Refutation of Aslams Proof that NP = P

Refutation of Aslam's Pro of that N P = P F rank F erraro Garrett Hall Andrew W o o d {erraro, ghall3, a w o o d8}u.ro  hester.edu Departmen t of Computer Siene Univ ersit y of Ro  hester Ro  hester, NY 14627 Otob er 22, 2018 Abstrat Aslam presen ts an algorithm he laims will oun t the n um b er of p er- fet mat hings in an y inomplete bipartite graph with an algorithm in the funtion-omputing v ersion of N C , whi h is itself a subset of FP . Coun ting p erfet mat hings is kno wn to b e # P -omplete; therefore if Aslam's algorithm is orret, then N P = P . Ho w ev er, w e sho w that Aslam's algorithm do es not orretly oun t the n um b er of p er- fet mat hings and oer an inomplete bipartite graph as a onrete oun ter-example. 1 In tro dution W e pro vide preliminary denitions from Aslam's pap er, whi h are neessary to understanding Aslam's algorithm and what it purp ortedly pro v es. After presen ting these denitions, laims and examples, w e lo ok at an o v erview of the algorithm and wh y it purp orts NP = P . In Setion 2, w e refute some ma jor argumen ts of his pap er whi h, under his urren t onstrution, in v alidate his urren t laim that NP = P . Then in Setion 3, w e pro vide a sound oun ter-example whi h demonstrates his algorithm do es not orretly 1 en umerate all p erfet mat hings in an y bipartite graph. In order to a v oid onfusion, our denitions, theorems, and lemmas are lab eled on tiguously , without regard to the setion in whi h they app ear. Unless noted otherwise, it an b e assumed that all other theorems and lemmas are from Aslam's pap er [1℄. W e w ould also lik e to note that w e are ritiquing v ersion 9 of Aslam's pap er. Although at time of writing this ritique, Aslam has released t w o additional v ersions, 10 and 11, b oth of those v ersions are simply summaries of v ersion 9. As a result, they rely hea vily on the laims made in v ersion 9 and th us it is suien t to analyze v ersion 9. Whenev er w e ite Aslam as [ 1℄, w e refer to v ersion 9 of his pap er. Aslam represen ts p erfet mat hings in bipartite graphs as p erm utations. These p erm utations are elemen ts of the symmetri group S n , the group of all p erm utations of n elemen ts. A review of general group theory an b e found in an y in tro dutory abstrat algebra b o ok su h as [2 ℄. P erfet Mat hings as P erm utations A p erfet mat hing in a bipartite graph B G n with v erties V ∪ W is a set of edges represen ted as n [ i =1 ij, where ea h ij represen ts the edge v i w j in B G n with ea h w j ∈ W and v i ∈ V o uring exatly one and | V | = | W | = n . W e an also represen t a p erfet mat hing as a p erm utation π = ( a 1 , a 2 , . . . , a n ) , 1 ≤ a r ≤ n for all r; in other w ords, ev ery elemen t in the p erm utation yle is an b e represen ted as n um b er b et w een 1 and n , inlusiv e. Letting a π r = a r +1 and a π n = a 1 , the p erfet mat hing orresp onding to π is E ( π ) = n [ i =1 ii π . F or instane, the p erm utation (2 , 3 , 1) orresp onds to the p erfet mat hing { 12 , 23 , 3 1 } , and (2 , 3 , 1 , 5 , 4) orresp onds to { 15 , 23 , 3 1 , 42 , 54 } . F uthermore 2 w e an deomp ose ea h p erm utation π in to a pro dut of transp ositions π = ψ n ψ n − 1 . . . ψ 1 where ψ i is the transp osition ( i, k ) and i ≤ k ≤ n . F rom the previous example, (2 , 3 , 1 , 5 , 4) = (5 , 5 )(4 , 5)(3 , 5)(2 , 3)(1 , 5) . Note that ev ery unique ψ n ψ n − 1 . . . ψ 1 represen ts a unique p erm utation, i.e., a unique p erfet mat hing for all n ! p erfet mat hings in a omplete bipartite graph. P erfet Mat hings from the Generating Graph Γ( n ) A generating graph Γ( n ) is a D A G (direted ayli graph) dened b y Aslam to ha v e O ( n ) v erties alled no des . Ea h no de on tains a unique pair of edges ( ik , j i ) su h that 1 ≤ i < k, j ≤ n or 1 ≤ i = j = k ≤ n . The graph Γ( n ) is designed so that tra v ersal of sp eial paths alled omplete v alid m ultipliation paths, CVMP s [1 , Denition 4.33℄, will en umerate ev ery n ! p erfet mat hing in a bipartite graph with 2 n v erties. There are t w o w a ys to nd the orresp onding p erfet mat hing giv en a CVMP . If p = a 1 a 2 ...a n is a CVMP in Γ( n ) , then one w a y to nd the p erfet mat hing is from the p erm utation π giv en b y π = ψ ( a n ) ψ ( a n − 1 ) . . . ψ ( a 1 ) . where ψ ( a i ) = ψ ( ik , j i ) = ( i, k ) . The seond w a y to nd a p erfet mat hing is through the union of all edge pairs in ea h a i without the surplus edges ( SE ), whi h will b e j k from ea h edge a i a j in p , whi h written formally is E ( p ) = E P ( p ) − ( S E ( p ) ∩ E P ( p )) = [ a i ∈ p E P ( a i ) ! −     [ a i ,a j ∈ p S E ( a i a j )   \ [ a i ∈ p E P ( a i ) !   where E P ( ik, j i ) = { ik , j i } and SE (( ik , j i )( j k , q j )) = { j k } , i < k , j < q or i < k = j = q . As an example, onsider the follo wing CVMP from the graph Γ(9) in Figure 1: 3                                                                                        Figure 1: A subgraph of Γ(9) on taining a CVMP along the dark single- arro w ed edges. Dotted edges are refered to as S-e dges , solid edges are R- e dges , and the edges b et w een nonadjaen t no des with double-arro ws are jump e dges . Ea h no de is lab eled with a pair of edges ( ik , j i ) . Surplus edges along the CVMP are indiated with a + j k ab o v e them. A CVMP an only b e along S -edges or R -edges whi h are not jump edges, so the CVMP pitured ab o v e is p = c 1 a 2 c 3 c 4 d 5 d 6 d 7 c 8 c 9 . F or ea h no de w e an easily nd ψ i , e.g. ψ ( c 1 ) = ψ (19 , 31) = (19) , giving the p erm utation π = (99)( 89)(78)( 69)(57)(49)(39)(24)(19) = (1 , 9 , 5 , 7 , 8 , 6 , 2 , 4 , 3) . W e an also easily ompute E ( p ) from the set of all edges in all the no des min us all of the surplus edges as E ( p ) = E P ( c 1 ) ∪ E P ( a 2 ) ∪ ... ∪ E P ( c 9 ) − S E ( c 1 ) ∪ S E ( a 2 ) ∪ ... ∪ S E ( c 9 ) = { 19 , 3 1 , 24 , 62 , ..., 98 , 99 } − { 39 , 64 , ..., 99 } = { 19 , 2 4 , 31 , 43 , 57 , 62 , 78 , 86 , 9 5 } . Clearly , E ( π ) = E ( p ) so b oth metho ds ha v e arriv ed at the same p erfet mat hing. CVMP s in Inomplete Graphs Ea h of the n ! CVMP s in Γ( n ) orresp onds to a unique p erfet mat hing in a omplete bipartite graph B G n . Of ourse, giv en an inomplete bipartite graph B G ′ n only a subset of all the CVMP s will on tain edges not in B G ′ n whi h annot b e oun ted as a p erfet mat hings. These edges not in B G ′ n 4 are refered to as the edge requiremen ts ( ER ) of a CVMP and are formally dened for a CVMP p as E R ( p ) = [ a i ∈ p E R ( a i ) ! −     [ a i ,a j ∈ p S E ( a i a j )   \ [ a i ∈ p E R ( a i ) !   , where ER ( a i ) = { e | e ∈ E P ( a i ) , e / ∈ B G ′ n } . Clearly , the ER of a CVMP will b e the empt y set if and only if the CVMP orresp onds to a v alid p erfet mat hing in B G ′ n . In other w ords, ER ( p ) = ∅ ⇐ ⇒ E ( p ) ⊆ B G ′ n . P erform- ing a valid enumer ation of CVMP s in Γ( n ) giv en a orresp onding B G ′ n , i.e. en umerating ev ery CVMP with ER ( p ) = ∅ , is equiv alen t to oun ting ev ery v alid p erfet mat hing. Ob viously , naiv e v alid en umeration of the n ! p ossible CVMP s annot b e done in p olynomial time. Ho w ev er, if Aslam's algorithm is orret, the sp eial prop erties of Γ( n ) allo w it to b e redued to sets of subpaths whi h an b e joined while preserving the ER of all CVMP s. Algorithm Ov erview A generating graph Γ( n ) on tains O ( n 3 ) no des [1, Prop ert y 4.21℄ and an b e reated in p olynomial time. VMPSet ( a i , a j ) is a data struture repre- sen ting a subset of VMP s b et w een no des a i and a j that ha v e the same ER . Within the algorithm, all of the VMPSet ( a i , a i +1 ) are initializated rst and all p ossible VMPSet ( a i , a j ) are stored in a matrix. During ea h iteration, the algorithm p erforms t w o redution op erations: adding and m ultiplying VMPSet s. If one m ultiplies t w o VMPSet s, VMPSet ( a, b ) and VMPSet ( b, c ) , eetiv ely doubles the length of the paths they represen t to get VMPSet ( a, c ) = VMP Set ( a, b ) × VMPSet ( b, c ) , and inreases the n um b er of VMP s in the new VMPSet ( a , c ) to | VMPSet ( a, c ) | = | VMPSet ( a, b ) | × | VMPSet ( b, c ) | . When t w o sets VMPSet ( a, b ) and VMPSet ′ ( a, b ) an b e om bined so as to satisfy onditions for m ultipliation they are added together, and the n um b er of VMP in the new VMPSet ′′ ( a, b ) = VMPSet ( a, b ) ∪ VMPSet ′ ( a, b ) is 5 | VMPSet ′′ ( a, b ) | = | VMPSet ( a, b ) | + | V MPSet ′ ( a, b ) | . Sine ev ery iteration doubles the length of VMP s represen ted in the VMPSet s, after O (lo g ( n )) iterations the en tire generating graph Γ( n ) is redued to sev eral disjoin t sets of VMPSet (1 , n ) represen ting all CVMP s. Summation o v er all | VMPSet (1 , n ) | that ha v e ER = ∅ then giv es the total n um b er CVMP s and lik ewise p erfet mat hings. The summation step is what neessitates k eeping ER the same for all VMPSet s. Sine the problem of oun ting p erfet mat hings in an y bipartite graph is # P -omplete and the lass of parallel algorithms running in (log n ) O (1 ) on a p olynomial n um b er of pro essors, whi h is analogous to and ommonly referred to as NC , is a subset of FP , the algorithm Aslam purp orts to ha v e will b e able to solv e an y # P problem in p olynomial time, meaning # P = F P and N P = P . 2 Refutation The main a w in Aslam's reasoning is that the ER of ev ery CVMP an b e preserv ed during deomp osition and subsequen t redution op erations o v er VMPSet s. Ho w ev er, the ER of a VMPSet do es not apture the SE of the VMP s it on tains, and ultimately SE will determine ER . W e will sho w that m ultipliation and addition of VMPSet s do es not alw a ys giv e a VMPSet with the same ER for all VMP s. As pro of this problem is inheren t in all suien tly large generating graphs w e giv e a oun ter-example in Setion 