A full complexity dichotomy for immanant families
AA full complexity dichotomy for immanant families
Radu Curticapean ∗ February 9, 2021
Abstract
Given an integer n ≥ and an irreducible character χ λ of S n for some partition λ of n , the immanant imm λ : C n × n → C maps matrices A ∈ C n × n to imm λ ( A ) = (cid:88) π ∈ S n χ λ ( π ) n (cid:89) i =1 A i,π ( i ) . Important special cases include the determinant and permanent , which are the immanants associatedwith the sign and trivial character, respectively.It is known that immanants can be evaluated in polynomial time for characters that are “close” to thesign character: Given a partition λ of n with s parts, let b ( λ ) := n − s count the boxes to the right of thefirst column in the Young diagram of λ . For a family of partitions Λ , let b (Λ) := max λ ∈ Λ b ( λ ) and write Imm(Λ) for the problem of evaluating imm λ ( A ) on input A and λ ∈ Λ .• If b (Λ) < ∞ , then Imm(Λ) is known to be polynomial-time computable. This subsumes the case ofthe determinant.• If b (Λ) = ∞ , then previously known hardness results suggest that Imm(Λ) cannot be solved inpolynomial time. However, these results only address certain restricted classes of families Λ .In this paper, we show that the parameterized complexity assumption FPT (cid:54) = W [ ] rules out polynomial-time algorithms for Imm(Λ) for any computationally reasonable family of partitions Λ with b (Λ) = ∞ .We give an analogous result in algebraic complexity under the assumption VFPT (cid:54) = VW [ ] . Furthermore,if b ( λ ) even grows polynomially in Λ , we show that Imm(Λ) is hard for P and VNP . This concludes aseries of partial results on the complexity of immanants obtained over the last 35 years. ∗ IT University of Copenhagen, Basic Algorithms Research Copenhagen. Supported by VILLUM Foundation grant [email protected] a r X i v : . [ c s . CC ] F e b Introduction
The determinant and permanent of an n × n matrix X = ( x i,j ) can be defined by the sum-product formulas det( X ) = (cid:88) π ∈ S n sgn( π ) n (cid:89) i =1 x i,π ( i ) , per( X ) = (cid:88) π ∈ S n n (cid:89) i =1 x i,π ( i ) . The similarity between these formulas is deceiving: While determinants admit polynomial-size circuits andcan be evaluated in polynomial time, only exponential-size circuits and exponential-time algorithms are knownfor permanents. Valiant [37] underpinned this divide by proving that evaluating permanents is P -hard: Anypolynomial-time algorithm for this problem would entail a polynomial-time algorithm for counting (and thusdeciding the existence of) satisfying assignments to Boolean formulas, thereby collapsing P and NP . With the
VNP -completeness of the permanent family [36], an analogous statement holds in algebraic complexity theory.Unconditional lower bounds for the complexity of permanents however remain elusive, with only aquadratic lower bound on the determinantal complexity of permanents known [28, 12]. That is, expressingthe permanent of an n × n matrix X as the determinant of an m × m matrix (whose entries are linear formsin the entries of X ) is known to require m = Ω( n ) . One of the core objectives in algebraic complexity theorylies in proving that m must grow super-polynomially [36, 11, 8], and this can be viewed as an algebraicversion of the P (cid:54) = NP problem. The family of immanants.
To understand the relationship between determinants and permanents better,it may help to recognize them as part of a larger family: The immanants are matrix forms that are arrangedon a spectrum in which the determinant and permanent represent extreme cases. These forms were studiedby Schur [32, 31] in the context of group character theory, and Littlewood and Richardson later explicitlyintroduced them as immanants [24].Given any class function f : S n → C , i.e., a function of permutations that depends only on the (multisetof) cycle lengths of the input permutation, the immanant imm f : C n × n → C is defined by replacing thepermutation sign sgn( π ) in the determinant expansion with f ( π ) : imm f ( X ) = (cid:88) π ∈ S n f ( π ) n (cid:89) i =1 x i,π ( i ) . In the literature, immanants are typically defined by requiring f to be an irreducible character of S n , i.e.,an element from a particular basis for the vector space of class functions. General f -immanants can then beexpressed as linear combinations of such immanants. Two extremal examples of irreducible characters are the trivial character : S n → { } and the sign character sgn : S n → {− , } , which induce det( X ) = imm sgn ( X ) , per( X ) = imm ( X ) . In general, the irreducible characters of S n correspond naturally to partitions of n , as outlined in Section 3.To see the existence of such a correspondence, note that the dimension of the space of class functions on S n is the number of different cycle length formats of n -permutations, that is, different partitions of the integer n .For now, let us remark that the refinement-wise minimal and maximal partitions (1 , . . . , and ( n ) naturallycorrespond to the sign and trivial character, respectively. As another example, we have χ (2 , ,..., ( π ) = sgn( π ) · ( { fixed points of π } − . (1) The resulting immanants are sometimes also called character immanants, as opposed to other types of immanants, such asthe Kazhdan-Lusztig immanants [29]. imm λ = imm χ λ , we have det = imm (1 ,..., and per = imm ( n ) . Likewise, imm (2 , ,..., sumsover row-column permutations of a matrix with weights as given in (1). For a more applied example, it isknown that the number of Hamiltonian cycles in a directed n -vertex graph G , i.e., the immanant associatedwith the indicator function for cyclic permutations (evaluated on the adjacency matrix of G ) can be writtenas a linear combination of the hook immanants imm ( r, n − r ) for ≤ r ≤ n .Beyond their theoretical origins in group character theory, immanants have been applied in combinatorialchemistry [13] and linear optics [34], and they feature in (conjectured) inequalities in matrix analysis [33]. Inthis paper, we focus on complexity-theoretic aspects of immanants and their role as an interpolating familybetween determinants and permanents. The complexity of immanants.
It is known that irreducible characters of S n can be evaluated inpolynomial time [9, 8]. Using this, any character immanant of an n × n matrix can be evaluated in n ! · n O (1) time by brute-force, or in n + o ( n ) time by a variant of the Bellman–Held–Karp dynamic programming approachfor Hamiltonian cycles. For some immanants however, among them the determinant, this exponential runningtime is far from optimal: Hartmann [23] gave an algorithm for evaluating imm λ in O ( n b ( λ )+4 ) time, where b ( λ ) := n − s for a partition λ = ( λ , . . . , λ s ) with s parts. In visual terms, the quantity b ( λ ) counts theboxes to the right of the first column in the Young diagram of λ , which is a left-aligned shape whose i -th rowcontains λ i boxes, when λ is ordered non-increasingly: λ = (4 , , , , , with b ( λ ) = 13 In particular, partitions λ with b ( λ ) = O (1) induce polynomial-time solvable immanants. Barvinok [1]and Bürgisser [10] later gave algorithms with improved running times O ( n d λ ) and O ( n s λ d λ ) , where s λ and d λ denote the numbers of standard and semi-standard tableaux of shape λ . These algorithms givebetter running times in the exponential-time regime, but they do not identify new polynomial-time solvableimmanants. One is therefore naturally led to wonder whether b ( λ ) is indeed the determining parameter forthe complexity of immanants. To investigate this formally, we consider families of partitions Λ and define Imm(Λ) as the problem of evaluating imm λ ( A ) on input a matrix A and a partition λ ∈ Λ . As discussedabove, the problem Imm(Λ) is polynomial-time solvable if the quantity b (Λ) := max λ ∈ Λ b ( λ ) is finite. On the other hand, for various families Λ with unbounded b (Λ) , the problem Imm(Λ) is indeedknown to be hard for the counting complexity class P and its algebraic analog VNP :• Bürgisser [9] showed
VNP -completeness and P -hardness of Imm(Λ) for any family Λ of hook partitions ( t ( n ) , n − t ( n ) ) , provided that t = Ω( n α ) with α > can be computed in polynomial time. A similarresult appears in Hartmann’s work [23].• In the same paper, Bürgisser showed similar hardness results for families of rectangular partitions ofpolynomial width. (The width is the largest entry in the partition.)In his 2000 monograph [8], Bürgisser conjectures that Imm(Λ) is hard for any reasonable family Λ ofpolynomial width. He also asks about the complexity status of partitions of width , and overall deems thecomplexity of immanants to be “still full of mysteries”. Some of these mysteries have since been resolved: Given a partition λ of n , a standard tableau of shape λ is an assignment of the numbers , . . . , n to the boxes in the Youngdiagram of λ such that all rows and columns are strictly increasing. In a semi-standard tableau, rows are only required to benon-decreasing. VNP -completeness for any family of partitions Λ with a gapof width Ω( n α ) for α > . Here, a gap is the difference between two consecutive rows.• In 2013, Mertens and Moore [27] proved P -hardness for the family Λ of all partitions of width , thatis, the partitions containing only entries and . They also proved ⊕ P -hardness for the more restrictedfamily of partitions containing only the entry .• In the same year, de Rugy-Altherre [20] gave a dichotomy for partition families Λ of constant width andpolynomial growth of b ( λ ) , confirming for such families that boundedness of b (Λ) indeed determines thecomplexity of Imm(Λ) .However, an exhaustive complexity classification of
Imm(Λ) for general partition families Λ still remained open,even 35 years after Hartmann’s initial paper [23] and despite several appearances as an open problem [27, 20],also in a monograph [8]. In fact, even very special cases like Imm(Λ) for the staircase partitions ( k, k − , . . . , remained unresolved [20]. We classify the complexity of the problems
Imm(Λ) for partition families Λ satisfying natural computabilityand density conditions that are satisfied by all families studied in the literature. Under the assumption FPT (cid:54) = W [ ] from parameterized complexity [21], we confirm that Imm(Λ) is polynomial-time solvable iff b (Λ) is unbounded. An algebraic analogue holds under the assumption VFPT (cid:54) = VW [ ] introduced by Bläserand Engels [6]. (Please consider Section 2.2 for a brief introduction to the relevant complexity classes.) Ourclassification holds even if b ( λ ) only grows sub-polynomially in Λ , which allows us to address families such as Λ log = { ( (cid:100) log n (cid:101) , n ) | n ∈ N } . (2)Note that Imm(Λ log ) can be solved in n O (log n ) time by the n O ( b ( λ )) time algorithms discussed before, whichlikely prevents hardness for P or VNP . At the same time, a polynomial-time algorithm seems unlikely.Thus, partition families like
Imm(Λ log ) fall into the “blind spot” of classical dichotomies, an issue that is alsoalluded to in [20].Our sanity requirements on Λ are encapsulated as follows: We say that Λ supports growth g : N → N ifevery n ∈ N admits a partition λ ( n ) ∈ Λ with b ( λ ( n ) ) ≥ g ( n ) and total size Θ( n ) . This ensures that Λ isdense enough and that Λ supplies sufficiently many boxes both in the first column and to the right of it.We may also require that λ ( n ) can be computed in polynomial time on input n ∈ N and then say that Λ computationally supports growth g . This condition is not required for the algebraic completeness results. Example.
