling polygon with rtngls, Pro. 33r Symp. ountions o omputr Sin, 992, pp Igor Pk n J Yng, Tiling simply onnt rgions with rtngls., J. om. Thory, Sr. 20 (203), no. 7, prtmnt o Mthmtis, Northstrn Univrsity, oston, Msshustts 025 -mil rss:

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