3. Lemma 1. The pro dut VMPSet A C found from m ultiplying t w o VMPSet s A = VMPSet ( a, b ) and C = VM PSet ( b, c ) m ust b e a single set and on tain only VMP s with the same ER . Pr o of. The ondition on ER follo ws from the denition of VMPSet and the indutiv e result of the algorithm itself. Sine after the nal iteration all VMPSet s with ER = ∅ will b e oun ted, if m ultipliation resulted in a VMPSet on taining some CVMP s with ER = ∅ (v alid p erfet mat hings) and some with ER 6 = ∅ (in v alid), then summation w ould result in an inorret n um b er of p erfet mat hings. 6 By Aslam's denition of m ultipliation, AC m ust b e a single VMPSet . W e note, ho w ev er, ev en if w e allo w ed m ultipliation to pro due more than one VMPSet , the n um b er of sets pro dued w ould ha v e to b e onstan t with re- sp et to | A | × | C | , sine m ultipliation is what allo ws the fast en umeration in O (lo g ( n )) iterations. Theorem 2. The  onditions for multiplying VMPSet s A × C = AC given in L emma 5.8 and L emma 5.9 of Aslam's pr o of ar e insuient  onditions for multipli ation. L emma 5.9 is not a ne  essary  ondition for multipli ation. Pr o of. Lemma 5.8 states that ∀ p ∈ A and ∀ q ∈ C , p m ust m ultiply (form a VMP ) with q . Ob viously , this is a neessary ondition sine w e are onsid- ering all pq to b e VMP s in AC . Lemma 5.9 states that ev ery no de o v ered b y a VMP in C from the same partition m ust ha v e the same ER . Note that partition n um b er is equal to the depth in Γ( n ) , so for ev ery x i , x ′ i ∈ q w e m ust ha v e ER ( x i ) = ER ( x ′ i ) . The pro of of this lemma is onspiuously omitted and w e found it to b e inorret. Assume the only no des with un- equal ER in C are x i ∈ q and x ′ i ∈ q ′ , all the VMP s in A ha v e the same SE , and let ER ( x i ) = e and ER ( x ′ i ) = ∅ . If e / ∈ SE ( A ) the resultan t AC will not ha v e ev ery VMP with the same ER violating the ondition laid out in Lemma 1. Ho w ev er, if e ∈ S E ( A ) then ER ( p q ′ ) = ER ( pq ) = ∅ and AC is a VMPSet with v alid VMP s all with ER = ∅ , so Lemma 5.9 is not a neessary ondition for m ultipliation. Note w e use the follo wing denitions: e ∈ S E ( A ) ⇐ ⇒ ( ∀ p ∈ A ) [ e ∈ S E ( p )] . e ∈ E R ( A ) ⇐ ⇒ ( ∀ p ∈ A ) [ e ∈ E R ( p )] . No w w e will sho w these lemmas are insuien t onditions for m ultipliation. Let p and p ′ b e VMP s in A su h that e ∈ SE ( p ) and e / ∈ SE ( p ′ ) . Let the no de x i o v ered b y all VMP s in C ha v e ER = e and all other no des ha v e ER = ∅ . Note that C satises Lemma 5.9 sine ev ery VMP in C o v ers x i there is no no de x ′ i in C and ev ery other no de has the same ER = ∅ . Pro of that A and C an satisfy Lemma 5.8 (ev ery path through a and c is a VMP ) relies on prop erties of the generating graph Γ( n ) itself and so w e will demonstrate that in our oun ter-example in Setion 3. F or no w w e assume Lemma 5.8 is satised. After m ultipliation of A and C , w e ha v e ER ( p q ) = ∅ and ER ( p ′ q ) = e for ev ery q ∈ C . The resultan t VMPSet AC do es not on tain VMP s with the same E R , violating the onditions for m ultipliation set forth in Lemma 1. 