The family of staircase partitions ( n, n − , . . . , supports growth Ω( n ) . The partition families ( (cid:100) log n (cid:101) , n ) and ( n, n ) for n ∈ N support growth Ω(log n ) , even though the second family is exponentiallysparse. On the other hand, partition families whose sizes grow doubly exponentially do not support anygrowth by our definition. It might still be possible to address such families via “infinitely often” versions of P or VNP , but we currently see no added value in doing so.In the polynomial growth regime for b ( λ ) , we obtain classical P -hardness and VNP -completeness results.As a bonus, we also obtain the expected quantitative lower bounds under the exponential-time hypothesis
ETH , which postulates that counting satisfying assignments to n -variable -CNFs takes exp(Ω( n )) time. Theorem 1.
For any family of partitions Λ : • If b (Λ) < ∞ , then Imm(Λ) ∈ FP and Imm(Λ) ∈ VP . • Otherwise, if Λ supports growth Ω( n α ) for some α > , then Imm(Λ) is VNP -complete. If Λ compu-tationally supports growth Ω( n α ) , then Imm(Λ) is P -hard and admits no exp( o ( n α )) time algorithmunless ETH fails.
VNP -hardness and P -hardness results for immanant families, confirmsthe conjecture from [27], and settles the case of staircases. Using parameterized complexity theory, wealso address the sub-polynomial growth regime for b ( λ ) . To this end, we consider parameterized problems ,whose instances ( x, k ) come with a parameter k . The corresponding objects in the algebraic setting are parameterized polynomial families ( p n,k ) , where the second index k is a parameter. A parameterized problem(or polynomial family) is said to be fixed-parameter tractable if it can be solved in f ( k ) · n O (1) time with n = | x | (or admits circuits of that size) for some computable function f . The problem (or polynomial family)is then said to be contained in FPT (or
VFPT ). The classes W [ ] ⊇ FPT (and VW [ ] ⊇ VFPT ) containproblems (and polynomial families) that are believed not to be fixed-parameter tractable.
Theorem 2.
For any family of partitions Λ : • If b (Λ) < ∞ , then Imm(Λ) ∈ FP and Imm(Λ) ∈ VP . • Otherwise, if Λ supports growth g ∈ ω (1) , then Imm(Λ) / ∈ VP unless VFPT = VW [ ] . If Λ computation-ally supports growth g , then Imm(Λ) / ∈ FP unless FPT = W [ ] . Note that we do not prove
Imm(Λ) to be hard for W [ ] or VW [ ] as a parameterized problem withparameter b ( λ ) , even though this might seem natural. Indeed, problems like Imm(Λ log ) are trivially fixed-parameter tractable in the parameter b ( λ ) . We only show that polynomial-time algorithms for Imm(Λ) wouldrender W [ ] -hard problems fixed-parameter tractable. We establish Theorems 1 and 2 by reduction from , the problem of counting k -matchings in bipartitegraphs H . When parameterized by k , this problem is W [ ] -complete [4, 15, 19, 18], with an analogousstatement in the algebraic setting [6]. When k grows polynomially in | V ( H ) | , counting k -matchings iscomplete for P and VNP by a trivial reduction from the permanent [37].To reduce counting matchings to immanants, we proceed in three stages: First, we identify two types of“exploitable resources” in partitions, then we show how to exploit them for a reduction, and finally we wrapthe proof up in complexity-theoretic terms.
Extracting resources (Section 4)
Our construction relies on two types of resources that can supplied by a given partition: A large staircase ora large number of non-vanishing tetrominos .To define these notions, consider successively “peeling” dominos and from λ , that is, removing themfrom the south-eastern border of λ while ensuring that the shape obtained after each step has non-increasingrow lengths. After peeling the maximum number of dominos this way, we reach some (possibly empty)staircase µ , which is easily seen to be unique. The domino number d ( λ ) is this maximum number of removabledominos, and µ is the staircase of λ ; we write w ( λ ) for its width.Now consider peeling two dominos from a partition λ . Some of the shapes that can arise this way areshown above in orange; we call them “non-vanishing tetrominos” for reasons that will become evident in theproof. For the four corner-connected domino pairs, we adopt the convention that their two dominos must bepeeled successively from disjoint rows and columns of λ . The non-vanishing tetromino number s ( λ ) then is5he maximum number of non-vanishing tetrominos that can be peeled from λ . Note that this number is forthe determinant-inducing partition (1 , . . . , and (cid:98) n/ (cid:99) for the permanent-inducing partition ( n ) .In Section 4, we establish a “win-win situation” for these resources: For any partition λ , at least one of w ( λ ) ∈ Ω( (cid:112) b ( λ )) or s ( λ ) ∈ Ω( b ( λ )) must hold. Exploiting resources (Sections 5 and 6)
Next, we outline how to exploit staircases and non-vanishing tetrominos in a partition λ for reductions fromcounting k -matchings to evaluating imm λ . Throughout this paper, the immanant of a directed graph G refersto the immanant of its adjacency matrix A . This way, we can view immanants as character-weighted sumsover the cycle covers of digraphs.Given an n -vertex graph H and k ∈ N , we construct a digraph G such that imm λ ( G ) counts the k -matchings in H up to a constant factor c λ,k that can be computed in polynomial time. In a second step, weshow that the constant factor c λ,k is non-zero if λ supplies enough resources. Both the construction of G andthe handling of c λ,k differ for staircases and tetrominos, as outlined below. Non-vanishing tetrominos (Section 6)
Each non-vanishing tetromino peeled from λ enables a particular edge gadget : To count the k -matchings in a graph H , we replace each edge uv ∈ E ( H ) by the gadget shownbelow; the weight w of uv appears on two edges of the gadget. Together with additional constructions detailedin Section 6, this results in a directed graph G . − u vw w The edge gadget effectively constrains the set of cycle covers counted by the immanant, as undesired cyclecovers cancel out in pairs of opposite signs. In the remaining cycle covers of G , each gadget is either in the passive state (shown below in cyan) or in one of four active states (two are shown below in green, two moreare symmetric versions thereof).This allows us to simulate matchings M in H via cycle covers in G : We interpret active gadgets asmatching edges e ∈ M and passive gadgets as edges e ∈ E ( H ) \ M . Intuitively speaking, each active gadget“uses up” one non-vanishing tetromino of λ , while passive gadgets only require a domino. Overall, if we canpeel O ( k ) non-vanishing tetrominos and some number of dominos from λ , then imm λ can be used to count k -matchings in H . Large staircase (Section 5)
If the staircase µ of λ contains Ω( k ) boxes, then properties of the staircasecharacter χ µ enable an ad-hoc reduction from counting k -matchings in bipartite graphs to the λ -immanant.More specifically, we observe and use that cycle covers with even cycles vanish in staircase characters χ µ .After discarding irrelevant dominos from λ , we can then use this fact together with a particular graphconstruction to compute a sum over cycle covers with one particular fixed cycle length format by reduction to imm λ . This way of exploiting staircase characters to avoid even-length cycles may also be useful for otheralgorithmic applications. 6 rap-up (Section 7) For a streamlined presentation, the two reductions above are encapsulated as mere mathematical formulasrelating the number of k -matchings in a graph H with the immanant of a digraph G constructed from H . InSection 7, we add the necessary “wrapper code” to obtain the (parameterized and polynomial-time, algebraicand computational) reductions required to prove Theorems 1 and 2. Most of our arguments rely on making non-vanishing tetrominos and staircases come together in just theright way. Additionally, the following can be pointed out:• In the tetromino-based reduction, the particular form of active and passive states in edge gadgets ensuresthat we only need to understand character values χ λ ( ρ ) on cycle formats ρ with cycle lengths , , and . This allows us to sidestep more involved representation-theoretic arguments that occur in relatedworks. It should be noted that equality and exclusive-or gadgets that enforce consistency properties ofcycle covers are common in algebraic and counting complexity, dating back to Valiant [36, 37]. Theparticular idea of repurposing an equality gadget into an edge gadget was also already used in theauthor’s very first paper with Bläser [3].• Parameterized complexity assumptions allow us to handle cases that cannot be addressed in classicalframeworks, such as the family Imm(Λ log ) discussed before. By basing hardness on the assumptions FPT (cid:54) = W [ ] and VFPT (cid:54) = VW [ ] , we can still argue about such families.• Two inconspicuous but crucial proof steps (Lemmas 22 and Fact 29) rely on cute arguments involving“dominos on chessboards” that one would rather expect in the context of recreational mathematics.This can again be credited to the edge gadget, as such arguments would likely fail for cycle lengthsother than and . We start with basic definitions for graphs and partitions in Section 2.1. In Sections 2.2 and 2.3, we presentthe complexity-theoretic preliminaries used in this paper. A bare-bones introduction to the relevant charactertheory of symmetric groups can be found in Section 3.
