AN UPPER BOUND FOR THE PERMANENT VERSUS DETERMINANT PROBLEM BRUNO GRENET

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1 AN UPPER BOUND FOR THE PERMANENT VERSUS DETERMINANT PROBLEM BRUNO GRENET Abstract. The Permaet versus Determat problem s the followg: Gve a matrx X of determates over a feld of characterstc dfferet from two, fd the smallest matrx M whose coeffcets are lear fuctos the determates such that the permaet of X equals the determat of M. We prove that the dmesos of M are at most 2 1. The determat ad the permaet of a ( ) matrx of determates X = (x j ) 1,j over a feld K are the two very smlar polyomals defed respectvely by det(x) = ε(σ) σ S =1 x σ() ad per(x) = σ S =1 x σ() where S s the set of permutatos of {1,..., } ad ε(σ) s the sgature of the permutato σ. The determatal complexty of the permaet (over K) s the fucto dc() defed as follows: Gve a teger, dc() s the dmeso of the smallest possble matrx M whose coeffcets are lear fuctos of the x j s ad such that det M = per X (as polyomals over K). The Permaet versus Determat problem s to fd the value of dc(). Ths value depeds o the feld K. I partcular, f the characterstc of the feld s two, permaet ad determat are the same polyomal ad dc() =. I ths followg, we cosder felds of characterstc dfferet from two. The Permaet versus Determat problem goes back to a questo by Pólya [8]. For 2 2 matrces, we have the equalty ( ) ( ) a b a b per = det. c d c d The atural questo of Pólya was to kow whether ths detty ca be exteded to hgher dmesos. Szegő [10] proved that ths s mpossble, that s dc() +1 for 3. More recetly, ths questo has bee revestgated due to ts relatos to the algebrac verso of the problem P versus NP [1]. It s cojectured that dc() = 2 Ω() but the best kow lower boud s Ω( 2 ) [6, 2]. See the latter referece for a more extesve descrpto o the hstory of the problem. Date: May 25,

2 2 BRUNO GRENET I ths ote, we are terested a upper boud for ths problem. Usg Valat s uversalty of the determat for arthmetc formulas [12] (see also [4]) we ca tur a formula for the permaet to a determat. Thus usg Ryser s [9] or Gly s [3] formula, we obta dc() O(2 ). Actually, Ca, Che ad L [2, Theorem 3] meto that usg Ryser s formula, oe ca obta the mproved boud dc() O(2 ). We show ths ote a drect costructo to buld a matrx of dmesos (2 1). The costructo s based o a Arthmetc Brachg Program (ABP) of sze 2. Note that the exstece of a ABP of ths sze ca also be foud as a cosequece of a result of Nsa [7]. Oe ca defe the strct the determatal complexty of the permaet dc s () by allowg oly costats or varables the matrx M stead of geeral lear fuctos. The dc s () dc() ad we ca actually show that dc s () poly(dc()). (For stace, dc(2) = 2 ad dc s (2) = 3.) I ths ote, we actually show that dc s () 2 1, whece dc() 2 1. I the same way, Nsa s result [7] apples to ABP whose arcs are weghted by lear forms. The ABP we gve s restrcted to havg oly varables ad costats as arc weghts. For small values of, lower ad upper bouds are ot very far from each other. Mgo ad Ressayre [6] proved that for felds of characterstc zero, dc() 2 /2. For = 3, ths proves that dc(3) 5, ad wth our upper boud boud, we get 5 dc(3) dc s (3) 7. It would be terestg to vestgate what the exact values of both dc(3) ad dc s (3) are. Defto. A arthmetc brachg program (ABP) over K[ x] s a drected acyclc graph wth two dstgushed vertces s (the source) ad t (the sk). Each arc s weghted by a lear form. The weght of a path from s to t s the product of the weghts of the arcs t uses. The value of the ABP s the sum of the weghts of all possble paths from s to t. A arthmetc brachg program s sad layered f ts set of vertces V s the dsjot uo of k sets V 1,..., V k such that V 1 = {s}, V k = {t}, ad each arc goes from V to V +1 for some < k. We shall costruct a ABP of sze 2 that computes the permaet of X. Lemma. There exsts a ABP of sze 2 wth weghts {x j : 1, j } whose value equals per X. Proof. The set of vertces of the ABP s dexed by the subsets of {1,..., }. The dstgushed vertces are s = ad t = {1,..., }. The ABP s layered: Each layer cossts of all the subsets of same cardalty. We deote by V 0,..., V the (+1) layers. There s a arc from a set S V of weght x j to the set S {j} wheever j / S. There s o other arc. Clearly, the ABP has sze 2. Cosder a path from s to t ad the varables that are weghts o ths path. The rage of the frst dex of the varables {1,..., } sce the path goes through each layer, ad a arc fallg the layer V has weght x j for some j. Moreover, the secod dex j of ths

