Complexity of the Cover Polynomial

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1 Complexity of the Coer Polynomial Marks Bläser and Holger Dell Comptational Complexity Grop Saarland Uniersity, Germany Abstract. The coer polynomial introdced by Chng and Graham is a twoariate graph polynomial for directed graphs. It conts the (weighted) nmber of ways to coer a graph with disjoint directed cycles and paths, it is an interpolation between determinant and permanent, and it is belieed to be a directed analoge of the Ttte polynomial. Jaeger, Vertigan, and Welsh showed that the Ttte polynomial is #P-hard to ealate at all bt a few special points and cres. It trns ot that the same holds for the coer polynomial: We proe that, in almost the whole plane, the problem of ealating the coer polynomial is #P-hard nder polynomial-time Tring redctions, while only three points are easy. Or constrction ses a gadget which is easier to analyze and more general than the XOR-gadget sed by Valiant in his proof that the permanent is #P-complete. 1 Introdction Graph polynomials map directed or ndirected graphs to polynomials in one or more ariables, sch that this mapping is inariant nder graph isomorphisms. Probably the most famos graph polynomials are the chromatic polynomial or its generalization, the Ttte polynomial. The chromatic polynomial is the polynomial in the ariable λ that conts the nmber of alid λ-colorings of a gien ndirected graph. The Ttte polynomial T in two ariables x and y has interpretations from different fields of combinatorics. For example, T(G, 1, 1) is the nmber of spanning trees, T(G, 1, 2) is the nmber of spanning sbgraphs of an ndirected graph G, and also the nmber of nowhere-zero flows or the Jones polynomial of an alternating link are contained in the Ttte polynomial. While the Ttte polynomial has been established for ndirected graphs, the coer polynomial by Chng and Graham [1] is an analoge for the directed case. Both graph polynomials satisfy similar identities sch as a contractiondeletion identity and prodct rle, bt the exact relation between the Ttte and the coer polynomial is not yet known. The coer polynomial has connections to rook polynomials and drop polynomials, bt from a complexity theoretic point of iew, we tend to see it as a generalization of the permanent and the determinant of a graph. The coer polynomial of a graph is the weighted sm of all of its spanning sbgraphs that consist of disjoint, directed, and simple cycles and paths. As it is the case for most graph polynomials, the coer polynomial is of interest becase it combines a ariety of combinatorial problems into one generalized theoretical framework.

2 2 Marks Bläser and Holger Dell Preios Reslts Jaeger, Vertigan, and Welsh [2] hae shown that, except along one hyperbola and at nine special points, compting the Ttte polynomial is #P-hard. In recent years, the complexity and approximability of the Ttte polynomial has receied increasing attention: Lotz and Makowsky [3] proe that the colored Ttte polynomial by Bollobás and Riordan [4] is complete for Valiant s algebraic complexity class VNP, Giménez and Noy [5] show that ealating the Ttte polynomial is #P-hard een for the rather restricted class of bicirclar matroids, and Goldberg and Jerrm [6] show that the Ttte polynomial is mostly inapproximable. A different graph inariant that seems related to the Ttte polynomial is the weighted sm of graph homomorphisms to a fixed graph H, so basically, it is the nmber of H-colorings. Blato and Grohe [7] and Dyer, Goldberg, and Paterson [8] proe that the complexity of compting this sm is #P-hard for most graphs H. Or Contribtion In this paper, we show that the problem of ealating the coer polynomial is #P-hard at all ealation points except for three points where this is easy. The big pictre of the proof is as follows: We se elementary identities of the coer polynomial in order to constrct simple redctions along horizontal lines. Frthermore, we establish an interesting identity along the y-axis. As it trns ot, there is qite a strong connection between coer polynomial and permanent, and or constrction ses an eqality gadget with a similar effect as the XOR-gadget which Valiant [9] ses in his proof that the permanent is #P-complete. Or gadget, howeer, is simpler to analyze and more general, in the sense that it satisfies additional properties abot the nmber of cycles contribted by or gadget to each cycle coer. In addition, we carry oer the hardness reslt to the geometric coer polynomial introdced by D Antona and Mnarini [10]. 2 Preliminaries Let IN = {0,1,... }. The graphs in this paper are directed mltigraphs D = (V,E) with parallel edges and loops allowed. We denote by G the set of all sch graphs. We write n for the nmber of ertices, and m for the nmber of edges. Two graphs are called isomorphic if there is a bijectie mapping on the ertices that transforms one graph into the other. A graph inariant is a fnction f : G F, mapping elements from G to some set F, sch that all pairs of isomorphic graphs G and G hae the same image nder f. In the case that F is a polynomial ring, f is called graph polynomial. Conting Complexity Basics. The class #P consists of all fnctions f : {0,1} IN for which there is a non-deterministic polynomial-time bonded Tring machine M which has exactly f(x) accepting paths on inpt x. For

