Computing the partition function for graph homomorphisms with multiplicities
aa r X i v : . [ m a t h . C O ] A ug COMPUTING THE PARTITION FUNCTION FORGRAPH HOMOMORPHISMS WITH MULTIPLICITIES
Alexander Barvinok and Pablo Sober´on
July 2015
Abstract.
We consider a refinement of the partition function of graph homomor-phisms and present a quasi-polynomial algorithm to compute it in a certain domain.As a corollary, we obtain quasi-polynomial algorithms for computing partition func-tions for independent sets, perfect matchings, Hamiltonian cycles and dense sub-graphs in graphs as well as for graph colorings. This allows us to tell apart inquasi-polynomial time graphs that are sufficiently far from having a structure of agiven type (i.e., independent set of a given size, Hamiltonian cycle, etc.) from graphsthat have sufficiently many structures of that type, even when the probability to hitsuch a structure at random is exponentially small.
1. Introduction and main results (1.1) Partition function of graph homomorphisms with prescribed mul-tiplicities.
Let G = ( V, E ) be an undirected graph with set V of vertices, set E ofedges, without loops or multiple edges. We denote by ∆( G ) the largest degree of avertex in G . In what follows, we assume that ∆( G ) ≥
1, so that the graph containsat least one edge.Let m = ( µ , . . . , µ k ) be a vector of positive integers such that µ + . . . + µ k = | V | and let A = ( a ij ) be a k × k symmetric real or complex matrix. We define the partition function of graph homomorphisms of G with multiplicities m by(1.1.1) P G,m ( A ) = X φ : V →{ ,... ,k } | φ − ( i ) | = µ i for i =1 ,... ,k Y { u,v }∈ E a φ ( u ) φ ( v ) . Mathematics Subject Classification . 05C30, 15A15, 05C85, 68C25, 68W25, 60C05, 82B20.
Key words and phrases. graph homomorphism, partition function, algorithm.The research of the first author was partially supported by NSF Grants DMS 0856640 andDMS 1361541. Typeset by
AMS -TEX φ : V −→ { , . . . , k } that map precisely µ i vertices of G into i for all i = 1 , . . . , k and the product is taken over all edgesof G . If we take the sum over all k | V | maps φ : V −→ { , . . . , k } , without themultiplicity restrictions, we obtain what is known as the partition function of graphhomomorphisms (1.1.2) P G ( A ) = X φ : V →{ ,... ,k } Y { u,v }∈ E a φ ( u ) φ ( v ) , see [BG05] and [C+13]. • One of the main results of our paper is a deterministic algorithm, which, givena graph G = ( V, E ), a vector m = ( µ , . . . , µ k ) of multiplicities, a k × k symmetricmatrix A = ( a ij ) such that(1.1.3) | a ij − | ≤ γ ∆( G ) for all i, j and a real ǫ >
0, computes P G,m ( A ) within relative error ǫ in ( | E | k ) O (ln | E |− ln ǫ ) time. Here γ > γ = 0 . complex matrices A satisfying (1.1.3) we necessarily have P G,m ( A ) = 0 and, given a real ǫ >
0, we compute in ( | E | k ) O (ln | E |− ln ǫ ) time a(possibly complex) number ˜ P G,m ( A ) = 0 such that (cid:12)(cid:12)(cid:12) ln P G,m ( A ) − ln ˜ P G,m ( A ) (cid:12)(cid:12)(cid:12) < ǫ (we choose a branch of the logarithm which is real when a ij = 1 for all i, j ).In [BS15], we construct a deterministic algorithm, which, given a graph G =( V, E ) and a k × k symmetric matrix A = ( a ij ) satisfying (1.1.3) with a better con-stant γ = 0 .
34 (we can choose γ = 0 .
45 if ∆( G ) ≥ γ = 0 .
54 for all sufficientlylarge ∆( G )) computes the graph homomorphism partition function (1.1.2) withinrelative error ǫ in ( | E | k ) O (ln | E |− ln ǫ ) time. Although the methods of [BS15] and thispaper are similar, it appears that neither result follows from the other.Specializing (1.1.1), we obtain various quantities of combinatorial interest. (1.2) Independent sets in graphs. Let G = ( V, E ) be a graph as above. A set S ⊂ V is called independent if { u, v } / ∈ E for every two vertices u, v ∈ S . Findingthe maximum size of an independent set in a given graph within a factor of | V | ǫ isan NP-hard problem for any 0 ≤ ǫ <
1, fixed in advance [H˚a99], [Zu07].Let us choose k = 2 and define the matrix A = ( a ij ) by a = 0 , a = a = a = 1 . m = ( µ , µ ) be an integer vector such that µ + µ = | V | . One can see that amap φ : V −→ { , } contributes 1 to (1.1.1) if φ − (1) is an independent set in G and contributes 0 otherwise. Hence P G,m ( A ) is the number of independent sets in G of size µ .Let us modify A ˜ A by choosing˜ a = 1 − γ ∆( G ) and ˜ a = ˜ a = ˜ a = 1 + γ ∆( G ) , where γ is the constant in (1.1.3). Then(1.2.1) (cid:18) γ ∆( G ) (cid:19) −| E | P G,m ( ˜ A ) = X S : S ⊂ V | S | = µ w ( S ) , where w ( S ) = (cid:18) γ ∆( G ) (cid:19) − e ( S ) (cid:18) − γ ∆( G ) (cid:19) e ( S ) and e ( S ) is the number of pairs of vertices of S that span an edge of G . Thusexp (cid:26) − γ e ( S )∆( G ) − γ e ( S )∆ ( G ) (cid:27) ≤ w ( S ) ≤ exp (cid:26) − γ e ( S )∆( G ) (cid:27) , so roughly w ( S ) ≈ exp (cid:26) − γ e ( S )∆( G ) (cid:27) . Hence (1.2.1) computes a weighted sum over all subsets S of vertices of cardinality µ , where w ( S ) = 1 if S is an independent set in G , while all other subsets areweighted down exponentially in the number of edges that the vertices of the subsetsspan.Computing (1.2.1) allows us to distinguish graphs with sufficiently many inde-pendent sets of a given size from graphs that are sufficiently far from having anindependent set of a given size. Indeed, if every subset S ⊂ V of µ vertices of G spans at least x edges of G , we get (cid:18) γ ∆( G ) (cid:19) −| E | P G,m ( ˜ A ) ≤ (cid:18) | V | µ (cid:19) exp (cid:26) − γ x ∆( G ) (cid:27) . If, on the other hand, a random subset of µ vertices of G is an independent setwith probability at least 2 exp (cid:26) − γx ∆( G ) (cid:27) then (cid:18) γ ∆( G ) (cid:19) −| E | P G,m ( ˜ A ) ≥ (cid:18) | V | µ (cid:19) exp (cid:26) − γ x ∆( G ) (cid:27) . P G,m ( ˜ A ) within a relative error of 0 .