3 AN UPPER BOUND FOR THE PERMANENT VERSUS DETERMINANT PROBLEM 3 varable correspods to the elemet that s ew the set. Sce the last set the path s t = {1,..., }, each teger betwee 1 ad appears as a secod dex of some varable. Sce there are varables o a path from s to t, ths shows that each path correspods to a permutato of {1,..., }. Coversely, to each permutato correspods a path from s to t. The value of the ABP s the sum over all possble paths from s to t of the weght of the path. Thus t equals the permaet of X. Ths ABP ca be tured to a matrx whose determat has the desred value. Theorem. There exsts a (2 1) (2 1) matrx M wth etres {x j : 1, j } { 1, 0, 1} such that det M = per X. Proof. We cosder the ABP obtaed Lemma, see as a dgraph. We buld a ew dgraph G obtaed by mergg the two vertces s ad t, ad by addg a loop (of weght 1) to all other vertces of the dgraph. Sce the vertex s G s the oly vertex wth o loop o t, a cycle cover of the graph G must cover s by a cycle of legth at least 2. The graph G was obtaed by mergg the source ad the sk of a DAG, thus ts oly cycles of legth at least 2 correspod to path from s to t the ABP. Ths shows that a cycle cover of G s made of a large cycle correspodg to a path from s to t the ABP ad of loops. I other words, f we defe the weght of a cycle cover by the product of the weghts of the arcs t uses, the sum of the weght of all the cycle covers of G equals the value of the ABP. Now t s a well-kow fact that the permaet of the adjacecy matrx M of G equals the sum of the weghts of all the cycle covers of G. Sce the cycle covers of G are made of oe large cycle ad loops, they correspod to permutatos of {1,..., } that have all the same sgature. If s odd, ths sgature s 1 ad per M = det M = per X ad we are doe. Otherwse, the sgature s 1. If we replace the frst layer (of sze ) the weghts of the loops by 1, the the weght of a cycle cover wll be multpled by ( 1) 1 = 1. Thus, we get a ew matrx N such that det N = per M = per X. Ths cocludes the proof. Whe s odd, we obtaed a matrx whch uses oly 0 ad 1 as costats. If s eve, we ca also obta such a matrx, but of dmesos 2 2. The determat of a matrx ca be computed by a arthmetc crcut of polyomal sze. The crcut ca eve be made skew, meag that oe argumet of each multplcato gate s a put [11, 5]. I partcular, a skew crcut s o commutatve the sese of Nsa [7]. We prove here that we ca obta here a qute small skew crcut for the permaet based o our costructo. Moreover, our costructo does ot use ay egatve costat ad our costructo therefore yelds a mootoe skew crcut. Corollary. There exsts a mootoe skew crcut of sze ( 1)(2 1) to compute the permaet.

4 4 BRUNO GRENET Proof. Ths proof s also based o the ABP obtaed Lemma. Sce the ABP s layered, we ca cosder the bpartte subgraphs duced by two cosecutve layers. For 1, let M be the badjacecy matrx of the bpartte subgraph duced by the layers V 1 ad V. The layer V cotas the subsets of {1,..., } of sze, therefore s of sze ( ). Ths mples that M has ( 1 ) rows ad ( ) colums. A vertex v V s dexed by a subset of sze, ad t receves arrows from the subsets obtaed by removg oe elemet v. Ths meas that each colum of M cotas exactly ozero etres. The value of the ABP s gve by the product M 1 M 2 M. We ca parethesze ths product ( ((M 1 M 2 )M 3 ) M ) to compute t. The for 2, M 1 M s a (1 ( )) matrx. Usg the ave matrx vector multplcato algorthm, the computato of (M 1 M 1 )M requres (2 1) arthmetc operatos per colum sce each colum of M has ozero etres, that s (2 1)( ) to compute every etres. Thus the total umber of arthmetc operatos requred to compute the complete product s ( ) (2 1) = 2 ( ) ( ) = (2 2) (2 1) = ( 1)(2 1). Each multplcato has oe of ts argumet beg a etry of a matrx M, therefore the crcut s skew. Moreover, the crcut s mootoe sce t does ot use ay costat. Refereces [1] Agrawal, M. Determat versus permaet. I Proc. 25th Iteratoal Cogress of Mathematcas (2006), vol. 3, pp [2] Ca, J., Che, X., ad L, D. A quadratc lower boud for the permaet ad determat problem over ay characterstc = 2. I Proc. STOC (2008), ACM, pp [3] Gly, D. The permaet of a square matrx. Europea J. Comb. 31, 7 (2010), [4] Greet, B., Kaltofe, E. L., Kora, P., ad Porter, N. Symmetrc Determatal Represetato of Formulas ad Weakly Skew Crcuts. I Radomzato, Relaxato, ad Complexty Polyomal Equato Solvg, o. 556 Cotemp. Math. Amer. Math. Soc., 2011, pp [5] Malod, G., ad Porter, N. Characterzg valat s algebrac complexty classes. J. Complexty 24, 1 (2008), [6] Mgo, T., ad Ressayre, N. A quadratc boud for the determat ad permaet problem. It. Math. Res. Notces 2004, 79 (2004), [7] Nsa, N. Lower bouds for o-commutatve computato. I Proc. STOC (1991), ACM, pp [8] Pólya, G. Augfabe 424. Arch. Math. Phys. 20 (1913), 271. [9] Ryser, H. Combatoral Mathematcs, vol. 14 of Carus Math. Moogr. Mathematcal Assocato of Amerca, Washgto, [10] Szegő, G. Zu aufgabe 424. Arch. Math. Phys. 21 (1913),

5 AN UPPER BOUND FOR THE PERMANENT VERSUS DETERMINANT PROBLEM 5 [11] Toda, S. Classes of arthmetc crcuts capturg the complexty of computg the determat. IEICE T. If. Syst. 75, 1 (1992), [12] Valat, L. Completeess classes algebra. I Proc. STOC (1979), ACM, pp LIP, UMR 5668, ÉNS de Lyo CNRS INRIA UCBL, Uversté de Lyo E-mal address: Bruo.Greet@es-lyo.fr