Graphs.
We consider undirected graphs (when counting matchings) and directed graphs (when evaluatingimmanants). Both graph types may feature (indeterminate) edge-weights, and directed graphs may featureself-loops. For a graph G with adjacency matrix A , we write imm( G ) instead of imm( A ) and view theimmanant as a sum over cycle covers: A partial cycle cover in a directed graph G is an edge-set C ⊆ E ( G ) such that each vertex has one incoming and one outgoing edge in C . It is a cycle cover if all of V ( G ) iscovered. Directed -cycles will also be called digons . Partitions. A partition of a positive integer n ∈ N is a multi-set λ of positive integers such that (cid:80) i ∈ λ i = n .Its elements are called parts , and we write λ (cid:96) n to indicate that λ is a partition of n . Several notations willbe used for partitions: sorted tuple compact notation Young diagram (4 , , , , ,
2) (4 , , )
7s depicted above, the
Young diagram of a partition λ = ( λ , . . . , λ s ) is a left-aligned shape consistingof λ i boxes in row i . We define the gap δ i of row i as δ i := λ i − λ i +1 , where we consider λ s +1 := 0 .Given partitions λ (cid:96) n and λ (cid:48) (cid:96) n (cid:48) , we sometimes abuse notation and write ( λ, λ (cid:48) ) for the partition λ ∪ λ (cid:48) of n + n (cid:48) . Sometimes we also specify an ordering for the elements in a partition: An ordered partition of n (also called composition ) is a tuple of positive integers that sum to n . We will state explicitly whenpartitions are considered to be ordered.Let us stress that any permutation π of n elements (and any cycle cover C of an n -vertex graph) naturallyinduces a partition λ (cid:96) n through its cycle lengths, the cycle format ρ ( π ) of π . For example, the identitypermutation has cycle format (1 , . . . , and cyclic permutations have cycle format ( n ) . We also say that acycle cover is a ρ -cycle cover if its format is ρ . We recall basic notions from complexity theory. For a more comprehensive overview, please consider [37, 21, 17]for (parameterized) counting complexity and [36, 6, 11, 8] for (parameterized) algebraic complexity.
Counting complexity.
We view functions f : { , } ∗ → Q as counting problems . For example, theproblem maps (binary encodings of) Boolean formulas ϕ to the number of satisfying assignments in ϕ .Properly encoded, the permanent of rational-valued matrices is a counting problem. A counting problem iscontained in FP if it can be solved in polynomial time.A polynomial-time Turing reduction from a counting problem to another counting problem is a polynomial-time algorithm that solves with an oracle for . We say that is P -hard (under Turing reductions) if admits a polynomial-time Turing reduction to . While more stringentreduction notions exist, they are not relevant for the purposes of this paper. Assuming FP (cid:54) = P , no P -hardproblem can be solved in polynomial time.A parameterized counting problem features inputs ( x, k ) for x ∈ { , } ∗ and k ∈ N . It is fixed-parameter tractable if it can be solved in time f ( k ) · | x | O (1) for some computable function f , and we write FPT for the class of such problems. A parameterized Turing reduction from a parameterized countingproblem to another parameterized problem is an algorithm that solves any instance ( x, k ) for intime f ( k ) · | x | O (1) with an oracle for that is only called on instances ( y, k (cid:48) ) with k (cid:48) ≤ g ( k ) . Here, both f and g are computable functions. We say that is W [ ] -hard if the problem of counting k -cliques in agraph admits a parameterized Turing reduction to . Assuming FPT (cid:54) = W [ ] , no W [ ] -hard problem isfixed-parameter tractable.The (counting version of the) exponential-time hypothesis ETH postulates that no exp( o ( n )) timealgorithm solves on n -variable formulas. If a problem cannot be solved in exp( o ( n )) time, andit admits a polynomial-time Turing reduction to a problem such that every invoked oracle query hassize O ( n c ) for c ≥ , then cannot be solved in exp( o ( n /c )) time. Likewise, if a parameterized problem cannot be solved in f ( k ) · n o ( k ) time and it admits a parameterized Turing reduction to a problem such that every invoked query ( y, k (cid:48) ) satisfies k (cid:48) ≤ O ( k ) , then also cannot be solved in f ( k ) · n o ( k ) time. The hypothesis ETH rules out an f ( k ) · n o ( k ) time algorithm for counting k -cliques and thus implies FPT (cid:54) = W [ ] . Algebraic complexity.
In the algebraic setting, p-families play the role of counting problems: A sequenceof multivariate polynomials f = ( f , f , . . . ) over some field is a p-family if, for all n ∈ N , the degree andthe number of variables of f n are bounded by n O (1) . For example, the sequences of determinants andpermanents of n × n matrices with indeterminates are p-families. A p-family is contained in VP if it admits apolynomial-size arithmetic circuit.A p-family f admits a c-reduction to another p-family g if there is an arithmetic circuit of polynomialsize that computes each f n with oracle gates for g , . . . g n O (1) . A p-family g is VNP -hard if the permanentfamily admits a c-reduction to g . Assuming VP (cid:54) = VNP , no
VNP -hard family is contained in VP .8 parameterized p-family is a family f = ( f n,k ) n,k ∈ N with two indices such that f n,k uses n O (1) variables and has degree ( n + k ) O (1) . A family f is contained in VFPT if it admits an arithmetic circuitof size h ( k ) · n O (1) for some function h . An example for a parameterized p-family is given by the partialpermanents per n,k = (cid:80) π ∈ S ( n,k ) (cid:81) ni =1 x i,π ( i ) for n, k ∈ N , where S ( n, k ) is the set of all permutations with n − k fixed points. A parameterized family f = ( f n,k ) n,k ∈ N admits a parameterized c-reduction to anotherfamily g = ( g n,k ) n,k ∈ N if there is an arithmetic circuit of size h ( k ) · n O (1) that computes f n,k with oraclegates for polynomials g n (cid:48) ,k (cid:48) satisfying n (cid:48) ≤ h ( k ) · n O (1) and k (cid:48) ≤ h ( k ) for some function h . We say that g is VW [ ] -hard if the partial permanents admit a parameterized c-reduction to g . Assuming VFPT (cid:54) = VW [ ] , no VW [ ] -hard problem is contained in VFPT . A matching in an undirected graph H is a set M ⊆ E ( H ) of pairwise disjoint edges. We write M k ( H ) forthe set of matchings with k edges in H . A matching M is a perfect matching if every vertex v ∈ V ( H ) iscontained in some edge of M . Given an n -vertex graph H with edge-weights w : E ( H ) → Q , we define H, k ) = (cid:88) M ∈M k ( H ) (cid:89) e ∈ M w ( e ) , H ) = H, | V ( H ) | / . Note that H ) is only defined for graphs H with an even number of vertices. We prove hardnessof immanants by reduction from (restrictions of) the problem of evaluating on bipartite graphs. Definition 3.
The counting problem asks to compute H ) for bipartite graphs H with edge-weights w : E ( H ) → Q .For any fixed polynomial-time computable function g : N → N , we define the counting problem ( g ) :Given a pair ( H, k ) consisting of an undirected bipartite graph H with edge-weights w : E ( H ) → Q and anumber k ≤ g ( | V ( H ) | ) , compute H, k ) .On the complete bipartite graphs K n,n with indeterminate edge-weights, induces the p-familyof permanent polynomials, and ( g ) likewise induces a restriction of the partial permanent family with ( g ) n,k = 0 for k > g ( n ) . Abusing notation, we call these p-families and ( g ) as well.Note that no graphs are given as inputs to these families; the numbers of k -matchings for given bipartitegraphs H can be obtained by evaluating the polynomials at points whose non-zero coordinates encode theedges of H .The hardness results for and ( g ) required in the remainder of the paper are eitherknown in the literature or can be derived easily. We collect the relevant results below. Theorem 4.
The following holds:1. The problem is VNP -complete and P -hard and admits no o ( n ) time algorithm under ETH , even on bipartite graphs of maximum degree .2. For any unbounded and polynomial-time computable function g , the problem ( g ) is W [ ] -hardand VW [ ] -complete.Proof. See [37, 36] for the P -hardness and VNP -completeness of and [16] for the lower boundunder
ETH . The W [ ] -hardness of the cardinality-unrestricted problem is shown in [19], andand VW [ ] -completeness is shown in [6, Lemma 8.7].For any polynomial-time computable function g , we give a parameterized reduction from to ( g ) : Any instance ( H, k ) with an n -vertex graph and k ≤ g ( n ) can be solved directly with a call to ( g ) . On the other hand, if k > g ( n ) , then we have n < g − ( k ) , so we can count k -matchings in H bybrute-force in g (cid:48) ( k ) time for some computable function g (cid:48) . This satisfies the requirements of a parameterized9eduction. In the algebraic setting, we need not branch on g ( n ) within a circuit, but instead hard-code intothe circuit family realizing whether to (i) perform a brute-force sum over matchings, or (ii) use theoracle gate for ( g ) . We give a minimal introduction to representations and characters of the symmetric group; this materialis covered thoroughly in classical textbooks [22, 26, 30, 25]. For our purposes, a representation of S n isa homomorphism f from S n to the group GL t ( C ) of invertible t × t matrices, for some dimension t ∈ N .Examples include the trivial representation that maps all of S n to , the sign representation that mapspermutations to their sign, and the permutation matrix representation that maps each permutation to its n × n permutation matrix.A representation f : S n → GL t ( C ) is irreducible if no proper subspace of C t is invariant under all thetransformations f ( π ) for π ∈ S n . Among the examples given before, this holds trivially for the trivial andsign representations. The permutation matrix representation however is not irreducible for n > , as everypermutation matrix maps the -dimensional subspace of C n spanned by (1 , . . . , to itself. Characters condense essential information about representations f : S n → GL t ( C ) into scalar-valued functions χ f : S n → C . For us, they play the role of “generalized signs” in the sum-product definition of immanants. Definition 5.