1, say, we can distinguish thesetwo cases in | E | O (ln | E | ) time.Let us fix some δ > ǫ > G thatsatisfy | E | ≥ δ | V | ∆( G ) (in other words, we consider graphs that are not very farfrom regular). Let us choose x = ǫ | E | for some ǫ ≥ ǫ >
0. In this case, G issufficiently far from having an independent set of size µ if every subset of verticesof size µ spans at least some constant proportion ǫ ≥ ǫ (arbitrarily small, butfixed in advance) of the total number | E | of edges whereas G has sufficiently manyindependent sets if the probability that a randomly selected set of µ vertices isindependent is at least2 exp (cid:26) − γǫ | E | ∆( G ) (cid:27) ≤ {− γǫ δ | V |} . We can distinguish these two cases in quasi-polynomial time, although in the lattercase the probability to hit an independent set at random is exponentially small inthe number of vertices of the graph.More generally, we can separate graphs where each subset S ⊂ V of cardinality µ spans at least x edges of G from graphs having sufficiently many subsets S ⊂ V of cardinality µ spanning at most y < x edges of G . (1.3) Hafnians, Hamiltonian permanents, and subgraph densities. Suppose that G is a union of n pairwise vertex-disjoint edges, so | V | = 2 n and ∆( G ) = 1. Suppose further that k = | V | and let us choose m = (1 , . . . , P G,m ( A ) is the hafnian of the matrix A , see,for example, Section 8.2 of [Mi78]. In particular, if A is the adjacency matrixof a simple undirected graph H with set { , . . . , k } of vertices, then the value of(2 n n !) − P G,m ( A ) is the number of perfect matchings in H , that is, the number ofcollections of edges of H covering every vertex of H exactly once. Hence we obtaina deterministic algorithm approximating the hafnian of a 2 n × n symmetric matrix A = ( a ij ) that satisfies(1.3.1) | − a ij | ≤ γ for all i, j within a relative error ǫ > n O (ln n − ln ǫ ) time. A similar algorithm, though witha better constant γ = 0 .
19 (which can be improved to 0 .
27, see Lemma 4.1 belowand the remark thereafter), was earlier constructed in [B15a].Suppose that G is a cycle with n > G ) = 2, and that k = n .Let us choose m = (1 , . . . , A is the adjacency matrix of a simple undirectedgraph H then the value of (2 n ) − P G,m ( A ) is the number of Hamiltonian cycles in H , that is, the number of walks that visit every vertex in H exactly once beforereturning to the starting point. For a general A , the value of n − P G,m ( A ) is aHamiltonian version of the permanent of A ,1 n P G,m ( A ) = X σ n Y i =1 a iσ ( i ) , n − σ of the set { , . . . , n } thatconsist of a single cycle, cf. [Ba15]. Hence we obtain a deterministic algorithmapproximating the Hamiltonian permanent of an n × n symmetric matrix A = ( a ij )that satisfies | − a ij | ≤ γ i, j within a relative error ǫ > n O (ln n − ln ǫ ) time. This result is new, althoughthere is a polynomial time algorithm, which, for, given a real A satisfying (1.3.1)with any γ <
1, fixed in advance, approximates the Hamiltonian permanent of (notnecessarily symmetric) A within a factor of n O (ln n ) (the implicit constant in the“ O ” notation depends on γ ), see [Ba15]. As in Section 1.2, we obtain a polynomialtime algorithm that distinguishes graphs that have sufficiently many (for example,at least ǫ n n ! for some 0 < ǫ <
1, fixed in advance) Hamiltonian cycles from graphsthat are sufficiently far from Hamiltonian (need at least ǫn new edges added tobecome Hamiltonian for some fixed 0 < ǫ < G with n vertices, let k ≥ n and let A be the adjacency matrixof a simple graph H with set { , . . . , k } of vertices. Let us consider a graph b G consisting of G and k − n isolated vertices. Let us choose a k -dimensional vector m = (1 , . . . , (cid:0) ( k − n )! (cid:1) − P b G,m ( A ) is the number of embeddings ψ : G −→ H that map distinct vertices of G into distinct vertices of H and the edges of G into edges of H . The case of a complete graph G is of a particular interest.Assuming that G is a complete graph with n vertices and (cid:0) n (cid:1) edges, we concludethat (cid:0) n !( k − n )! (cid:1) − P b G,m ( A ) is the number of cliques of size n in H . In this case wehave ∆( b G ) = n −
1. Given H with vertex set { , . . . , k } , let us modify A ˜ A by˜ a ij = (cid:26) γn − if { i, j } is an edge of H − γn − if { i, j } is not an edge of H .Given a set S of vertices of H , let t ( S ) be the number of pairs of distinct verticesof S that do not span an edge of H . Then(1.3.2) ( n !( k − n )!) − (cid:18) γn − (cid:19) − ( n ) P b G,m ( ˜ A ) = X S : S ⊂{ ,... ,k }| S | = n w ( S ) , where w ( S ) = (cid:18) γn − (cid:19) − t ( S ) (cid:18) − γn − (cid:19) t ( S ) = exp (cid:26) − γ t ( S ) n − O (cid:18) t ( S )( n − (cid:19)(cid:27) . Hence we obtain a deterministic algorithm of ( kn ) O (ln n ) complexity, which, givena graph H with k vertices, computes (within a relative error of 0 .
1, say) the sum(1.3.2) of the weights of n -subsets S of the set of vertices, where each weight w ( S )5s exponentially small in the number of edges of H that the vertices of S fail tospan. A similar algorithm was constructed earlier in [B15b] (in [B15b] a worseconstant γ = 0 .