7 W e see this is b eause the VMPSet represen tation do es not apture ases where SE ( p ) 6 = SE ( p ′ ) . F urther Disussion Although Aslam do es not giv e the suien t onditions for p erforming m ulti- pliation to redue all VMPSet s in O (lo g ( n )) iterations, if his pro of is v alid un til that p oin t, then pro ving whether su h onditions exist or do not exist ma y b e equiv alen t to pro ving whether P = N P . In the rest of the setion w e explore wh y p erforming m ultipliation while preserving ER o v er VMPSet s is diult. In the follo wing lemma w e sho w wh y the v arious SE s of all the paths in a VMPSet are signian t. Lemma 3. There are at least ( n − 1)! CVMP s where SE ( p ) ( E P ( p ) for ev ery CVMP p . In these CVMP s, if a i ∈ p and all edges SE ( a i ) are not presen t in the bipartite graph B G ′ n , then  hanging an y one no de in p results in ER ( p ) 6 = ∅ . Pr o of. Ea h CVMP p in Γ( n ) is n no des long. F or simpliit y , w e only on- sider the ( n − 1) ! ases where p is omp osed of non-iden tit y no des (exept the last no de). Reall E ( p ) = E P ( p ) − ( S E ( p ) ∩ E P ( p )) = [ a i ∈ p E P ( a i ) ! −     [ a i ,a j ∈ p S E ( a i a j )   \ [ a i ∈ p E P ( a i ) !   . Ev ery no de a i = ( ik, j i ) , 1 ≤ i < j, k < n on tributes t w o unique edges ik , j i to E P ( p ) and the last no de on tributes one so that | E P ( p ) | = 2 n − 1 . Ev ery no de exept the last on tributes an edge to SE ( p ) so | S E ( p ) | = n − 1 . Sine | E ( p ) | = n , ev ery edge in SE ( p ) m ust b e unique in p , eliminate one edge from E P ( p ) , and SE ( p ) ( EP ( p ) . Note that ea h a i = ( ik, j i ) ∈ Γ( n ) is unique, so for all no des SE ( a i ) 6 = S E ( b i ) . Therefore if the edge SE ( a i ) is not presen t in the bipartite graph B G ′ n , then  hanging the no de a i ∈ p to an y b i ∈ Γ( n ) results in an ER ( p ) 6 = ∅ , where 8 E R ( p ) = [ a i ∈ p E R ( a i ) ! −     [ a i ,a j ∈ p S E ( a i a j )   \ [ a i ∈ p E R ( a i ) !   . It follo ws there are at least ( n − 1)! CV MP s with orresp onding B G ′ n s in whi h satisfying ER = ∅ ma y b e dep enden t on the SE of ev ery no de in p . If w e are onerned with satisfying the onditions on m ultipliation outlined in Lemma 1, w e an k eep all VMP s with diering SE in separate VMPSet s, but this leads to a large n um b er of VMPSet s. Lemma 4. The n um b er of VMPSet s with the same SE from partition 1 to i is at least  n i  . Pr o of. By denition, the generating graph Γ( n ) on tains n ! / ( n − i )! VMP s from partitions 1 to i . Note this is neessary for Γ( n ) to b e able to en umerate n ! p erfet mat hings. Sine all these VMP s are unique, for an y t w o VMP s p 1 and p 2 , ea h will on tain at least one dieren t no de x i ∈ p 1 and y i ∈ p 2 , with SE ( a i ) 6 = SE ( b i ) . Consequen tly , no t w o VMP s ha v e surplus edges o uring in exatly the same order, so the maxim um size of an y VMPSet with the same set of SE edges from 1 to i will b e i ! whi h is the n um b er of p erm utations of SE edges from i no des. T o get a lo w er b ound on the n um b er of VMPSet s with the same SE w e divide the n um b er of VMP s b y the upp er b ound on set size: n ! ( n − i )! i ! =  n i  . 