The character of a representation f is the function χ f : S n → C that maps π ∈ S n to thetrace of the matrix f ( π ) .The trivial and sign representations coincide trivially with their characters. The character of thepermutation matrix representation counts the fixed points of a permutation.Characters of representations are class functions , which are functions f : S n → C that depend only onthe cycle format of the input. These functions form a vector space by point-wise linear combinations, and aparticularly useful basis for this space is given by the irreducible characters , which are the characters ofirreducible representations. The set of irreducible characters corresponds bijectively to the partitions of n .For S , the only irreducible characters are the trivial character χ (2) and sign character χ (1 , . There arefive irreducible characters for S ; their values χ λ ( ρ ) are shown below as a character table. ρ (1 ) (2 , ) (2 ) (3 , ) (4 ) λ (4 ) 1 1 1 1 1(3 , ) 3 1 − − ) 2 0 2 − , ) 3 − − ) 1 − − In the next subsections, we describe the Murnaghan-Nakayama rule, a combinatorial method for calculatingcharacter values χ λ ( ρ ) , together with a simple extension thereof that applies to particular linear combinationsof character values. To state these rules, we need to introduce several types of tableaux . Recall that partitions λ can be described by Young diagrams, such asGiven such a diagram, a tableau is obtained by writing numbers (or other objects) into the boxes, subjectto some specified rules. The representation theory of S n abounds in different types of tableaux—we introduceyet another such type, the skew shape tableaux. , , ) . Definition 6.
Let λ, µ be partitions such that µ i ≤ λ i for all rows i of λ . We say that µ is contained in λ and define the skew shape λ/µ by removing the diagram of µ from λ .Consider the two examples below. The right example shows that skew shapes need not be connected; wecall a skew shape connected if each pair of boxes can be reached by a path in the interior of the shape. (4 , , ) / (2 , ) (4 , , ) / (3 , ) Table 1 lists the connected skew shapes on boxes. Any general skew shape on b boxes is obtained bychoosing connected skew shapes with a total of b boxes and arranging them in disjoint rows and columns.We will often peel skew shapes γ from other skew shapes λ/µ . Our definition of this process allows forpeeling different components of γ from different places. Definition 7.
A skew shape γ can be peeled from λ/µ if there is a partition λ (cid:48) contained between µ and λ such that λ/λ (cid:48) equals γ after deleting empty rows and columns from both λ/λ (cid:48) and γ .A skew shape tableau is obtained by successively peeling skew shapes from λ . Definition 8.
Let ˜Γ = (Γ , . . . , Γ s ) be such that Γ i for i ∈ [ s ] is a set of skew shapes on the same number n i of boxes. Let λ be a partition of n = (cid:80) i n i . A skew shape tableau of λ with format ˜Γ is obtained bysuccessively peeling skew shapes γ , . . . , γ s with γ i ∈ Γ i from λ and labeling γ i with i in λ . We write S ( λ, ˜Γ) for the set of such tableaux.Three skew shape tableaux with different formats are shown below. The boxes are colored rather thannumbered. The third tableau is even a border strip tableau, as defined in the next subsection. The Murnaghan-Nakayama rule expresses the character value χ λ ( ρ ) for partitions λ and ρ as a signed sumover particular skew shape tableaux of λ . The sign of a tableau is determined by the parity of odd-heightshapes. In our later arguments, ρ will always be the format of a cycle cover. Definition 9. A border strip is a connected skew shape not containing any × square. A border striptableau of λ (cid:96) n is a skew shape tableau consisting only of border strips. Given a partition λ and an orderedpartition κ , we write B ( λ, κ ) for the set of border strip tableaux of λ in which the i -th shape has κ i boxes.Table 1: All connected skew shapes on boxes.11he height ht( γ ) of a border strip γ is the number of occupied rows in γ minus . Given a skew shapetableau T with skew shapes γ , . . . , γ s , the height sign ht( T ) is defined as ht( T ) = (cid:81) si =1 ( − ht( γ i ) . For the border strip tableau in the above example, the heights are (green), (yellow), (red), and (blue), resulting in an overall height sign of +1 . Having established these preliminaries, we can state theMurnaghan-Nakayama rule. For proofs, we refer to textbooks [22, 25]. Theorem 10 (Murnaghan-Nakayama rule) . For partitions λ and ρ and any ordering κ of ρ , we have χ λ ( ρ ) = (cid:88) T ∈B ( λ,κ ) ht( T ) . Remark . When invoking this rule, it makes sense to choose a useful ordering κ of ρ . For example, to show χ (5 , , , , (1 , ) = 0 , we can• sum over a rather large number of border strip tableaux to observe that their signs cancel, or alternatively• reorder ρ = (1 , ) to κ = (2 , ) and realize directly that no border strip on boxes can be peeledfrom the border of the staircase (5 , , , , .More generally, this observation shows that χ µ ( ρ ) = 0 whenever µ is a staircase and ρ is a partition thatcontains at least one even part.The above remark will be crucial for the staircase-based reduction in Section 5. For the tetromino-basedreduction in Section 6, we extend Theorem 10 to character evaluations on products of partition sets. Definition 12.
Let F , . . . , F t be sets of partitions such that each set F i collects partitions of the same integer d i . The partition product F × . . . × F t is the multi-set consisting of the (cid:81) i | F i | partitions of d + . . . + d t obtained by choosing one partition from each set F i and concatenating those t partitions.We linearly extend class functions f to multi-sets S of partitions by declaring f ( S ) = (cid:80) ρ ∈ S f ( ρ ) andshow how to calculate χ λ ( F × . . . × F t ) combinatorially by an extension of the Murnaghan-Nakayama rule.To this end, we define admissible skew shapes for each F i , each with a particular coefficient. In analogy withthe original rule, admissible skew shapes play the role of border strips, and the coefficients of admissible skewshapes play the role of heights of border strips. Definition 13.
Given a set of partitions F , let Γ F be the set of all skew shapes γ that admit a border striptableau T ∈ B ( γ, ρ ) for some ρ ∈ F . For γ ∈ Γ F , we define the coefficient α F ( γ ) = (cid:88) ρ ∈ F (cid:88) T ∈B ( γ,ρ ) ht( T ) . In preparation of Section 6, we exemplify this definition with the cycle formats of active edge gadgets.For F = { (2 ) , (4) } , we observe that the set Γ F consists of all skew shapes that can be covered with twodisjoint dominos. Note that there are shapes γ ∈ Γ F with α F ( γ ) = 0 : For example, the vertical -box lineadmits a border strip tableau for (2 ) and one for (4) , shown below. These tableaux have opposite heightsigns and thus cancel. ht( T ) = 1 ht( T (cid:48) ) = − ⇒ α F ( γ ) = 0 -box shapes γ with α F ( γ ) (cid:54) = 0 serve as a resourcefor establishing hardness of immanants. If we can peel many such shapes from a partition λ , then we canreduce a large permanent to the λ -immanant. Thus, if the vertical -box line γ satisfied α F ( γ ) (cid:54) = 0 , ourreductions would allow us to establish hardness of the determinant.With all relevant notions introduced, we can now turn to our generalization of the Murnaghan-Nakayamarule. The proof is almost syntactic; it essentially requires us to group border strips into skew shapes andcollect terms accordingly. Nevertheless, a little care is required to ensure that the grouped border strips areindeed proper skew shapes. Lemma 14.
Let λ be a partition. Given a partition product F × . . . × F t , abbreviate Γ i = Γ F i and α i = α F i for i ∈ [ t ] . Writing ˜Γ = (Γ , . . . , Γ t ) , we have χ λ ( F × . . . × F t ) = (cid:88) S ∈S ( λ, ˜Γ)with shapes γ ...γ t t (cid:89) i =1 α i ( γ i ) . (3) Proof.
Write k i := | F i | and enumerate F i = { ρ i, , . . . , ρ i,k i } for i ∈ [ t ] . We write A = [ k ] × . . . × [ k t ] for theset of multi-indices into F × . . . × F t . For any multi-index a ∈ A , we define the ordered partition ρ a := ( ρ ,a (1) , . . . , ρ t,a ( t ) ) by concatenating the partitions ρ i,a ( i ) for i ∈ [ t ] . Here, the order within ρ i,a ( i ) is not relevant, but thepartitions need to be concatenated in the specified order. We then have χ λ ( F × . . . × F t ) = (cid:88) a ∈ A (cid:88) T ∈B ( λ,ρ a ) ht( T ) . (4)Given a ∈ A and a border strip tableau T ∈ B ( λ, ρ a ) , we define the skew shape tableau S ( T ) by grouping,for each i ∈ [ t ] , the different border strips corresponding to block ρ i,a ( i ) into a skew shape. Since these borderstrips are peeled consecutively in T , the tableau S ( T ) is indeed a skew shape tableau, and it is contained in S ( λ, ˜Γ) by definition of the sets Γ , . . . , Γ t . We then partition the set of tableaux T ∈ B ( λ, ρ a ) according to S ( T ) and obtain (cid:88) a ∈ A (cid:88) T ∈B ( λ,ρ a ) ht( T ) = (cid:88) a ∈ A (cid:88) S ∈S ( λ, ˜Γ) (cid:88) T ∈B ( λ,ρ a ) with S ( T )= S ht( T ) . (5)Let us change the order of summation and sum over S ∈ S ( λ, ˜Γ) first. For a fixed skew shape tableau S with shapes γ , . . . , γ t , each border strip tableau T ∈ (cid:83) a ∈ A B ( λ, ρ a ) with S ( T ) = S is obtained by choosing aborder strip tableau independently for each of the shapes γ , . . . , γ t . We obtain (cid:88) a ∈ A (cid:88) T ∈B ( λ,ρ a ) with S ( T )= S ht( T ) = t (cid:89) i =1 k i (cid:88) j =1 (cid:88) T ∈B ( γ i ,ρ i,j ) ht( T ) = t (cid:89) i =1 α i ( γ i ) . By summing both sides over all S ∈ S ( λ, ˜Γ) and combining the result with (5) and (4), the lemma follows. We associate various quantities with partitions λ (cid:96) n to measure to what extent the λ -immanant lends itselfto a reduction from counting matchings. Then we investigate their interplay with the number b ( λ ) of boxesto the right of the first column of λ .The diagrams and of (2 ) and (1 ) will be denoted as domino s. The domino number d ( λ ) is themaximum number of dominos that can be peeled successively from λ . As we show below, the result of peeling13hese dominos from λ is a partition of the form µ = ( k, . . . , for some k ∈ N . Such partitions and theirassociated shapes are called staircases , and we define z ( λ ) = k + . . . + 1 as the staircase size and w ( λ ) = k asthe staircase width of λ . Note that d ( λ ) + z ( λ ) = n and that z ( λ ) can be zero; this happens when λ can becovered fully by dominos. Fact 15.