07 is achieved in the general case; however, in the most interestingcase of n = o ( k ) and n ≥
10, [B15b] achieves a better constant of γ = 0 . n from graphs having sufficiently many dense induced subgraphs of size n , even when“many” still allows for the probability to hit a dense induced subgraph at randomto be exponentially small in n , see [B15b]. Finding the densest induced subgraphof a given size n in a given graph with k vertices is a notoriously hard problem.The best known algorithm of k O (ln k ) complexity approximates the highest densityof an n -subgraph up to within a multiplicative factor of k / [B+10], [Bh12].Although an independent set in a graph corresponds to a clique in the comple-mentary graph on the same set of vertices, there appears to be no direct relationbetween the partition functions (1.2.1) and (1.3.2). In (1.2.1), the weight of a sub-set depends on the maximum degree ∆( G ) while in (1.3.2) it depends on the sizeof the subset. (1.4) Graph colorings. Let G = ( V, E ) be a graph as in Section 1.1 and let φ : V −→ { , . . . , k } be a map. The map φ is called a proper k -coloring if φ ( u ) = φ ( v ) whenever { u, v } ∈ E . The smallest k for which a proper k -coloring of G exists is called the chromaticnumber of G . Approximating the chromatic number of a given graph within a factorof | V | − ǫ for any 0 < ǫ <
1, fixed in advance, is NP-hard [FK98], [Zu07]. Let usdefine a matrix A = ( a ij ) by a ij = (cid:26) i = j i = j. Given an integer vector m = ( µ , . . . , µ k ), we observe that P G,m ( A ) is the numberof proper colorings of G , where the i -th color is used exactly µ i times. Coloringswith prescribed number of vertices of a given color were studied in the case of equitable colorings , where any two multiplicities µ i and µ j differ by at most 1, see[K+10] and references wherein.As above, let us modify A ˜ A by letting˜ a ij = ( γ ∆( G ) if i = j − γ ∆( G ) if i = j. Then we obtain (cid:18) γ ∆( G ) (cid:19) −| E | P G,m ( ˜ A ) = X φ : V →{ ,... ,k } | φ − ( i ) | = µ i for i =1 ,... ,k w ( φ ) , w ( φ ) = (cid:18) γ ∆( G ) (cid:19) − e ( φ ) (cid:18) − γ ∆( G ) (cid:19) e ( φ ) and e ( φ ) is the number of miscolored edges of G (that is, edges whose endpointsare colored with the same color under the coloring φ ). Thusexp (cid:26) − γ e ( φ )∆( G ) − γ e ( φ )∆ ( G ) (cid:27) ≤ w ( e ) ≤ exp (cid:26) − γ e ( φ )∆( G ) (cid:27) and hence (1.4.1) represents a weighted sum over all colorings into k colors withprescribed multiplicity of each color and the weight of each coloring being exponen-tially small in the number of miscolored edges. As before, we can compute (1.4.1)within a relative error of 0 . k | E | ) O (ln | E | ) time. (1.5) Partition function of edge-colored graph homomorphisms with mul-tiplicities. It turns out that instead of the partition function P G,m ( A ) defined by(1.1.1), it is more convenient to consider a more general expression. Let G = ( V, E )be a graph as in Section 1.1 and let B = (cid:0) b uvij (cid:1) be | E | × k ( k +1)2 real or complexmatrix with entries indexed by edges { u, v } ∈ E and unordered pairs 1 ≤ i, j ≤ k .Technically, we should have written b { u,v }{ i,j } , but we write just b uvij , assuming that b uvij = b vuij = b vuji = b uvji . We define(1.5.1) Q G,m ( B ) = X φ : V →{ ,... ,k } | φ − ( i ) | = µ i for i =1 ,... ,k Y { u,v }∈ E b uvφ ( u ) φ ( v ) . Let H be a simple undirected graph with set { , . . . , k } of vertices and supposethat the edges of G and H are colored. Let us define b uvij = { u, v } and { i, j } are edges of the same colorof G and H respectively0 otherwise.Then Q G,m ( B ) is the number of maps V −→ { , . . . , k } such that for every edge { u, v } of G , the pair { φ ( u ) , φ ( v ) } spans an edge of H of the same color and precisely µ i vertices of V are mapped into the vertex i of H .Clearly, (1.1.1) is a specialization of (1.5.1) since P G,m ( A ) = Q G,m ( B ) provided b uvij = a ij for all { u, v } ∈ E and all 1 ≤ i, j ≤ k . The advantage of working with Q G,m ( B ) instead of P G,m ( A ) is that Q G,m is a multi-affine polynomial, that is, thedegree of each variable in Q G,m is 1. 7f in (1.5.1) we consider the sum over all k | V | maps φ : V −→ { , . . . , k } weobtain the partition function of edge-colored graph homomorphisms (1.5.2) Q G ( B ) = X φ : V →{ ,... ,k } Y { u,v }∈ E b uvφ ( u ) φ ( v ) introduced in [BS15], cf. also [AM98]. • The main result of our paper is a deterministic algorithm, which, given agraph G = ( V, E ), a vector m = ( µ , . . . , µ k ) of multiplicities, a | E | × k ( k +1)2 matrix B = (cid:0) b uvij (cid:1) such that(1.5.3) (cid:12)(cid:12) b uvij − (cid:12)(cid:12) ≤ γ ∆( G ) for all { u, v } ∈ E and all 1 ≤ i, j ≤ k and a real ǫ >
0, computes Q G,m ( B ) within relative error ǫ in ( | E | k ) O (ln | E |− ln ǫ ) time. Here γ > γ = 0 . G =( V, E ) and a matrix B satisfying (1.5.3) with a better constant γ = 0 .
34 (wecan choose γ = 0 .
45 if ∆( G ) ≥ γ = 0 .
54 for all sufficiently large ∆( G ))computes the partition function (1.5.2) within relative error ǫ in ( | E | k ) O (ln | E |− ln ǫ ) time. Although the methods of [BS15] and this paper are similar, it appears thatneither result follows from the other. (1.6) Partition functions in combinatorics. Partition functions are success-fully used to count deterministically various combinatorial structures such as inde-pendent sets [BG08] and matchings [B+07] in graphs. The approach of [BG08] and[B+07] is based on the “correlation decay” idea motivated by statistical physics.Our approach is different (and the partition functions we compute are also different)but one can argue that our method is also inspired by statistical physics. Roughly,we use that the logarithm of the partition function is well-approximated by a lowdegree Taylor polynomial in the region that is sufficiently far away from the phasetransition. Phase transitions are associated with complex zeros of partition func-tions [LY52], see also [SS05] for connections to combinatorics, and the main effortof our method is in isolating the complex zeros of Q G,m .While our estimate of γ = 0 . P G,m ( A ) can be efficiently approximated aslong as | a ij − | < γ for any γ <
1, fixed in advance, is unlikely to be true evenwhen k = 2 and ∆( G ) = 3. It is argued in [BS15] that computing the partitionfunction (1.1.2) in quasi-polynomial time in such a large domain would have ledto a quasi-polynomial algorithm in an NP-hard problem. The argument of [BS15]transfers almost verbatim to the partition function (1.1.1). Hence it appears to bean interesting problem to find the best possible values of γ in (1.1.3) and (1.5.3).8 . The algorithm In this section, we describe the algorithm for computing (1.5.1). (2.1) The algorithm.