3 Coun ter-example W e presen t γ , a subgraph of ev ery Γ( n ) with n ≥ 9 , whi h en umerates v e p erfet mat hings. W e then oer an inomplete bipartite graph B G ′ n and sho w that Aslam's algorithm will inorretly oun t some n um b er of p erfet mat hings in B G ′ n using γ . This serv es as an example that the graph 9                                                                                                                                                                                                   Figure 2: The ab o v e gure represen ts graph γ . Double-arro w ed solid-lines are jump edges, single-arro w ed solid-lines are R-edges, and dotted-lines are S-edges. desrib ed in Theorem 2 an b e realized in Γ( n ) , satisfying Lemma 5.8 of Aslam's pap er. Graphial Represen tation of the Example In Figure 2 w e pro vide a graphial represen tation of the oun ter-example. Lemma 5. γ , in Figure 2, represen ts a subgraph of ev ery Γ( n ) , where n ≥ 9 . Pr o of. The generating graph Γ( n ) is dened o v er all 1 ≤ i < k ≤ n , where v erties are a i ∈ g ( i ) , R -edges are a i a j su h that | a i Ra j | = 1 , jump edges are R -edges su h that j 6 = i + 1 , and S -edges are a i a i +1 su h that a i S a i +1 [1 , Denition 4.12℄. F rom this denition w e should note that the generating graph Γ( n ) is a subgraph of Γ( n + 1) . No w w e will sho w that γ from is a v alid subgraph of Γ(9) . It an b e easily v eried ev ery no de a i = ( ik 1 , k 2 i ) in γ has either k 1 , k 2 > i or i = k 1 = k 2 , 10 so a i ∈ g ( i ) is true for all no des. An y relation R in γ b et w een no des a i and a j = ( j t 1 , t 2 j ) app ears i t 1 = k 2 , t 2 = k 1 , and i < j , whi h satises | a i Ra j | = 1 . The relation app ears as an R -edge if j = i + 1 , otherwise it app ears as a jump-edge. An y S -edge app ears if and only if j = i + 1 , and either k 1 , t 2 < k 2 , t 1 , k 2 , t 1 < k 1 , t 2 , or k 1 = k 2 < t 1 , t 2 . The disjoin tness of a i S a j giv en these onditions is made lear from Remark 4.5 [1℄ sine no R -yle C a i a j an b e onstruted whi h has a stritly inreasing or dereasing tra v ersal. Sine for all no des in γ w e ha v e a i ∈ g ( i ) , and all edges app ear if and only if they satisfy their resp etiv e denitions (at least for the no des app earing in γ ), γ is a subgraph of Γ(9) and ev ery Γ( n ) , where n ≥ 9 . CVMPS in γ . Let P ( a, b ) denote the path in γ starting at c 1 going through a, b and ending at c 9 . Then γ only on tains v e CVMP s: p aa = P ( a 2 , a 5 ) , p ab = P ( a 2 , b 5 ) , p ba = P ( b 2 , a 5 ) , p bb = P ( b 2 , b 5 ) , and p ad = P ( a 2 , d 5 ) whi h orresp ond to v e unique p erfet mat hings. Pr o of. By denition the p erm utation π ( p ) that realizes the p erfet mat h- ing orresp onding to CVMP p is giv en b y ψ n . . . ψ 1 , where for ev ery no de x i = ( ik , j i ) ∈ p , ψ i = ( ik ) . The set of edges in the p erfet mat hing orre- sp onding to CVMP p is giv en b y E ( p ) = ( S { e | e ∈ x i } ) − ( S S E ( x i )) , where SE ( x i ) giv es the surplus edges in p . It is triv al to v erify π ( p ) and E ( p ) are onsisten t: π ( p aa ) = ( 99)(89)( 79)(69)(57)(49)(39)(24)(19) E ( p aa ) = { 19 , 24 , 31 , 43 , 57 , 62 , 76 , 85 , 98 } π ( p ab ) = (99 )(89)(79) (69)(58)(49)(39)(24)(19) E ( p ab ) = { 19 , 24 , 31 , 43 , 58 , 62 , 76 , 87 , 95 } π ( p ba ) = (99 )(89)(79) (69)(57)(49)(39)(26)(19) E ( p ba ) = { 19 , 26 , 31 , 43 , 57 , 64 , 72 , 85 , 98 } π ( p bb ) = ( 99)(89)( 79)(69)(58)(49)(39)(26)(19) E ( p bb ) = { 19 , 26 , 31 , 43 , 