The shape µ obtained by peeling d ( λ ) dominos from λ is the unique staircase on z ( λ ) boxes.Proof. By maximality of d ( λ ) , no domino can be peeled from µ , so all gaps between consecutive rows in µ are . This requires µ to be a staircase. It has n − d ( λ ) = z ( λ ) boxes, and the number of boxes uniquelydetermines a staircase.We define s ( λ ) to be the maximum number of non-vanishing tetrominos (depicted on page 5 and Table 2)that can be peeled successively from λ . As we establish in Sections 5 and 6, any partition λ with large w ( λ ) or s ( λ ) induces an immanant to which the hard problem of counting k -matchings can be reduced. In thissubsection, we show that at least one of these numbers is large if b ( λ ) is large. Towards this, we first observethat partitions λ with s ( λ ) = 0 are strongly restricted. Lemma 16.
For any partition λ with s ( λ ) = 0 , at least one of the following holds:1. The shape λ is a staircase up to some number of additional boxes in the first column.2. There is an index i ∗ such that, after peeling a horizontal domino from each row i ≤ i ∗ , the remainingshape is the staircase of λ .Proof. There is at most one row i with gap δ i > : If δ i , δ i (cid:48) ≥ for i (cid:54) = i (cid:48) , then we could peel fromthese two rows. As these dominos share no rows or columns, they constitute a non-vanishing tetromino,contradicting s ( λ ) = 0 . Similarly, we see that the set of rows i with δ i = 0 , if non-empty, forms a contiguousinterval [ j, k ] : If the set contained two disjoint intervals, we could peel two from their ends.In the following, we distinguish whether λ contains a row with gap or not. If such a row exists, we showthat the first case of the lemma statement applies. Otherwise, we establish the second case.1. If there is a row i with gap δ i = 0 , let [ j, k ] (cid:51) i be the single interval of rows with gap . We rule outthe existence of a row i (cid:48) with δ i (cid:48) > , as it would lead to the following contradictions:• If i (cid:48) < j or i (cid:48) > k + 1 , then we could peel from row i (cid:48) and from rows k − and k . Thiscontradicts s ( λ ) = 0 .• If i (cid:48) = k + 1 , then we could peel from rows k and k + 1 .It follows that δ i (cid:48) = 1 for all i (cid:48) / ∈ [ j, k ] . Next, we show that λ i = 1 for all i ∈ [ j, k ] : If λ j = λ k > , thencould be peeled from rows k to k + 2 for a contradiction. (Indeed, we would have δ k +1 = 1 , since δ i (cid:48) = 1 for all rows i (cid:48) / ∈ [ j, k ] , and λ k > would that row k + 2 exists.) Hence, the first case of thelemma applies.2. Otherwise, we have δ i > for all rows i . As discussed in the first paragraph of the proof, there is atmost one row i ∗ with δ i ∗ > .• If δ i ∗ ≥ , we could peel from row i ∗ , contradicting s ( λ ) = 0 .• If δ i ∗ = 2 and row i ∗ + 1 exists, we could peel from rows i ∗ and i ∗ + 1 .It follows that δ i ∗ = 3 , or that i ∗ is the last row and δ i ∗ = 2 . As i ∗ is the only row with gap δ i ∗ (cid:54) = 1 ,the second case of the lemma applies.Since at least one of the two above cases applies, the lemma is proven.It follows that partitions λ with s ( λ ) = 0 contain only few boxes outside their first column and staircase. Corollary 17.
Let λ be a partition with s ( λ ) = 0 and staircase µ of width w ∈ N . Then b ( λ/µ ) ≤ w + 1 . roof. If the second case of Lemma 16 applies to λ , then b ( λ/µ ) ≤ w + 1 . If the first case applies, let µ (cid:48) denote the largest staircase contained in λ . (This need not be the staircase of λ .) Then λ is µ (cid:48) with somenumber a ∈ N of additional boxes in the first column. When peeling dominos from λ , the first dominos mustbe contained in the first column.• If a is even, then µ (cid:48) is the staircase of λ , so b ( λ/µ ) = 0 : Peeling from the first column results in µ (cid:48) .• If a is odd, then the lowest box of µ (cid:48) is contained in the last that can be peeled from the first column.We can then peel a horizontal domino from each of the remaining w + 1 rows. The remainder is thestaircase of λ . It follows that b ( λ/µ ) ≤ w + 1 .In both cases, we obtain b ( λ/µ ) ≤ w + 1 , thus proving the corollary.We are ready to prove the main lemma of this section. Lemma 18.
For any partition λ , at least one of s ( λ ) ≥ b ( λ ) / or w ( λ ) ≥ √ b − holds.Proof. Let µ be the staircase of λ and w = w ( λ ) . The number of boxes to the right of the first column in λ/µ is b ( λ/µ ) = b ( λ ) − b ( µ ) = b ( λ ) − (cid:0) w (cid:1) . By applying the contraposition of Corollary 17 repeatedly, it followsthat s ( λ ) ≥ b ( λ/µ ) − (2 w + 1)4 = b ( λ ) − (cid:0) w (cid:1) − (2 w + 1)4 = b ( λ )4 − w + 3 w + 28 . We obtain s ( λ ) + w + 3 w + 28 ≥ b ( λ )4 , and hence, at least one of the two terms is larger than b ( λ ) / . This implies the lemma. We show how to count k -matchings in n -vertex graphs H with access to the λ -immanant for a partition λ with staircase size z ( λ ) ∈ Ω( k ) . If k (cid:28) n , we also require a large domino number d ( λ ) . Our reduction relieson the intermediate problem of counting cycle covers with a particular “onion” format, outlined in Section 5.1,which we then reduce to the λ -immanant using a graph construction that has a favorable interplay withstaircase characters, as established in Section 5.2. We collect the steps in Section 5.3. We first describe a particular “onion partition” derived from λ , depicted in Figure 1. To define this partition,recall that w ( λ ) and z ( λ ) denote the staircase width and size of λ , respectively. Definition 19.
Let λ (cid:96) n (cid:48) be a partition with staircase µ . For (cid:96) ≤ w ( λ ) / , let θ be the partition obtainedby peeling (cid:96) maximal-length border strips from µ and recording their lengths. We write (cid:107) θ (cid:107) for the numberof boxes in θ . Then the (cid:96) -layer onion ρ of λ is the partition (2 d ( λ ) , θ, z ( λ ) −(cid:107) θ (cid:107) ) . For k ∈ N , we say that ρ accommodates k edges if (cid:107) θ (cid:107) ≥ k + (cid:96) .In words, the (cid:96) -layer onion ρ of λ is obtained as follows: First peel all d ( λ ) dominos from λ to expose thestaircase µ . Then peel (cid:96) maximal-length border strips from µ . Finally, peel the remaining z ( λ ) − (cid:107) θ (cid:107) boxesof µ as singletons. If ρ accommodates k edges, then the border strips contain k boxes and one extra box foreach peeled border strip; such partitions allow us to reduce counting k -matchings to counting ρ -cycle covers.Next, we describe two related ways of constructing a digraph G whose cycle covers of onion formatcorrespond to k -matchings in H . Given an edge-weighted digraph G , let G, ρ ) = (cid:88) ρ -cycle cover C in G (cid:89) e ∈ C w ( e ) . λ with d ( λ ) = 8 dominos, staircase width w ( λ ) = 10 and size z ( λ ) = 55 . The -layeronion ρ = (2 , , , ) of λ is shown as a border strip tableau; parts corresponding to dominos andsingletons are not distinguished in this figure. All border strip tableaux for ρ in λ have the depicted form.This onion accommodates edges; the two layers contain boxes, and one box per layer is required for“closing” the layer. xxxx xxxx Figure 2: The graphs from Lemma 20. To reduce clutter, edges from R to L are only hinted, and only onetransit vertex is shown. In the left graph, a cycle corresponding to a -matching is displayed in yellow color.Furthermore, we write [ x s ] p for the coefficient of x s in a polynomial p ∈ Q [ x ] . We also say that a vertex-set T in a graph G is an odd-cycle transversal if G − T is bipartite up to self-loops. (Note that not requiringself-loops to be removed by T may not be the standard way of defining odd-cycle transversals.) Lemma 20.
Let H be a bipartite n -vertex graph and let λ be an integer partition.1. If there are onions of λ that accommodate n/ edges, let ρ be such an onion with the minimal number oflayers. In polynomial time, we can construct a digraph G with G, ρ ) = ( n/ · H ) .