Let J be an | E | × k ( k +1)2 matrix filled with 1s. Given a | E | × k ( k +1)2 matrix B = (cid:0) b uvij (cid:1) , where { u, v } ∈ E and 1 ≤ i, j ≤ k , we consider theunivariate function f ( t ) = ln Q G,m (cid:0) J + t ( B − J ) (cid:1) , so that f (0) = ln Q G,m ( J ) = ln | V | ! µ ! · · · µ k ! and f (1) = ln Q G,m ( B ) . Hence our goal is to approximate f (1) and we do it by using the Taylor polynomialapproximation of f at t = 0:(2.1.1) f (1) ≈ f (0) + n X j =1 j ! d j dt j f ( t ) (cid:12)(cid:12)(cid:12) t =0 . We claim that the right hand side can be computed in ( | E | k ) O ( n ) time. Indeed, let(2.1.2) g ( t ) = Q G,m (cid:0) J + t ( B − J ) (cid:1) = X φ : V →{ ,... ,k } | φ − ( i ) | = µ i for i =1 ,... ,k Y { u,v }∈ E (cid:16) t (cid:16) b uvφ ( u ) φ ( v ) − (cid:17)(cid:17) , so that f ( t ) = ln g ( t ), f ′ ( t ) = g ′ ( t ) g ( t ) and g ′ ( t ) = g ( t ) f ′ ( t ) . Therefore,(2.1.3) d j dt j g ( t ) (cid:12)(cid:12)(cid:12) t =0 = j − X i =0 (cid:18) j − i (cid:19) (cid:18) d i dt i g ( t ) (cid:12)(cid:12)(cid:12) t =0 (cid:19) (cid:18) d j − i dt j − i f ( t ) (cid:12)(cid:12)(cid:12) t =0 (cid:19) , where we agree that the 0-th derivative of g is g . We note that(2.1.4) g (0) = | V | ! µ ! · · · µ k ! . If we compute the values of(2.1.5) d j dt j g ( t ) (cid:12)(cid:12)(cid:12) t =0 for j = 1 , . . . , n, d i dt i f ( t ) (cid:12)(cid:12)(cid:12) t =0 for i = 1 , . . . , n, from the triangular system (2.1.3) of linear equations with the coefficients (2.1.4)on the diagonal. Hence our goal is to compute (2.1.5).Using (2.1.2), we obtain d j dt j g ( t ) (cid:12)(cid:12)(cid:12) t =0 = X φ : V →{ ,... ,k } | φ − ( i ) | = µ i for i =1 ,... ,k X I = (cid:0) { u ,v } , ···{ u j ,v j } (cid:1) (cid:16) b u v φ ( u ) φ ( v ) − (cid:17) · · · (cid:16) b u j v j φ ( u j ) φ ( v j ) − (cid:17) , where the inner sum is taken over all ordered collections I of j distinct edges { u , v } , . . . , { u j , v j } of G . For such a collection I , let S ( I ) = { u , v } ∪ . . . ∪ { u j , v j } be the set of all distinct endpoints of the edges. Then we can write d j dt j g ( t ) (cid:12)(cid:12)(cid:12) t =0 = X I = (cid:0) { u ,v } , ···{ u j ,v j } (cid:1) X φ : S ( I ) →{ ,... ,k } | φ − ( i ) | ≤ µ i for i =1 ,... ,k (cid:0) | V | − | S ( I ) | )!( µ − | φ − (1) | )! · · · ( µ k − | φ − ( k ) | )! × (cid:16) b u v φ ( u ) φ ( v ) − (cid:17) · · · (cid:16) b u j v j φ ( u j ) φ ( v j ) − (cid:17) . In words: we enumerate at most | E | j ordered collections I = (cid:0) { u , v } , . . . , { u j , v j } (cid:1) of j distinct edges of G , for each such a collection, we enumerate atmost k j maps φ defined on the set of the endpoints of the edges from I into the set { , . . . , k } , multiply the term (cid:16) b u v φ ( u ) φ ( v ) − (cid:17) · · · (cid:16) b u j v j φ ( u j ) φ ( v j ) − (cid:17) by the numberof ways to extend the map φ to the whole set V of vertices so that the multiplicityof i is µ i and add the results over all choices of I and φ . Since j ≤ n , the complexityof computing (2.1.5) is indeed ( | E | k ) O ( n ) as claimed.The quality of the approximation (2.1.1) depends on the location of complex zeros of Q G,m . (2.2) Lemma. Suppose that there is a real β > such that Q G,m (cid:0) J + z ( B − J ) (cid:1) = 0 for all z ∈ C satisfying | z | ≤ β. hen the right hand side of (2.1.1) approximates f (1) within an additive error of | E | ( n + 1) β n ( β − . Proof.
The function g ( t ) defined by (2.1.2) is a polynomial in t of degree at most d ≤ | E | and g (0) = 0 by (2.1.4), so we can factor g ( z ) = g (0) d Y i =1 (cid:18) − zα i (cid:19) , where α , . . . , α d ∈ C are the roots of g ( z ). In addition, | α i | ≥ β > i = 1 , . . . , d. Thus(2.2.1) f ( z ) = ln g ( z ) = ln g (0) + d X i =1 ln (cid:18) − zα i (cid:19) for all | z | ≤ , where we choose the branch of ln g ( z ) for which ln g (0) is real. Using the standardTaylor series expansion, we obtainln (cid:18) − α i (cid:19) = − n X j =1 j (cid:18) α i (cid:19) j + ξ n , where | ξ n | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ∞ X j = n +1 j (cid:18) α i (cid:19) j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ n + 1) β n ( β − . Hence from (2.2.1) we obtain f (1) = f (0) + n X j =1 − j d X i =1 (cid:18) α i (cid:19) j ! + η n , where | η n | ≤ | E | ( n + 1) β n ( β − . It remains to notice that − j d X i =1 (cid:18) α i (cid:19) j = 1 j ! d j dt j f ( t ) (cid:12)(cid:12)(cid:12) t =0 . (cid:3) For a fixed β >
1, to achieve an additive error of 0 < ǫ <
1, we can choose n = O (ln | E | − ln ǫ ), in which case the algorithm of Section 2.1 computes Q G,m ( B )within a relative error ǫ in ( | E | k ) O (ln | E |− ln ǫ ) time. Hence it remains to identifymatrices B for which the number β > There exists an absolute constant α > (one can choose α =0 . ) such that for any undirected graph G = ( V, E ) , for any vector m = ( µ , . . . , µ k ) of positive integers such that µ + . . . + µ k = | V | and for anycomplex | E | × k ( k +1)2 matrix Z = (cid:0) z uvij (cid:1) satisfying (cid:12)(cid:12) − z uvij (cid:12)(cid:12) ≤ α ∆( G ) for all { u, v } ∈ E and ≤ i, j ≤ k, one has Q G,m ( Z ) = 0 . Theorem 2.3 implies that if B satisfies (1.5.3) with γ = 0 . β = αγ = 0 . . . > Q G,m ( B ) withinrelative error ǫ in ( | E | k ) O (ln | E |− ln ǫ ) time.In the rest of the paper, we prove Theorem 2.3.