3 Complexity of the Coer Polynomial 3 two conting problems f,g : {0,1} Q not necessarily in #P, we say f Tring-redces to g in polynomial time (f p T g), if there is a deterministic oracle Tring machine M which comptes f in polynomial time with oracle access to g. If the oracle is sed only once, we say f many-one redces to g (f p m g), and if the oracle otpt is the otpt of the redction, we speak of a parsimonios many-one redction (f p g). The notions of #P-hardness and #P-completeness (nder polynomial-time Tring redctions) are defined in the sal way. Polynomials. Polynomials p(x 1,...,x m ) are elements of the polynomial ring Q[x 1,...,x m ], and, in this context, the ariables are abstract objects. Lagrange interpolation can be sed to compte the following problem in polynomial time. Inpt Point-ale pairs (a 1,p 1 ),...,(a d,p d ) Q 2, encoded in binary. Otpt The coefficients of the polynomial p(x) with deg(p) d and p(a j ) = p j. The Coer Polynomial. The coer polynomial basically conts a relaxed form of cycle coers, namely path-cycle coers. For a directed graph D = (V,E) and some sbset C E, we denote the sbgraph (V,C) again by C. A pathcycle coer of D is a set C E, sch that in C eery ertex V has an indegree and an otdegree of at most 1. A path-cycle coer ths consists of disjoint simple paths and simple cycles. Note that also an independent ertex conts as a path, and an independent loop conts as a cycle. By the graph inariant c D (i,j), we denote the nmber of path-cycle coers of D that hae exactly i paths and j cycles. It is not hard to proe that the fnction D c D (i,j) is #P-complete (cf. [11]). The coer polynomial by Chng and Graham [1] is a graph polynomial in the ariables x and y, and it is defined by the eqation m m C(D,x,y) := c D (i,j)x i y j, (1) i=0 j=0 where x i := x(x 1)... (x i + 1) denotes the falling factorial. Writing i(c) and j(c) for the nmber of paths and cycles of a path-cycle coer C, the coer polynomial is the weighted sm oer all coers of D: 3 Oeriew C(D,x,y) = path-cycle coer C x i(c) y j(c). The problem of ealating the coer polynomial, written C(a, b), is parameterized by the coordinates (a,b). Formally, C(a,b) is the fnction from G Q with D C(D, a, b) where we assme graphs and rationals to be represented explicitly in a standard way. Or main theorem is the following.

4 4 Marks Bläser and Holger Dell Main Theorem. Let (a,b) Q 2. It holds: If (a,b) {(0,0),(0, 1),(1, 1)}, then C(a,b) is #P-hard. Otherwise, C(a, b) is comptable in polynomial time. Proof (otline). The proof is in seeral steps (cf. Fig. 1). We begin by classifying the polynomial-time comptable points in Section 4. Frthermore, we point ot that C(0,1) is the permanent and C(1,0) is the nmber of Hamiltonian paths, which both are #P-complete conting problems. In Section 5, sing elementary identities of the coer polynomial and interpolation, we redce along horizontal lines, that means we proe C(0,b) p T C(a,b) for all a,b and C(1,0) p T C(a,0) for all a 0. This implies the hardness of C(a,1) for all a and of C(a,0) for a 0. To proe the remaining hardness part where b { 1,0,1}, we redce the permanent to C(0, b). Section 6 is the core part of or proof, establishing this redction C(0,1) p T C(0,b) along the y-axis. There we introdce and analyze the eqality gadget, se it to establish a new identity for the weighted coer polynomial, and show how to derie a redction for the standard coer polynomial from this. In Section 7, we briefly show how to carry oer or reslt to the geometric ersion of the coer polynomial. y 1 p T Permanent p T 1 1 #Hamiltonian Paths p T x e e Determinant e 1 e 2 e 3 e α 1 e α = = = Fig. 1. The big pictre: three points (black discs) are easy to ealate, and the rest of the plane is #P-hard. The crosses indicate the points for the redctions. Fig.2. Top: Shows how the eqality gadget connects two edges e, e. Bottom: Shows how the mltieqality gadget connects α edges with eqality gadgets from the top (large diamonds). 4 Special Points A Hamiltonian path is a path-cycle coer with exactly one path and zero cycles, and a cycle coer is a path-cycle coer withot paths. The permanent Perm(D) is the permanent of the adjacency matrix of D, and it conts the nmber of cycle coers of D. The determinant det(d) is the determinant det(a). Remarkably, both the determinant and the permanent can be fond in the coer polynomial.