2. For any k ∈ N with d ( λ ) ≥ n − k : If there are onions of λ that accommodate k edges and contain ≥ k copies of , let ρ be such an onion with the minimal number of layers. In polynomial time, we canconstruct a digraph G (whose edge-weights may be constant multiples of an indeterminate x ) such that [ x k ] G, ρ ) = k ! · H, k ) .In both cases, let ρ = (2 d ( λ ) , θ, z ( λ ) −(cid:107) θ (cid:107) ) be the relevant (cid:96) -layer onion, for (cid:96) ∈ N . Then the following holds: • G admits an odd-cycle transversal T ⊆ V ( G ) of cardinality (cid:96) . • Every cycle cover in G (in the second case, every cycle cover whose weight is a constant multiple of x k ) contains exactly z ( λ ) − (cid:107) θ (cid:107) self-loops and at least d ( λ ) digons.Proof. Let V ( H ) = L ∪ R and write ρ = (2 d ( λ ) , θ, z ( λ ) −(cid:107) θ (cid:107) ) for the relevant (cid:96) -layer onion of λ . By addingisolated edges to H , we assume that n/ (in the first case) or k (in the second case) is the maximum numberof edges accommodated by λ . For the first part of the lemma, the graph G is defined as follows:16. Direct all edges in H from L to R and add all edges R × L .2. Add a set of transit vertices T = { t , . . . t (cid:96) } and all edges in R × T and T × L .3. Add d ( λ ) disjoint padding digons and z ( λ ) − (cid:107) θ (cid:107) padding vertices with self-loops.Any ρ -cycle cover of G uses all padding digons and loops. Any remaining odd-length cycle must use a vertexof T , as it would otherwise be contained in a bipartite graph. This implies that T is an odd-cycle transversal.It follows that the (cid:96) (odd-length) parts in θ can only be accommodated by (cid:96) disjoint cycles in G that eachinclude exactly one transit vertex. Hence, any ρ -cycle cover C in G induces a perfect matching in H whenrestricted to edges from L to R : Deleting transit vertices results in a cover of V ( H ) with paths of odd length ≥ that start in L , and deleting the edges from R to L then leaves us with a matching.Conversely, every perfect matching in H induces exactly ( n/ cycle covers of format ρ in G : For θ = (2 r + 1 , r + 1 , . . . ) , there are (cid:0) n/ r ,...,r (cid:96) (cid:1) ways of partitioning the n/ matching edges into the cyclescorresponding to transit vertices t , . . . t (cid:96) , and there are r i ! ways of choosing an ordering of the edges and the i -th transit vertex within the i -th cycle. This concludes the first part of the lemma.For the second part , we construct G by performing steps 1 and 2 from above, followed by these steps:3. For each vertex v ∈ V ( H ) , add a switch vertex s v with a self-loop of weight x , and a switch digon between v and s v .4. Add d ( λ ) − ( n − k ) padding digons and z ( λ ) − (cid:107) θ (cid:107) − k padding vertices with self-loops.Again, any cycle cover C of G uses all padding elements. The weight of C is x k iff it includes exactly k self-loops at switch vertices; it then includes n − k switch digons touching the remaining switch vertices, soit contains at least d ( λ ) digons and z ( λ ) − (cid:107) θ (cid:107) self-loops. As before, if C has format ρ , then the (cid:96) remainingodd-length cycles induce a k -matching in H when restricted to edges from L to R , and any k -matching in H can be extended to k ! such cycle covers.The odd-cycle transversal of G and special properties of staircase characters ensure that we can determine G, ρ ) for an onion ρ of λ by reduction to imm λ ( G ) . We prove this in the next subsection. Given a partition λ with staircase µ , consider any partition ρ = (2 d ( λ ) , ρ (cid:48) ) obtained by peeling the maximumnumber of dominos from λ , followed by some other partition ρ (cid:48) of the staircase µ thusly exposed. We showhow to relate χ λ ( ρ ) to χ µ ( ρ (cid:48) ) by means of domino tilings of λ/µ , which are border strip tableaux that onlycontain the dominos and . Definition 21. A domino tiling of a skew shape λ/µ is a border strip tableau T of λ/µ with format (2 (cid:107) λ/µ (cid:107) / ) .The parity of a domino tiling T is the parity of the number of in T .Note that the parity of a domino tiling T is essentially its height sign ht( T ) : The parity of T is even/oddiff the height sign of T is positive/negative. Curiously, these numbers do not depend on T . Lemma 22.
All domino tilings of a fixed shape λ/µ have the same parity.Proof.
Paint the rows of λ/µ black and white in an alternating way. In any domino tiling T , every vertical(horizontal) domino contains an odd (even) number of white boxes. Hence, the number of vertical dominos in T agrees in parity with the number of white boxes in λ/µ , which does not depend on T .This elementary yet crucial observation gives the desired connection between χ λ ( ρ ) and χ µ ( ρ (cid:48) ) . Each such ordering corresponds to a pair ( j, σ ) for j ∈ [ r i ] and a cyclic permutation σ of [ r j ] . The index j indicates that thetransit vertex t i is visited immediately after the j -th edge, and σ describes the order of the edges along the cycle. There are r i · ( r i − r i ! such pairs ( j, σ ) . emma 23. Let λ be a partition with staircase µ , and let ρ = (2 d ( λ ) , ρ (cid:48) ) for a partition ρ (cid:48) . Then we have χ λ ( ρ ) (cid:54) = 0 iff χ µ ( ρ (cid:48) ) (cid:54) = 0 .Proof. As any way of peeling d ( λ ) dominos from λ results in µ , the Murnaghan-Nakayama rule (Theorem 10)shows that χ λ ( ρ ) = c λ/µ · χ µ ( ρ (cid:48) ) with c λ/µ = (cid:88) domino tiling T of λ/µ ht( T ) , (6)By Lemma 22, each term in (6) has the same sign. Since λ/µ is obtained by peeling d ( λ ) dominos from λ ,there is at least one domino tiling, and hence there is at least one term in the sum. It follows that c λ/µ > ,thus proving the lemma.The last missing piece is to recall from Remark 11 that the staircase character χ µ ( ρ (cid:48) ) vanishes whenever ρ (cid:48) contains an even part. This allows us to analyze cycle covers in the particular graphs G constructed in thelast subsection: Any such cycle cover (with the right number of self-loops and digons) is counted by imm λ ( G ) iff its format is an onion. Lemma 24.
Let λ (cid:96) n (cid:48) be a partition and let ρ = (2 d ( λ ) , θ, z ( λ ) −(cid:107) θ (cid:107) ) be the (cid:96) -layer onion of λ . Let G be an n (cid:48) -vertex digraph with an odd-cycle transversal of size (cid:96) . For any cycle cover C in G that contains at least d ( λ ) digons and exactly z ( λ ) − (cid:107) θ (cid:107) self-loops: If χ λ ( C ) (cid:54) = 0 , then the format of C is ρ .Proof. Let C be a cycle cover in G with format β = (2 d ( λ ) , ρ (cid:48) ) and χ λ ( β ) (cid:54) = 0 . By Lemma 23, we have χ µ ( ρ (cid:48) ) (cid:54) = 0 , where µ is the staircase of λ . Since G has an odd-cycle transversal of size (cid:96) , there are at most (cid:96) non-singleton odd parts in ρ (cid:48) , as every non-singleton odd cycle uses exactly one transversal vertex. Since ρ (cid:48) contains z ( λ ) − (cid:107) θ (cid:107) singletons, the non-singleton odd parts contain (cid:107) θ (cid:107) boxes in total. This can only beachieved by peeling (cid:96) maximal-length border strips from µ , so the cycle format of the non-singleton parts is θ .Thus, we have ρ (cid:48) = ( θ, z ( λ ) −(cid:107) θ (cid:107) ) . Corollary 25.
Let λ (cid:96) n (cid:48) and let ρ = (2 d ( λ ) , θ, z ( λ ) −(cid:107) θ (cid:107) ) be the (cid:96) -layer onion of λ . Let G be an n (cid:48) -vertexgraph with an odd-cycle transversal of size (cid:96) . For t ∈ N , if every cycle cover whose weight is a constantmultiple of x t contains at least d ( λ ) digons and z ( λ ) − (cid:107) θ (cid:107) self-loops, then [ x t ] imm λ ( G ) = χ λ ( ρ ) · [ x t ] G, ρ ) with χ λ ( ρ ) (cid:54) = 0 .Proof. The equation follows directly from Lemma 24 and the requirement on G . To show χ λ ( ρ ) (cid:54) = 0 , itsuffices by Lemma 23 to show χ µ ( ρ (cid:48) ) (cid:54) = 0 for ρ (cid:48) = ( θ, z ( λ ) −(cid:107) θ (cid:107) ) . There is exactly one way of peeling θ from ρ (cid:48) , and each of the z ( λ ) − (cid:107) θ (cid:107) singletons peeled from the resulting staircase incurs positive height sign. Itfollows that all border strip tableaux of format ρ (cid:48) in µ contribute to χ µ ( ρ (cid:48) ) with the same sign.Note that we can consider G to be unweighted and invoke the corollary with t = 0 . This yields theunweighted setting (in x ) from the first part of Lemma 20. We combine the results from the previous sections into reductions from counting matchings to evaluatingimmanants for partitions with large staircases.
Lemma 26.