3. Recurrence relations
We consider the polynomials Q G,m ( Z ) within a larger family of polynomials. (3.1) Definitions. Let us fix the graph G and the integer vector m = ( µ , . . . , µ k ).We say that a sequence W = ( v , . . . , v n ) of vertices of G is admissible if thevertices v , . . . , v n are distinct.Let I = ( i , . . . , i n ) be a sequence of (not necessarily distinct) indices i j ∈{ , . . . , k } for j = 1 , . . . , n . For i ∈ { , . . . , k } , we define the multiplicity ν i ( I ) of i in I by ν i ( I ) = |{ j : i j = i }| , the number of times that i occurs in I .A sequence I = ( i , . . . , i n ) of (not necessarily distinct) indices i j ∈ { , . . . , k } is called admissible provided ν i ( I ) ≤ µ i for i = 1 , . . . , k. For an admissible sequence W = ( v , . . . , v n ) of vertices and an admissible sequence I = ( i , . . . , i n ) of indices such that | W | = | I | , we define Q WI ( Z ) = X φ : V →{ ,... ,k } | φ − ( i ) | = µ i for i =1 ,... ,kφ ( v j )= i j for j =1 ,... ,n Y { u,v }∈ E z uvφ ( u ) φ ( v ) .
12n other words, to define Q WI ( Z ), we restrict the sum (1.5.1) defining Q G,m ( Z ) tomaps φ : V −→ { , . . . , k } that map prescribed vertices v , . . . , v n of the graph tothe prescribed indices i , . . . , i n . Note that if W = ( v , . . . , v n ) and I = ( i , . . . , i n )are admissible sequences, then there is a map φ : V −→ { , . . . , k } which mapsprecisely µ i distinct vertices of V onto i for i = 1 , . . . , k and such that φ ( v j ) = i j for j = 1 , . . . , n .We suppress the graph G and the vector of multiplicities m in the notation for Q WI ( Z ) and note that when W and I are both empty, then Q WI ( Z ) = Q G,m ( Z ).For a sequence W of vertices and a vertex v of the graph, we denote by ( W, v )the sequence W appended by v . For a sequence I of indices and another index i ∈ { , . . . , k } , we denote by ( I, i ) the sequence I appended by i . We denotesimilarly sequences appended by several vertices or indices. (3.2) Recurrence relations. We will use the following two recurrence relations.First, if W and I are admissible sequences such that | W | = | I | and the sequence( W, v ) is also admissible (that is, v is distinct from the vertices in W ), then(3.2.1) Q WI ( Z ) = X i ∈{ ,... ,k } :( I,i ) is admissible Q ( W,v )( I,i ) ( Z ) . Second, if W and I are admissible sequences such that | W | = | I | and if ( I, i ) isan admissible sequence of indices then(3.2.2) Q WI ( Z ) = 1 µ i − ν i ( I ) X v ∈ V :( W,v ) is admissible Q ( W,v )( I,i ) ( Z )(recall that ν i ( I ) is the multiplicity of i in I so that µ i − ν i ( I ) > I, i ) isadmissible).
4. Angles in the complex plane
In what follows, we measure angles between non-zero complex numbers, consid-ered as vectors in R identified with C .We start with two geometric calculations. (4.1) Lemma. Let z , . . . , z n ∈ C be non-zero numbers such that the angle betweenany two numbers z i and z j does not exceed θ for some ≤ θ < π/ . Then, for z = z + . . . + z n , we have | z | ≥ (cid:18) cos θ (cid:19) n X i =1 | z i | . Proof.
We observe that 0 is not in the convex hull of any three vectors z i , z j , z k , sinceotherwise the angle between some two of those three vectors would have been at13east 2 π/
3. By the Carath´eodory Theorem, 0 is not in the convex hull of z , . . . , z n .Therefore, the vectors are enclosed in an angle of at most θ with vertex at 0. Letus project each vector z i orthogonally onto the bisector of the angle. The lengthof the projection is at least | z i | cos( θ/
2) and hence the length of the projection of z + . . . + z n is at least ( | z | + . . . + | z n | ) cos( θ/ (cid:3) Lemma 4.1 was suggested by Boris Bukh [Bu15]. It replaces a weaker bound of √ cos θ ( | z | + . . . + | z n | ) of an earlier version of the paper. (4.2) Lemma. Let a , . . . , a n and b , . . . , b n be complex numbers such that all a , . . . , a n are non-zero. Let a = n X i =1 a i and b = n X i =1 b i . Suppose that for some real > τ > ǫ > we have (cid:12)(cid:12)(cid:12)(cid:12) b i a i − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ for i = 1 , . . . , n and | a | ≥ τ n X i =1 | a i | . Then a = 0 , b = 0 and the angle between a and b does not exceed arcsin ǫτ . Proof.
Clearly, a = 0. We can write b i = (1 + ǫ i ) a i where | ǫ i | ≤ ǫ for i = 1 , . . . , n. Hence b = n X i =1 (1 + ǫ i ) a i = a + n X i =1 ǫ i a i and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n X i =1 ǫ i a i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ n X i =1 | a i | ≤ ǫτ | a | , from which b = 0 and (cid:12)(cid:12)(cid:12)(cid:12) ba − (cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫτ . Therefore, the argument of the complex number b/a lies in the interval h − arcsin ǫτ , arcsin ǫτ i and the proof follows. (cid:3) The main result of this section concerns the angles between various numbers Q WI ( Z ) introduced in Section 3. 14 Let us us fix an admissible sequence W of vertices, an admis-sible sequence I of indices such that ≤ | W | = | I | ≤ | V | − , a complex | E | × k ( k +1)2 matrix Z , a real ǫ > and a real ≤ θ < π/ such that ǫ < cos( θ/ . Suppose thatfor any two vertices u, v ∈ V and any two i, j ∈ { , . . . , k } such that the sequences ( W, u, v ) and ( I, i, j ) are admissible, we have Q ( W,u,v )( I,i,j ) ( Z ) = 0 , Q ( W,u,v )( I,j,i ) ( Z ) = 0 and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q ( W,u,v )( I,i,j ) ( Z ) Q ( W,u,v )( I,j,i ) ( Z ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ. (1) Suppose that for any two vertices u, v ∈ V and any three i, j , j ∈ { , . . . , k } such that the sequences ( W, u, v ) and ( I, i, j ) and ( I, i, j ) are admissible,the angle between two complex numbers Q ( W,u,v )( I,i,j ) ( Z ) and Q ( W,u,v )( I,i,j ) ( Z ) doesnot exceed θ . Then for any two vertices u, v ∈ V and for any i ∈ { , . . . , k } such that the sequences ( W, u ) , ( W, v ) and ( I, i ) are admissible, we have Q ( W,u )( I,i ) ( Z ) = 0 , Q ( W,v )( I,i ) ( Z ) = 0 and the angle between two complex numbers Q ( W,u )( I,i ) ( Z ) and Q ( W,v )( I,i ) ( Z ) does not exceed arcsin ǫ cos( θ/ . (2) Suppose that for any three u, v , v ∈ V and any two indices i, j ∈ { , . . . , k } such that the sequences ( W, u, v ) , ( W, u, v ) and ( I, i, j ) are admissible, theangle between two complex numbers Q ( W,u,v )( I,i,j ) ( Z ) and Q ( W,u,v )( I,i,j ) ( Z ) does notexceed θ . Then for any u ∈ V and any i, j ∈ { , . . . , k } such that thesequences ( W, u ) , ( I, i ) and ( I, j ) are admissible, we have Q ( W,u )( I,i ) ( Z ) = 0 , Q ( W,u )( I,j ) ( Z ) = 0 and the angle between two complex numbers Q ( W,u )( I,i ) ( Z ) and Q ( W,u )( I,j ) ( Z ) does not exceed arcsin ǫ cos( θ/ . Proof.