5 Lemma 1. For any nonempty directed graph D, we hae Complexity of the Coer Polynomial 5 (i) C(D,0,0) = 0, (ii) C(D, 1, 0) = #HamiltonianPaths(D), (iii) C(D,0,1) = Perm(D), (i) C(D,0, 1) = ( 1) n det(d), () C(D,1, 1) = C(D,0, 1) C(D,0, 1) where D is deried from D by adding a new ertex 0 with all edges to and from the nodes of D, Proof (sketch). (For a detailed proof, cf. [11]) The proof of the first three claims is simple. The proof of the forth claim ses Laplace expansion and the mltilinearity of the determinant to show that the contraction-deletion identity from (1) (3) in [1] is satisfied by the fnction D ( 1) n(d) det(d). Since the contraction-deletion identities are defining eqations of the coer polynomial, the claim follows. For the last claim, notice that C(D, 1, 1) conts all path-cycle coers with at most one path (weighted with ( 1) j(c) ), while the determinant C(D,0, 1) conts only cycle coers. The idea is that C(D,1, 1) C(D,0, 1) is the nmber of coers of D with exactly one path, and can be expressed by C(D,0, 1), the nmber of cycle coers of D. This is becase eery path-cycle coer of D with one path becomes a cycle coer in D where the path gets closed by the 0 -edges to form a cycle. As a conseqence, C(0,1) and C(1,0) are #P-hard [9, 12], while C(0,0), C(0, 1) and C(1, 1) are polynomial-time comptable. The #P-hardness of C(2, 1) follows from the following lemma and reqires a sophisticated proof. e b b a D a Fig.3. Shows the graph D constrcted in the proof of Lemma 2. The two edges e,(, ) together with the corresponding crossing edges form the crossing gadget. Fig.4. Shows the change in the nmber of paths and cycles in a path-cycle coer if we switch one crossing gadget. Lemma 2. It holds #HamiltonianPaths p m C(2, 1). Proof. Let D be a directed graph with ertex set V = {1,...,n}. We constrct a graph D from D as follows (cf. Fig. 3). ncontract all ertices from D, into two ertices, with one edge (, ) in between, and moe all otgoing edges (,w) to (,w),

6 6 Marks Bläser and Holger Dell add fresh ertices a,a to D, and add the edges (a,) and (,a ) for all, add an independent directed path of length n with edges e 1,...,e n, and add the crossing edges (,w),(, ) for all (, ) and e = (,w). Note that ncontracting edges as aboe does not change the strctre of the path-cycle coers. Therefore, we refer to the graph indced by the ertices and again by D. Let b be the start ertex of e 1, and let b be the end ertex of e n. Since a and b hae no incoming edges, a and b are starting points of paths, and similarly a and b are ending points of paths in eery path-cycle coer. For the coer polynomial of D, we hae C(D,2, 1) = 2 ( 1) j(c), where the sm is only oer those path-cycle coers of D that hae exactly two paths and an arbitrary nmber of cycles. In the following, we proe C(D,2, 1) = 2 n+1 #HamiltonianPaths(D). From a gien path-cycle coer C, related path-cycle coers can be constrcted by switching the presence of the edges e,(, ) and their corresponding crossing edges. Let C 0 be the set of path-cycle coers C 0 of D that hae exactly two paths, se no crossing edge, and hae at least one cycle. We define the set of bad cycle coers C b as the closre of C 0 nder switching arbitrary crossing gadgets. Let C C b be arbitrary. By switching, we can niqely trn C into a pathcycle coer C 0 C 0. Let be the smallest ertex of D that occrs in a cycle of C 0. We define the partner of C as the path-cycle coer p(c) C b which is deried from C by switching the crossing gadget e,(, ). As depicted in Fig. 4, the nmbers of cycles in C and in p(c) differ by exactly 1. Since p is a permtation on C b with the property ( 1) j(c) = ( 1) j(p(c)), the weights ( 1) j(c) of the bad cycle coers sm p to 0 in C(D,2, 1). This means that C(D,2, 1) is jst 2 times the nmber of path coers of D with exactly two paths. Any sch 2-path coer C of D translates to an Hamiltonian path of D (by switching all gadgets to C 0, recontracting the edges (, ), and remoing a,a and the b-b -path), and this procedre does not add any cycles. Since there are 2 n possible gadget states, we get C C(D,2, 1) = 2 2 n #HamiltonianPaths(D). Note that in the proof aboe, we basically examined an operation necessary for Gassian elimination: Exchanging two rows or colmns in the adjacency matrix switches the sign of the determinant, bt as soon as we allow more than one path, this is no longer tre for the coer polynomial. 5 Horizontal Redctions Let s consider redctions along the horizontal lines L b := {(a,b) : a Q}. For a directed graph D, let D (r) be the graph obtained by adding r independent ertices. Corollary 4 in [1] is the core part of the horizontal-line redctions: C(D (r),x,y) = x r C(D,x r,y). (2)