The following can be achieved in polynomial time and with polynomial-sized arithmetic circuits:1. Given a bipartite n -vertex graph H and a partition λ with w ( λ ) ≥ √ n , compute a digraph G and anumber c ∈ Q such that H ) = c · imm λ ( G ) . . Given a bipartite n -vertex graph H and k ∈ N , as well as a partition λ with w ( λ ) ≥ √ k and d ( λ ) ≥ n − k , compute a digraph G and a number c ∈ Q such that H, k ) = c · [ x k ] imm λ ( G ) .Proof. For the first part , note that w ( λ ) ≥ √ n implies that λ admits a t -layer onion ρ = (2 d ( λ ) , θ, z ( λ ) −(cid:107) θ (cid:107) ) for t = (cid:98)√ n (cid:99) . For large enough n , the number of edges accommodated by this onion is at least (cid:0) t +12 (cid:1) − t t ≥ n − √ n + 1 ≥ n/ . Hence, Lemma 20 allows us to construct a digraph G with G, ρ ) = ( n/ · H ) such that G admits an odd-cycle transversal of cardinality t , and every cycle cover in G contains z ( λ ) − (cid:107) θ (cid:107) self-loopsand at least d ( λ ) digons. By Corollary 25, we then have imm λ ( G ) = χ λ ( ρ ) · G, ρ ) with χ λ ( ρ ) (cid:54) = 0 . Itfollows from the last two equations that H ) = imm λ ( G )( n/ · χ λ ( ρ ) . Note that χ λ ( ρ ) can be computed in polynomial time and with polynomial-sized arithmetic circuits, seeProposition 7.4 in [8].For the second part , note that w ( λ ) ≥ √ k implies that λ admits an onion ρ = (2 d ( λ ) , θ, z ( λ ) −(cid:107) θ (cid:107) ) that accommodates ≥ k edges. Additionally, w ( λ ) ≥ √ k also implies that z ( λ ) − (cid:107) θ (cid:107) ≥ k . Since d ( λ ) ≥ n − k holds by assumption, all conditions for the second part of Lemma 20 are thus fulfilled, andwe can construct a graph G such that [ x k ] G, ρ ) = k ! · H, k ) and any cycle cover of weight x k in x contains exactly z ( λ ) − (cid:107) θ (cid:107) self-loops and at least d ( λ ) digons. Corollary 25 then implies that [ x k ] imm λ ( G ) = χ λ ( ρ ) · [ x k ] G, ρ ) with χ λ ( ρ ) (cid:54) = 0 . We conclude that H, k ) = [ x k ] imm λ ( G ) k ! · χ λ ( ρ ) . This proves the lemma.
We show how to count k -matchings with access to λ -immanants for partitions λ with large non-vanishingtetromino number s ( λ ) : In Section 6.1, we use edge gadgets to construct particular immanants imm λ ( G ) that count matchings up to a multiplicative constant factor. We then show, in Section 6.2, that the factorarising in the above construction is non-zero. To this end, we prove that the character χ λ does not vanish ona particular partition product. Finally, we combine these insights in Section 6.3 to obtain a reduction fromcounting matchings. Consider the edge gadget Q depicted on page 6. Intuitively speaking, this graph fragment ensures thatunwanted cycle covers in G annihilate in imm λ ( G ) . In the remaining consistent cycle covers C , the edgeswithin each edge gadget Q cover either both endpoints (in an active state, as listed in Figure 3) or none ofthem (in a passive state). In particular, this allows us to interpret an active gadget Q as a matching edgebetween its endpoints, since the active state prevents any edge outside of Q from being incident with theendpoints of Q . A similar approach was taken by the author together with Bläser [3] to establish hardness ofthe so-called cover polynomial . Lemma 27.
Let G be a directed graph containing copies Q , . . . , Q t of the edge-gadget Q , with endpoints e i = { u i , v i } for i ∈ [ t ] , such that distinct edge-gadgets intersect only at endpoints. Let C ∗ ( G ) denote the set w w − − − w − Figure 3: Each column lists the edge-sets C i that can arise for fixed combinations of endpoint in/out-degreesat gadget Q i in E ( G ) \ E ( Q i ) . The first column lists active states, the second column shows the passive state.All other states come with annihilating partners and cancel out, as shown in Lemma 27. of consistent cycle covers C in G : In such cycle covers, the restriction C ∩ E ( Q i ) for i ∈ [ t ] is an active orpassive state, as depicted in Figure 3. Then we have imm ∗ λ ( G ) := (cid:88) C ∈C ∗ ( G ) χ λ ( C ) (cid:89) e ∈ C w ( e ) = imm λ ( G ) . Proof.
For i ∈ [ t ] , we say that edges from E ( G ) \ E ( Q i ) are i -external. For any cycle cover C ∈ C ( G ) ,the total number of i -external incoming edges at the endpoints u i and v i of Q i equals the total number of i -external outgoing edges at u i and v i : Otherwise, the vertices of Q i cannot be covered by cycles. This givessix possible in/out-degree combinations at u i and v i . For each of these combinations, the possible restrictions C ∩ E ( Q i ) are listed in Figure 3.Using this list, we show that the set C \ C ∗ of non-consistent cycle covers can be partitioned into pairs { C, C } whose constituents have equal cycle formats ρ ( C ) = ρ ( C ) and opposite weights w ( C ) = − w ( C ) .Indeed, given C ∈ C \ C ∗ , let i ∈ [ t ] be minimal such that C i = C ∩ E ( Q i ) is not consistent. Then C i corresponds to one of the edge-sets in the last four columns of Figure 3. Exchanging C i for the other edge-setin its column yields a cycle cover C ∈ C \ C ∗ of the same cycle format as C , but with opposite weight. Wethus have χ λ ( C ) · w ( C ) = − χ λ ( C ) · w ( C ) . As the map from C to C induces a fixed-point free involution on C \ C ∗ , the pairs { C, C } indeed partition C \ C ∗ , and we conclude that imm λ ( G ) = (cid:88) C ∈C ∗ χ λ ( C ) · w ( C ) + (cid:88) C ∈C\C ∗ χ λ ( C ) · w ( C ) (cid:124) (cid:123)(cid:122) (cid:125) =0 , which proves the lemma.Using this lemma, we show how to transform instances ( H, k ) for into digraphs G such that imm λ ( G ) equals H, k ) up to a multiplicative constant. Our constructions only rely on non-vanishingtetrominos and dominos; the staircase of λ is “discarded” by introducing z ( λ ) padding vertices with self-loops.The constant arising in this construction needs some attention. To define it, consider the partition products { (2 ) , (4) } × s induced by the cycle formats of active states; we pad the partitions in this product to partitionsof n (cid:48) with d − s dominos and z ( λ ) singletons. Formally, for fixed λ (cid:96) n (cid:48) and s ≤ s ( λ ) , define θ s := { (2 ) , (4) } × s × { (2 d ( λ ) − s , z ( λ ) ) } . (7)In the next subsection, we then establish the crucial fact that χ λ ( θ s ) (cid:54) = 0 for relevant choices of s .20 emma 28. Let H be a graph with n vertices and m edges and let λ (cid:96) n (cid:48) be a partition.1. If λ has skew tetromino number s ( λ ) ≥ n/ and m − n/ additional dominos, i.e., d ( λ ) ≥ n/ m ,then we can construct an n (cid:48) -vertex graph G in polynomial time such that imm λ ( G ) = ( − m − n/ · n/ · χ λ ( θ n/ ) (cid:124) (cid:123)(cid:122) (cid:125) =: c · H ) . (8)
2. For any k ≤ s ( λ )3 such that d ( λ ) ≥ m + n + 2 kn − k : We can construct an n (cid:48) -vertex graph G inpolynomial time such that imm λ ( G ) = ( − m +2 kn − k · (2 k )! · k · χ λ ( θ k ) (cid:124) (cid:123)(cid:122) (cid:125) =: c · H, k ) . Proof.
For the first part , we define G as follows:1. Replace each edge uv ∈ E ( H ) with a fresh copy Q uv of the edge gadget Q . Identify u and v with theendpoints of Q .2. Add d (cid:48) = d ( λ ) − ( n/ m ) padding digons and z ( λ ) isolated padding vertices with self-loops.By Lemma 27 we have imm λ ( G ) = imm ∗ λ ( G ) , where imm ∗ λ ( G ) sums over the set C ∗ of consistent cycle covers.Any cycle cover C ∈ C ∗ includes all padding elements. Apart from padding elements, C consists of the activestates of some gadgets and the passive states of the remaining gadgets; let M ( C ) ⊆ E ( H ) denote the set ofpairs uv such that Q uv is active in C . Since active gadget states must be vertex-disjoint, and all verticesof G must be covered by cycles in C , the set M ( C ) induces a perfect matching M ( C ) ∈ M n/ ( H ) in H .Conversely, given a perfect matching M of H , let C ∗ M ⊆ C ∗ denote the set of consistent cycle covers with M ( C ) = M . By grouping the terms in imm ∗ λ ( G ) , we obtain imm ∗ λ ( G ) = (cid:88) M ∈M n/ ( H ) (cid:88) C ∈C ∗ M χ λ ( C ) · w ( C ) (cid:124) (cid:123)(cid:122) (cid:125) =: a ( M ) . (9)To calculate a ( M ) , we investigate the set C ∗ M : Each cycle cover C ∈ C ∗ M is obtained by• choosing an active state for each of the n/ gadgets Q uv with uv ∈ M , each inducing the weight w ( uv ) ,• adding the passive state (of weight − ) at the remaining gadgets, for a total weight of ( − m − n/ , and• adding all padding elements, all of weight .The total weight is thus ( − m − n/ . As choices can only be made at active states, the formats of cycle coversin C ∗ M are given by the partition product { (2 ) , (2 ) , (4) , (4) } × n/ × { (2 d ( λ ) − n , z ( λ ) ) } . Viewed as a multiset of partitions, this product amounts to n/ copies of θ n/ : After choosing one of theformats { (2 ) , (4) } for each of n/ entries, we can choose the first or second copy of this format. This showsthat a ( M ) = ( − m − n/ · n/ · χ λ ( θ n/ ) and thus proves (8).For the second part , we construct the graph G in a similar way, but we need to add some additionalstructures to account for the fact that most vertices are unmatched in a k -matching for k (cid:28) n .1. Replace all edges of H by edge gadgets.2. For each vertex v ∈ V ( H ) , add a switch vertex s v and connect it to v with a switch digon .21. Add k receptor vertices . Add a receptor edge between each pair of receptor and switch vertex, thenreplace these edges by edge gadgets.4. Add d (cid:48) = d ( λ ) − ( m + n + 2 kn − k ) isolated digons and z ( λ ) isolated vertices.Any cycle cover C ∈ C ∗ then consists of the following cycles:• Each receptor vertex must be covered by an active edge gadget. Then the other endpoint of that gadgetis the switch vertex of some vertex in H . The remaining k · ( n − edge gadgets incident with receptorvertices are passive. There are (2 k )! ways of matching the k receptor vertices to k fixed switch verticeswith active gadgets.• The n − k switch vertices not touched by active gadgets from receptor vertices must be covered byswitch digons.• By the previous item, k vertices in H are left to be covered by k active edge gadgets that representedges in H . As active edge gadgets are vertex-disjoint, they induce a k -matching M ( C ) in H , and theycontribute weight (cid:81) uv ∈ M ( C ) w ( uv ) .• Overall, there are m − k + 2 k · ( n −
1) = m + 2 kn − k passive edge gadgets in C , each contributingweight − . With padding digons and loops in G , there are d ( λ ) − k digons and z ( λ ) loops in C .The third item describes how C induces a k -matching M ( C ) ∈ M k ( H ) . Conversely, we can observe (as inthe first case) that any k -matching M of H induces consistent cycle covers with a total contribution of ( − m +2 kn − k · (2 k )! · k · w ( M ) · χ λ ( θ k ) . Note that the factor (2 k )! stems from the different ways receptor vertices can match to switch vertices. Remark.