To prove Part (1), we note that by (3.2.1), we have Q ( W,u )( I,i ) ( Z ) = X j ∈{ ,... ,k } :( I,i,j ) is admissible Q ( W,u,v )( I,i,j ) ( Z ) and Q ( W,v )( I,i ) ( Z ) = X j ∈{ ,... ,k } :( I,j,i ) is admissible Q ( W,v,u )( I,i,j ) ( Z ) . Q ( W,u,v )( I,i,j ) ( Z ) and Q ( W,u,v )( I,i,j ) ( Z )does not exceed θ , by Lemma 4.1 we have (cid:12)(cid:12)(cid:12) Q ( W,u )( I,i ) ( Z ) (cid:12)(cid:12)(cid:12) ≥ (cid:18) cos θ (cid:19) X j ∈{ ,... ,k } :( I,i,j ) is admissible (cid:12)(cid:12)(cid:12) Q ( W,u,v )( I,i,j ) ( Z ) (cid:12)(cid:12)(cid:12) . The proof of Part (1) follows by Lemma 4.2 applied to the numbers a j = Q ( W,u,v )( I,i,j ) ( Z ) , b j = Q ( W,v,u )( I,i,j ) ( Z ) , a = Q ( W,u )( I,i ) ( Z ) , b = Q ( W,v )( I,i ) ( Z )and τ = cos( θ/ i = j . Then the sequence ( I, i, j )is admissible provided (
I, i ) and (
I, j ) are both admissible. Applying (3.2.2), weobtain Q ( W,u )( I,i ) ( Z ) = 1 µ j − ν j ( I, i ) X v ∈ V :( W,u,v ) is admissible Q ( W,u,v )( I,i,j ) ( Z )and Q ( W,u )( I,j ) ( Z ) = 1 µ i − ν i ( I, j ) X v ∈ V :( W,u,v ) is admissible Q ( W,u,v )( I,j,i ) ( Z ) . (4.3.1)For a vertex v ∈ V such that the sequence ( W, u, v ) is admissible, let us denote a v = Q ( W,u,v )( I,i,j ) ( Z ) and b v = Q ( W,u,v )( I,j,i ) ( Z )and let a = X v ∈ V :( W,u,v ) is admissible a v and b = X v ∈ V :( W,u,v ) is admissible b v . Since the angle between any two numbers a v and a v does not exceed θ , by Lemma4.1 we have | a | ≥ (cid:18) cos θ (cid:19) X v ∈ V :( W,u,v ) is admissible | a v | , and by Lemma 4.2, the angle between a and b does not exceedarcsin ǫ cos( θ/ . Since by (4.3.1) we have Q ( W,u )( I,i ) ( Z ) = 1 µ j − ν j ( I, i ) a and Q ( W,u )( I,j ) ( Z ) = 1 µ i − ν i ( I, j ) b, µ j − ν j ( I, i ) > µ i − ν i ( I, j ) >
0, the result follows.It remains to handle the case of i = j , in which case we just have to provethat Q ( W,u )( I,i ) ( Z ) = 0 whenever ( W, u ) and (
I, i ) are admissible sequences. Since | W | = | I | ≤ | V | −
2, there is an index l ∈ { , . . . , k } for which the sequence ( I, i, l )is admissible. Using the first equation in (4.3.1) with j replaced by l throughoutand Lemma 4.1, we conclude that Q ( W,u )( I,i ) ( Z ) = 0. (cid:3) (4.4) Remark. The intuition behind Proposition 4.3 is as follows. We are interestedin how the value of Q WI ( Z ) may change if we change one vertex in the sequence W or one index in the sequence I , while keeping the sequences admissible. Supposethat the value of Q WI ( Z ) does not change much if we just permute two indicesin the sequence I . Part (1) asserts that if for sequences W and I of some length | W | = | I | , the complex number Q WI ( Z ) rotates by a small angle when one index in I is changed, then for shorter sequences W ′ and I ′ such that | W ′ | = | W | − | I ′ | = | I |−
1, the complex number Q W ′ I ′ ( Z ) rotates by a small angle when one vertexin W ′ is changed. Part (2) asserts that if for sequences W and I of some length | W | = | I | , the complex number Q WI ( Z ) rotates by a small angle when one vertexin W is changed, then for shorter sequences W ′ and I ′ such that | W ′ | = | W | − | I ′ | = | I | −
1, the complex number Q W ′ I ′ ( Z ) rotates by a small angle when oneindex in I ′ is changed.For the subsequent proof, we would like this condition of Q WI ( Z ) being rotatedby a small angle if one vertex in W is changed or one index in I is changed topropagate for shorter and shorter sequences W and I . For that, we will needto find the “fixed point” of the conditions of Proposition 4.3, that is, a solution0 ≤ θ < π/ θ = arcsin ǫ cos( θ/ . It is not hard to see that for all sufficiently small ǫ > α in Theorem 2.3 as large as possible, we would like to choose ǫ as large as possible. Numerical computations show that we can choose ǫ = 0 . θ ≈ .
5. Derivatives
The first goal of this section is to relate how the value of Q WI ( Z ) changes whentwo indices in I are permuted, with partial derivatives of Q WI ( Z ). (5.1) Definition. For a 0 < δ <
1, we define the polydisc U ( δ ) ⊂ C k ( k +1) | E | / by U ( δ ) = (cid:8) Z = (cid:0) z uvij (cid:1) : (cid:12)(cid:12) − z uvij (cid:12)(cid:12) ≤ δ for all { u, v } ∈ E and 1 ≤ i, j ≤ k (cid:9) . Thus U ( δ ) is the closed polydisc of radii δ centered at the matrix J of all 1’s. Wewill be interested in the situations when Q WI ( Z ) = 0 for all Z ∈ U ( δ ), in which casewe can choose a branch of ln Q WI ( Z ) for Z ∈ U ( δ ) in such a way that ln Q WI ( J ) isa real number. 17 Let us fix an integer ≤ r ≤ | V | and let τ > and < δ < bereal. Suppose that for any admissible sequences W and I such that | W | = | I | = r and for any Z ∈ U ( δ ) we have Q WI ( Z ) = 0 and the following condition is satisfied:if W = ( W ′ , v ) and I = ( I ′ , i ) then (cid:12)(cid:12) Q WI ( Z ) (cid:12)(cid:12) ≥ τ ∆( G ) X w : { w,v }∈ El : 1 ≤ l ≤ k | z vwil | (cid:12)(cid:12)(cid:12)(cid:12) ∂∂z vwil Q WI ( Z ) (cid:12)(cid:12)(cid:12)(cid:12) . Then for any admissible W and I such that | W | = | I | = r and for any Z ∈ U ( δ ) the following condition is satisfied: if W = ( W ′ , u, v ) and I = ( I ′ , j, i ) , then for ξ = 4 δ ∆( G )(1 − δ ) τ we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q ( W ′ ,u,v )( I ′ ,j,i ) ( Z ) Q ( W ′ ,u,v )( I ′ ,i,j ) ( Z ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ e ξ − . Proof.