7 Complexity of the Coer Polynomial 7 From this eqation, it is not hard to proe the next lemma (also cf. [11]). Lemma 3. For all (a,b) Q 2, we hae C(0,b) p T C(a,b). Proof (sketch). For a IN, it follows directly. For a IN, we can compte the ales C(D,a 1,b),...,C(D,a m,b) and interpolate to get C(D,x,b). In a similar fashion, one can also proe C(1,0) p T C(a,0) for a 0 and C(2, 1) p T C(a,0) for a 0,1. Please note that C(a,b) is now known to be #P-hard for eery point (a,b) on the lines L 1, L 0, and L 1, except for the three easy points (0,0), (0, 1), and (1, 1). 6 Vertical Redction In this section, we redce the permanent along the y-axis: Theorem 1. Let b Q with 1 b 0. It holds C(0,1) p T C(0,b). Proof (otline). For some graph D, we compte C(D,0,1) with oracle access to C(0,b), and we se interpolation to do so. In order to interpolate the polynomial C(D,0,y), we need to compte some ales C(D,0,b 1 ),...,C(D,0,b m ). This can be done by sing the oracle for some ales C(D 1,0,b),...,C(D m,0,b) instead. More specifically, we constrct graphs D α containing α copies of a graph D, sch that there is a polynomial-time comptable relation between C(D,0,b α ) and C(D α,0,b). Compting C(D,0,b α ) for α = 1,...,m and applying interpolation, we get C(D,0,y). Constrction details are spelled ot in the remainder of this section. The constrcted graph D α is a graph in which eery cycle coer ideally has α times the nmber of cycles a corresponding cycle coer of D wold hae. This way, the terms y j in the coer polynomial ideally become y αj, and some easily comptable relation between C(D,0,b α ) and C(D α,0,b) can be established. In the constrction, we therefore dplicate the graph α times, and we connect the dplicates by eqality gadgets. These eqality gadgets make sre that eery cycle coer of D α is a cycle coer of D copied α times, and ths has roghly α times the nmber of cycles. Let s constrct the graph D α explicitly. Start with the inpt graph D. Create α copies D 1,...,D α of D. Let e i be the copy of e in the graph D i. Replace eery tple of edges (e 1,...,e α ) by the mltieqality gadget on α edges, which is drawn in Fig. 2. The D 1 -part of eery cycle coer of D α is isomorphic to a cycle coer of D and has to be imitated by the other sbgraphs D i becase of the eqality gadgets. What we are left with is to proe that the eqality gadget indeed ensres that, in eery cycle coer, two edges are either both present or both absent. Note that some edges in the mltieqality gadget become paths of length three. This is becase we add the eqality gadget first to, say, e 1 and e 2 which gies two new edges each. Next we apply it to one of the new edges for e 2 and to e 3 which explains why e 2 now has three edges. Please also note that the eqality gadget can be adapted easily to work as an XOR-gadget [11], easier to analyze than that fond in [9].