58 , 64 , 72 , 87 , 95 } 11 π ( p ad ) = (99 )(89)(78) (69)(57)(49)(39)(24)(19) E ( p ad ) = { 19 , 24 , 31 , 43 , 57 , 62 , 78 , 86 , 95 } . Note that P ( b 2 , d 5 ) is not a CVMP b eause it do es not on tain the mdag M D G ( b 2 , c 3 , c 6 ) . Cho osing B G ′ The initialization of PTM , whi h Aslam denes as the matrix on taining all VMPSet s, is v ague regarding ho w it inorp orates infor- mation from the adjaeny matrix BGX of the bipartite graph, onsidering there is no lear relation b et w een remo v al of S -edges or R -edges, and edges from the bipartite graph. T o allo w this am biguit y , w e ha v e  hosen a B G ′ so that no edge migh t b e remo v ed from γ without losing a p erfet mat hing. F ormally , B G ′ =  [ E ( p )  − { 76 } In other w ords, all CVMP s in γ are v alid mat hings with ER ( p ) = ∅ exept for those on taining the edge { 76 } whi h will ha v e ER ( p ) = { 7 6 } . Clearly only p ba , p bb , and p ad are CVMP s whi h represen t v alid p erfet mat hings so the result of en umerating with γ should b e 3. In addition, these CVMP s o v er ev ery edge of γ , so no edge ma y b e remo v ed without losing at least one CVMP . Iterations of A dd and Join on γ During the rst iteration of the algorithm, the VMPSet s VMPSet A ( c 1 , c 3 ) , VMPSet B ( c 1 , c 3 ) , VMPSet A ( c 4 , c 6 ) , VMPSet B ( c 4 , c 6 ) , and VMPSet s o v er all other pairs of no des are reated. In the next iteration t w o addi- tions b et w een the orresp onding A and B sets o ur: VMPSet A ( c 1 , c 3 ) and VMPSet B ( c 1 , c 3 ) are added together, and VMPSet A ( c 4 , c 6 ) and VMPSet B ( c 4 , c 6 ) are added. So w e ha v e VMPSet ( c 4 , c 6 ) = VM PSet A ( c 4 , c 6 ) + VMPSet B ( c 4 , c 6 ) VMPSet ( c 1 , c 3 ) = VM PSet A ( c 1 , c 3 ) + VMPSet B ( c 1 , c 3 ) . Cruially , | VMPSet ( c 4 , c 6 ) | = | V MPSet ( c 1 , c 3 ) | = 2 and the set of surplus edges SE of VMPSet ( c 4 , c 6 ) equals the ER of VMPSet ( c 1 , c 3 ) whi h is { 76 } . 12 One these t w o sets are m ultiplied to get 4 and the additional VMPSet represen ting p ad is oun ted, the n um b er of p erfet mat hings the algorithm will return will b e 2 more than the atual n um b er of p erfet mat hings b eause the SE of the VMP s in VMPSet ( c 4 , c 6 ) w as om bined. Th us, as the ab o v e example demonstrates, Aslam's algorithm do es not or- retly en umerate all p erfet mat hings for all ases. Therefore, his urren t pro of that # P ⊆ F P (and hene that P = N P and the p olynomial-time hierar h y ollapses) do es not v alidly establish that laim. 4 A  kno wledgemen ts This w ork w as ompleted in partial fulllmen t of the requiremen ts for an Honors Ba helor of Siene Degree in Computer Siene from the Depart- men t of Computer Siene at the Univ ersit y of Ro  hester, in Ro  hester, NY, USA. This pap er w as also written as the Honors Pro jet for the ourse CSC200H during the Spring 2009 semester. W e w ould lik e to thank Profes- sor Lane A. Hemaspaandra, the ourse T.A. A dam Sadilek and others in the omm unit y for their feedba k, supp ort and suggestions. Referenes [1℄ Aslam, Ja v aid. The Collapse of the P olynomial Hierar h y: NP = P . 9 Mar. 2009. < h df/0812.1385v9 > . [2℄ F raleigh, John B. A First Course in A bstr at A lgebr a . 7th ed. New Y ork: A ddison W esley . 2003. 13