Let us choose admissible W and I such that | W | = | I | = r and suppose that W = ( W ′ , u, v ) and I = ( I ′ , j, i ). If i = j the result is trivial, so we assume that i = j . Since ∂∂z vwil ln Q WI ( Z ) = (cid:18) ∂∂z vwil Q WI ( Z ) (cid:19) / Q WI ( Z )and | z vwil | ≥ − δ for all Z ∈ U ( δ ) , by the conditions of the lemma, we obtain X w : { w,v }∈ El : 1 ≤ l ≤ k (cid:12)(cid:12)(cid:12)(cid:12) ∂∂z vwil ln Q WI ( Z ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ∆( G )(1 − δ ) τ and X w : { w,u }∈ El : 1 ≤ l ≤ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂∂z uwjl ln Q WI ( Z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∆( G )(1 − δ ) τ . (5.2.1)Given a matrix A ∈ U ( δ ), we define a matrix B ∈ U ( δ ) as follows: b uwjl = a uwil for all w = v such that { u, w } ∈ E and all l = 1 , . . . , kb vwil = a vwjl for all w = u such that { v, w } ∈ E and all l = 1 , . . . , k, whereas all other entries of B are equal to the corresponding entries of A . Sinceswapping the values of φ : V −→ { , . . . , k } on two vertices u and v does not changethe multiplicities (cid:12)(cid:12) φ − ( l ) (cid:12)(cid:12) for l = 1 , . . . , k , we have Q ( W ′ ,u,v )( I ′ ,j,i ) ( B ) = Q ( W ′ ,u,v )( I ′ ,i,j ) ( A )18nd using (5.2.1), we obtain (cid:12)(cid:12)(cid:12) ln Q ( W ′ ,u,v )( I ′ ,j,i ) ( A ) − ln Q ( W ′ ,u,v )( I ′ ,i,j ) ( A ) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ln Q ( W ′ ,u,v )( I ′ ,j,i ) ( A ) − ln Q ( W ′ ,u,v )( I ′ ,j,i ) ( B ) (cid:12)(cid:12)(cid:12) ≤ sup Z ∈ U ( δ ) X w : { u,w }∈ El : 1 ≤ l ≤ k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂∂z uwjl ln Q WI ( Z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + X w : { v,w }∈ El : 1 ≤ l ≤ k (cid:12)(cid:12)(cid:12)(cid:12) ∂∂z vwil ln Q WI ( Z ) (cid:12)(cid:12)(cid:12)(cid:12) × max w ∈ V ≤ l ≤ k (cid:12)(cid:12) a uwjl − b uwjl (cid:12)(cid:12) , | a vwil − b vwil | ≤ (cid:18) ∆( G )(1 − δ ) τ + ∆( G )(1 − δ ) τ (cid:19) × (2 δ ) = 4 δ ∆( G )(1 − δ ) τ = ξ. Let us denote ζ = Q ( W ′ ,u,v )( I ′ ,j,i ) ( Z ) Q ( W ′ ,u,v )( I ′ ,i,j ) ( Z ) . Hence | ln ζ | ≤ ξ. Since | e z − | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + ∞ X m =1 z m m ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ + ∞ X m =1 | z | m m ! = e | z | − z ∈ C , we conclude that | ζ − | ≤ (cid:12)(cid:12) e ln ζ − (cid:12)(cid:12) ≤ e | ln ζ | − ≤ e ξ − (cid:3) (5.3) Remark. We will be interested in the situation when (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q ( W ′ ,u,v )( I ′ ,j,i ) ( Z ) Q ( W ′ ,u,v )( I ′ ,i,j ) ( Z ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ . , cf. Proposition 4.3 and Remark 4.4. To ensure the estimate, it suffices to have ξ ≤ .
565 in Lemma 5.2.Our next goal is to relate the inequality for partial derivatives of Lemma 5.2 toanother angle condition of Proposition 4.3. Namely, we show that if we can boundthe angle by which the number Q WI ( Z ) rotates when one index in I is changed,we obtain the inequality of Lemma 5.2 for shorter sequences W ′ and I ′ such that | W ′ | = | I ′ | = | W | − | I | −
1. 19
Let ≤ θ < π/ be a real number, let W be an admissible sequenceof vertices and let I be an admissible sequence of indices such that ≤ | I | = | W | ≤ | V | − . Suppose that for any Z ∈ U ( δ ) , for every w such that ( W, w ) isadmissible and for every ≤ l, j ≤ k such that ( I, l ) and ( I, j ) are admissible, wehave Q ( W,w )( I,l ) ( Z ) = 0 , Q ( W,w )( I,j ) ( Z ) = 0 and the angle between the complex numbers Q ( W,w )( I,l ) ( Z ) and Q ( W,w )( I,j ) ( Z ) does not exceed θ .Let W = ( W ′ , v ) and I = ( I ′ , i ) . Then (cid:12)(cid:12) Q WI ( Z ) (cid:12)(cid:12) ≥ τ ∆( G ) X w : { w,v }∈ Ej : 1 ≤ j ≤ k (cid:12)(cid:12) z vwij (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂∂z vwij Q WI ( Z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , where τ = cos θ . Proof.