8 8 Marks Bläser and Holger Dell 6.1 The Weighted Coer Polynomial Unfortnately, the eqality gadget cannot 1 enforce eqality in eery possible cycle coer, and we call those cycle coers good that satisfy eqality (we later change this notion slightly). As yo noticed, we introdce weights w e { 1,1} on the edges. These weights make sre that, in the weighted coer polynomial C w (D,0,y) := w(c)y j(c) := w e, cycle coer C cycle coer C y j(c) e C the bad cycle coers sm p to 0, so effectiely only the good cycle coers are isible. We define the ealation fnction C w (a,b) in the weighted case again as D C w (D,a,b) and show that both ealation complexities are eqal. Lemma 4. For all b Q, it holds C(0,b) p m C w (0,b) and C w (0,b) p m C(0,b). Proof (sketch). The first claim is triial. For the second claim we adapt the proof of Valiant [9] to the coer polynomial: We replace the 1-edges by an nweighted N-gadget that simlates a sfficiently large weight N, and we compte modlo N + 1. Howeer, or N-gadget is different from Valiant s since it has to contribte a constant amont of cycles to cycle coers. To handle rational ales, we mltiply with the common denominator, and for negatie ales, we choose N een larger. (for details, cf. [11]) 6.2 Partner Elimination In Fig. 5, we hae drawn all possible ways for an eqality gadget to be coered by cycles if the srronding graph allows it. The state of the gadget is the set of edges chosen to coer it. Notice that the good states as well as state ± 1 and state ± 2 hae the eqality property, that is, the left path is completely present or absent if and only if the same holds for the right one. As stated aboe, the problem is that the eqality gadget does not preent states that do not hae the eqality property. Therefore, sing Lemma 4, we switch to the weighted coer polynomial for which or eqality gadget is constrcted in sch a way, that the good cycle coers contribte a weight 0 to the sm C w (D α,0,b) while the bad cycle coers do not contribte to the sm, that is, the weights of the bad cycle coers sm p to zero. This is achieed by the fact that eery bad cycle coer of weight +1 has a niqe bad cycle coer of weight 1 as a partner. More specifically, eery cycle coer C has a partner C sch that the corresponding smmands in the weighted coer polynomial anish, that is, w(c)y j(c) + w(c )y j(c ) = 0. Note that not only the weights mst be of different sign, bt also the nmbers of cycles mst be eqal! This is the crcial factor why we cannot simply adapt the XOR-gadget of Valiant to form an eqality gadget for the coer polynomial. 1 Fortnately! If it were possible withot weights, we cold adapt the gadget for a parsimonios redction from #SAT to the permanent and thereby proe P = NP.

9 Complexity of the Coer Polynomial 9 1 good 1 state + 1 state 1 1 good 2 state + 2 state good 3 state + 3 state 3 1 state + 4 state 4 Fig. 5. All possible states (=partial cycle coers) of the eqality gadget except that the states state ± 3 and state± 4 also hae symmetric cases, which we hae not drawn. We call good1, good 2, and good 3 good states as they hae no partners and satisfy the eqality property. Note that the choice of the good states is arbitrary as long as the eqality property is satisfied and all state ± i hae partners.

10 10 Marks Bläser and Holger Dell The nmber of cycles contribted by his XOR-gadget aries a lot, and ths the smmands corresponding to the bad cycle coers do not cancel ot. (If we plg in y = 1 to get the permanent, the condition on the cycles is not needed, and the XOR-gadget works, of corse.) Now let s qickly smmarize and proe the properties of the eqality gadget. We now call a cycle coer bad if an eqality gadget is in a state state ± i. Lemma 5. Eery bad cycle coer C of D α has a partner C with the properties (i) C is again bad, and its partner is C, (ii) for the weights, it holds w(c ) = w(c), and (iii) for the nmber of cycles, it holds j(c ) = j(c). Proof. We choose an arbitrary ordering on the eqality gadgets of D α. Let C be a bad cycle coer and g be its smallest gadget in state state ± i. We define its partner C to be the same cycle coer bt with gadget g in state state i instead. Verifying the three properties proes the claim. It immediately follows that only the good cycle coers of D α remain in C w (D α ): C w (D α,0,y) = cycle coer C, C is good w(c)y j(c). 6.3 Conting the Good Cycle Coers In order to finish the proof of Theorem 1, and ths also the proof of the Main Theorem, it remains to express C w (D α ) in an appropriate way in terms of C w (D). Lemma 6. It holds C w (D α,0,y) = ( y n+4m (1 + y) m n) α 1 C w (D,0,y α ). Proof (sketch). Eery cycle coer C of D indces seeral possible good cycle coers C of D α. Since C = n, a nmber of (α 1)n gadgets hae state good 3 in C. The other m n gadget hae the free choice between the states good 1 and good 2. In addition to the αj(c) large cycles, eery gadget contribtes small cycles to C. More precisely, j(c ) = αj(c)+4 #good #good #good 3. Using the fact that there are ( m n) i possible C corresponding to C and with the property #good 1 (C ) = i, the claim can be erified by a simple comptation (cf. [11]). The easily comptable relation between C w (D α,0,y) and C w (D,0,y α ) from the last lemma proes (a weighted formlation of) Theorem 1. Finally, the Main Theorem follows, for 1 b 0, from the redction chain C(0,1) p T Cw (0,1) p T Cw (0,b) p T C(0,b) p T C(a,b).

11 7 The Geometric Coer Polynomial Complexity of the Coer Polynomial 11 The geometric coer polynomial C geom (D,x,y) introdced by D Antona and Mnarini [10] is the geometric ersion of the coer polynomial, that is, the falling factorial x i is replaced by the sal power x i in (1). We are able to gie complexity reslts also for this graph polynomial, except for the line L 1. Theorem 2. Let (a,b) Q 2. It holds: If (a,b) {(0,0),(0, 1)}, then C geom (a,b) is #P-hard. Otherwise, C geom (a,b) is comptable in polynomial time. Proof (sketch). The reslts on the y-axis follow from C(D,0,y) = C geom (D,0,y). For the other points, a new horizontal redction mst be established. This can be done by considering the α-fattening of the graph, in which eery edge is replaced by α parallel edges (for details, cf. [11]). 8 Conclsion and Frther Work In this paper, we completely characterized the complexity of ealating the coer polynomial in the rational plane. Or redctions shold also work for complex or algebraic nmbers, bt we do not hae any interpretation for sch points yet. The next natral step along the path is to characterize the complexity of approximately ealating the coer polynomial, in analogy to recent reslts by Goldberg and Jerrm [6] for the Ttte polynomial. Very recently, Foz had sccess in characterizing the approximability of the geometric coer polynomial in large regions of the plane. One might conjectre that the connection between the Ttte and the coer polynomial is soled becase both are #P-hard to ealate at most points, so they can be compted from each other (where they are in #P). From a combinatorial point of iew, howeer, #P-redctions are not really satisfying steps towards a deeper nderstanding of the connection between the two graph polynomials. One promising approach might be to consider generalizations of the coer polynomial. Very recently, Corcelle analyzed general contraction-deletion identities analogos to those of the colored Ttte polynomial, and he fond conditions for the ariables which are necessary and sfficient for the well-definedness of the polynomial. Unfortnately, these conditions seem too restrictie. The contraction-deletion identities of Ttte and coer polynomial look astonishingly similar, bt contractions work differently for directed and for ndirected graphs. Explaining the connection between Ttte and coer polynomial means oercoming this difference. Acknowledgements. We wold like to thank Mahmod Foz, Thomas Jansen and Moritz Hardt for fritfl discssions and comments on earlier ersions of this paper.

12 12 Marks Bläser and Holger Dell References 1. Chng, F.R., Graham, R.L.: On the coer polynomial of a digraph. Jornal of Combinatorial Theory Series B 65 (1995) Jaeger, F., Vertigan, D.L., Welsh, D.J.: On the comptational complexity of the Jones and Ttte polynomials. Mathematical Proceedings of the Cambridge Philosophical Society 108 (1990) Lotz, M., Makowsky, J.A.: On the algebraic complexity of some families of colored Ttte polynomials. Adances in Applied Mathematics 32 (2004) Bollobás, B., Riordan, O.: A Ttte polynomial for colored graphs. Combinatorics, Probability and Compting 8 (1999) Giménez, O., Noy, M.: On the complexity of compting the Ttte polynomial of bicirclar matroids. Combinatorics, Probability and Compting 15 (2006) Goldberg, L.A., Jerrm, M.: Inapproximability of the Ttte polynomial (2006) 7. Blato, A., Grohe, M.: The complexity of partition fnctions. Theoretical Compter Science 348 (2005) Dyer, M.E., Goldberg, L.A., Paterson, M.: On conting homomorphisms to directed acyclic graphs. In Bgliesi, M., Preneel, B., Sassone, V., Wegener, I., eds.: ICALP (1). Volme 4051 of Lectre Notes in Compter Science., Springer (2006) Valiant, L.G.: The complexity of compting the permanent. Theoretical Compter Science 8 (April 1979) D Antona, O.M., Mnarini, E.: The cycle-path indicator polynomial of a digraph. Ad. Appl. Math. 25 (2000) Bläser, M., Dell, H.: Complexity of the coer polynomial (jornal ersion). (to appear) 12. Dyer, M.E., Frieze, A.M., Jerrm, M.: Approximately conting Hamilton paths and cycles in dense graphs. SIAM Jornal on Compting 27 (1998)