Recall that Q WI ( Z ) is a multi-affine function in Z .Let us choose a vertex w such that { v, w } ∈ E . If w is an element of the sequence W ′ , then z vwij ∂∂z vwij Q WI ( Z ) = (cid:26) Q WI ( Z ) if j is the element of I ′ that corresponds to w w is not an element of W ′ , then ( W, w ) is an admissible sequence and z vwij ∂∂z vwij Q WI ( Z ) = ( Q ( W,w )( I,j ) ( Z ) if ( I, j ) is admissible,0 otherwise.Denoting by d the number of vertices w in W ′ such that { w, v } ∈ E , we obtain X w : { w,v }∈ Ej : 1 ≤ j ≤ k (cid:12)(cid:12) z vwij (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂∂z vwij Q WI ( Z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = d (cid:12)(cid:12) Q WI ( Z ) (cid:12)(cid:12) + X w,j : w not in W ′ , { w,v }∈ E ( I,j ) is admissible (cid:12)(cid:12)(cid:12) Q ( W,w )( I,j ) ( Z ) (cid:12)(cid:12)(cid:12) . (5.4.1)On the other hand, by (3.2.1) for all w not in W ′ , we have Q WI ( Z ) = X j : 1 ≤ j ≤ k ( I,j ) is admissible Q ( W,w )( I,j ) ( Z )20nd hence by Lemma 4.1,(5.4.2) (cid:12)(cid:12) Q WI ( Z ) (cid:12)(cid:12) ≥ (cid:18) cos θ (cid:19) X j : 1 ≤ j ≤ k ( I,j ) is admissible (cid:12)(cid:12)(cid:12) Q ( W,w )( I,j ) ( Z ) (cid:12)(cid:12)(cid:12) . Denoting by d the number of vertices w such that { v, w } ∈ E and w are not in W ′ , we deduce from (5.4.1) and (5.4.2) thatcos( θ/ G ) X w : { w,v }∈ Ej : 1 ≤ j ≤ k (cid:12)(cid:12) z vwij (cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ∂∂z vwij Q WI ( Z ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = d cos( θ/ G ) (cid:12)(cid:12) Q WI ( Z ) (cid:12)(cid:12) + cos( θ/ G ) X w,j : w not in W ′ , { w,v }∈ E ( I,j ) is admissible (cid:12)(cid:12)(cid:12) Q ( W,w )( I,j ) ( Z ) (cid:12)(cid:12)(cid:12) ≤ d cos( θ/ G ) (cid:12)(cid:12) Q WI ( Z ) (cid:12)(cid:12) + d ∆( G ) (cid:12)(cid:12) Q WI ( Z ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) Q WI ( Z ) (cid:12)(cid:12) , as desired. (cid:3)
6. Proof of Theorem 2.3
First, we define some constants. Let ǫ = 0 . θ be the solution of the equation θ = arcsin ǫ cos( θ/ , so, θ ≈ . , cf. Remark 4.4. Let τ = cos θ ≈ . α = 0 . τ . τ ≈ . . In particular, as long as ∆( G ) ≥
1, we have ξ = 4 α (cid:16) − α ∆( G ) (cid:17) τ ≤ . , e ξ − ≤ ǫ, cf. Remark 5.3. Finally, let δ = α ∆( G ) . Recall that U ( δ ) is the polydisc of radii δ centered at the matrix of all 1’s, cf.Definition 5.1.We prove by descending induction for r = | V | , | V | − , . . . , r )–(6.5. r ) are satisfied for all Z ∈ U ( δ ).(6.1. r ) Let W be an admissible sequence of vertices and let I be an admissiblesequence of indices such that | W | = | I | = r . Then Q WI ( Z ) = 0.(6.2. r ) Let W be an admissible sequence of vertices and let I be an admissiblesequence of indices such that | W | = | I | = r . Assuming that W = ( W ′ , v ) and I = ( I ′ , i ), we have (cid:12)(cid:12) Q WI ( Z ) (cid:12)(cid:12) ≥ τ ∆( G ) X w : { w,v }∈ El : 1 ≤ l ≤ k | z vwil | (cid:12)(cid:12)(cid:12)(cid:12) ∂∂z vwil Q WI ( Z ) (cid:12)(cid:12)(cid:12)(cid:12) . (6.3. r ) Let W be an admissible sequence of vertices and let I be an admissiblesequence of indices such that | W | = | I | = r . Assuming that W = ( W ′ , u, v ) and I = ( I ′ , j, i ), we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q ( W ′ ,u,v )( I ′ ,j,i ) ( Z ) Q ( W ′ ,u,v )( I ′ ,i,j ) ( Z ) − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ. (6.4. r ) Let W be an admissible sequence of vertices and let I be an admissiblesequence of indices such that | W | = | I | = r −
1. Let w be a vertex such that thesequence ( W, w ) is admissible and let i and j be indices such that the sequences( I, i ) and (
I, j ) are admissible. Then Q ( W,w )( I,i ) ( Z ) = 0, Q ( W,w )( I,j ) ( Z ) = 0 and the anglebetween the complex numbers Q ( W,w )( I,i ) ( Z ) and Q ( W,w )( I,j ) ( Z ) does not exceed θ .(6.5. r ) Let W be an admissible sequence of vertices and let I be an admissiblesequence of indices such that | W | = | I | = r −
1. Let u and v be vertices such thatthe sequences ( W, u ) and (
W, v ) are admissible and let i be an index such that thesequences ( I, i ) is admissible. Then Q ( W,u )( I,i ) ( Z ) = 0, Q ( W,v )( I,i ) ( Z ) = 0 and the anglebetween the complex numbers Q ( W,u )( I,i ) ( Z ) and Q ( W,v )( I,i ) ( Z ) does not exceed θ .Let W and I be admissible sequences such that | W | = | I | = r , so that W =( v , . . . , v r ) and I = ( i , . . . , i r ). If r = | V | then Q WI ( Z ) = Y ≤ j 22o (6.1. r ) holds for r = | V | . Moreover, denoting by deg ( v r ) the degree of v r , weobtain in this case X w : { w,v }∈ El : 1 ≤ l ≤ k | z vwil | (cid:12)(cid:12)(cid:12)(cid:12) ∂∂z vwil Q WI ( Z ) (cid:12)(cid:12)(cid:12)(cid:12) = deg ( v r ) (cid:12)(cid:12) Q WI ( Z ) (cid:12)(cid:12) , so (6.2. r ) holds as well. Lemma 5.2 then implies that (6.3. r ) holds for r = | V | , while(6.4. r ) and (6.5. r ) hold trivially, since for W ′ = ( v , . . . , v r − ) the only admissi-ble extension is W = ( v , . . . , v r ) and for I ′ = ( i , . . . , i r − ) the only admissibleextension is I = ( i , . . . , i r ). Hence (6.1. r )–(6.5. r ) are satisfied for r = | V | .From formula (3.2.1) and Lemma 4.1 we get the implication(6 . .r ) and (6 . .r ) = ⇒ (6 . .r − . From Lemma 5.4 we get the implication(6 . .r ) = ⇒ (6 . .r − . From Lemma 5.2 we get the implication(6 . .r − 1) and (6 . .r − 1) = ⇒ (6 . .r − . From Part (1) of Proposition 4.3 we get the implication(6 . .r ) and (6 . .r ) = ⇒ (6 . .r − . From Part (2) of Proposition 4.3 we get the implication(6 . .r ) and (6 . .r ) = ⇒ (6 . .r − . This proves (6.1.2)–(6.5.2). From Part (2) of Proposition 4.3, we get the implication(6 . . 2) and (6 . . 2) = ⇒ (6 . . . 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