Why not use self-loops to handle the switch vertices of matched vertices? It seemed to work finewhen dealing with staircases in Lemma 20? The reason is that doing so may require us to peel less than d ( λ ) dominos from λ to accommodate these self-loops. As a toy example for the complications that can arisethis way, consider λ = (3 , ) with d ( λ ) = 2 and ρ = (2 , ) . There is only one domino in ρ , and we have χ λ ( ρ ) = 0 , since this single domino can be peeled as or , leading to a cancellation. If we peel all d ( λ ) dominos from λ , then no such cancellations can occur, since Lemma 22 guarantees the same parity of thepeeled domino tilings. In this section, we analyze χ λ ( θ s ) for the partition product θ s defined in (7). First, we collect a few factson F that can be checked by simple manual calculations. In the following, recall that a domino tiling of a t -box skew shape γ is a border strip tableau of format (2 t ) . We call it even/odd if its number of verticalpieces is even/odd. By Lemma 22, all domino tilings of γ have the same parity. Also recall Section 3.3 forthe required definitions related to partition products. Fact 29.
For F = { (2 ) , (4) } , we have α F ( γ ) (cid:54) = 0 iff γ is a non-vanishing tetromino. Furthermore, for anynon-vanishing tetromino, the sign of α F ( γ ) is given by the parity of its domino tilings, see Table 2: α F ( γ ) = (cid:40) +2 γ has even domino tilings , − γ has odd domino tilings.Proof. Let Γ F denote the skew shapes that admit a border strip tableau with a format from F = { (2 ) , (4) } .We observe that every shape in Γ F admits a domino tiling with two dominos, and that Γ F therefore is asubset of the nine connected -box skew shapes from Table 1 and the four disconnected two-domino skewshapes. Furthermore: 22ositive:negative:Table 2: The shapes γ with α F ( γ ) (cid:54) = 0 , grouped by sign.• We have α F ( γ ) ∈ {− , } if γ consists of two disconnected dominos, and the sign is positive iff bothdominos have the same orientation. Indeed, γ admits exactly two border strip tableaux T , T withformats from F , which are domino tilings with the same height sign.• We have α F ( ) = 2 , since the square admits two border strip tableaux with block formats from F .Both tableaux are domino tilings with the same height sign.• The remaining connected -box shapes γ are border strips. As such, they admit one border strip tableauof format (2 ) and (4) each, implying that α F ( γ ) ∈ {− , , } . We have α F ( γ ) (cid:54) = 0 iff the height of γ and the unique domino tiling of γ agree in parity. (Recall that the height ht( γ ) is the number oftouched rows minus one. ) The sign is positive iff the tiling is even.These observations together prove the statement.We can now prove that the relevant characters in the tetromino-based reduction do not vanish. Lemma 30.
For any partition λ (cid:96) n (cid:48) and integer s ≤ s ( λ ) , we have χ λ ( θ s ) (cid:54) = 0 .Proof. By Fact 29, we have γ ∈ Γ F and α F ( γ ) (cid:54) = 0 for a -box skew shape γ iff γ is a non-vanishing tetromino.Let D = { (2) } and S = { (1) } ; then Γ D = { , } and Γ S = { (cid:31) } . Define the set of skew shape tableaux S = S ( λ, Γ F , . . . , Γ F (cid:124) (cid:123)(cid:122) (cid:125) s times , Γ D , . . . , Γ D (cid:124) (cid:123)(cid:122) (cid:125) d ( λ ) − s times , Γ S , . . . , Γ S (cid:124) (cid:123)(cid:122) (cid:125) z ( λ ) times ) and let t = d ( λ ) − s + z ( λ ) be the number of sets of skew shapes in the above list. For i ∈ [ t ] , let α i ∈ { α F , α D , α S } be the coefficient function for the i -th set in the list. By Lemma 14, we have χ λ ( θ s ) = (cid:88) S ∈S with shapes γ ...γ t t (cid:89) i =1 α i ( γ i ) . (10)It follows that every tableau S ∈ S with non-zero weight in the above sum peels s non-vanishing tetrominosfrom λ , followed by d ( λ ) − s dominos, and z ( λ ) singleton boxes. The tetrominos and dominos tile λ/µ ,where µ is the staircase of λ .Since s ≤ s ( λ ) , at least one tableau S ∈ S exists, and we can prove the lemma by showing that all tableauxare counted with the same sign in (10). Towards this, note that any skew shape tableau S ∈ S can be turnedinto a border strip tableau B ( S ) of λ that peels d ( λ ) dominos and z ( λ ) singleton boxes from λ : Simply tileeach tetromino γ in S arbitrarily with two dominos. By Fact 29, we know that α F ( γ ) is positive/negative iffthe tiling of γ is even/odd, so (cid:81) ti =1 α i ( γ i ) is positive/negative iff the domino tiling of λ/µ induced by B ( S ) iseven/odd. (Singleton boxes can be ignored, as they contribute the factor +1 .) But by Lemma 22, all dominotilings of λ/µ have the same parity. Therefore, all terms in (10) have the same sign. Corollary 31.
The coefficients c and c defined in Lemma 28 are both non-zero. .3 Reductions As in Section 5.3, we collect the previous arguments to obtain a reduction from counting matchings toimmanants for partitions with large non-vanishing tetromino number.
Lemma 32.
The following can be achieved in polynomial time and with polynomial-sized arithmetic circuits:1. Given an n -vertex graph H of maximum degree and a partition λ with s ( λ ) ≥ n/ and d ( λ ) ≥ n ,compute a digraph G and a number c ∈ Q such that H ) = c · imm λ ( G ) .2. Given an n -vertex graph H and k ∈ N , and a partition λ with s ( λ ) ≥ k and d ( λ ) ≥ n + n + 2 kn − k ,compute a digraph G and a number c ∈ Q such that H, k ) = c · imm λ ( G ) .Proof. For the first part , let H be a graph with n vertices and maximum degree , so that m ≤ n . Note that d ( λ ) ≥ n + m by assumption. We construct a graph G via Lemma 28 such that imm λ ( G ) = c · H ) .For the second part , we can use Lemma 28 to construct a graph G such that imm λ ( G ) = c · H, k ) .We have c , c (cid:54) = 0 by Corollary 31. Let Λ be a family of partitions with unbounded b (Λ) . We compose the constructions from the precedingsections to an overall hardness proof for Imm(Λ) . This requires us to find sequences of partitions within Λ thatare dense enough and supply sufficiently many boxes to the right of the first column. In the sub-polynomialgrowth regime, we also need to ensure a sufficiently large supply of dominos; this can be achieved by having alarge number of boxes in the first column. Definition 33.
Given a polynomial-time computable function g : N → N , a family of partitions Λ supportsgrowth g if there is a sequence Λ (cid:48) = ( λ (1) , λ (2) , . . . ) in Λ such that λ ( n ) satisfies b ( λ ( n ) ) ≥ g ( n ) and (cid:13)(cid:13) λ ( n ) (cid:13)(cid:13) =Θ( n ) . We also say that Λ supports growth g via Λ (cid:48) . We say that Λ computationally supports growth g if λ ( n ) can be computed in polynomial time from n .We are ready to prove the main theorems of this paper. To this end, we distinguish whether Λ supportspolynomial growth (for a reduction from ) or only sub-polynomial growth (for a reduction from ( g ) for any growth g supported by Λ ). Recall that, by known algorithms [23, 8], we have Imm(Λ) ∈ FP and Imm(Λ) ∈ VP for any family Λ with b (Λ) < ∞ . Proof of Theorem 1.
The tractability part is known. For hardness, we reduce from : Let H bean n -vertex bipartite graph for which we want to compute H ) . Let α > be maximal suchthat Λ supports growth Ω( n α ) . We find a partition λ ∈ Λ with (cid:107) λ (cid:107) = Θ( n /α ) such that b ( λ ) ≥ n . ThenLemma 18 guarantees that at least one of s ( λ ) ≥ . n or w ( λ ) ≥ √ n − holds. In either case, using thefirst cases of Lemmas 26 and 32, we compute a graph G on (cid:107) λ (cid:107) vertices and a number c ∈ Q such that H ) = c · imm λ ( G ) holds. Overall, this yields polynomial-time and c-reductions from to Imm(Λ) , showing P -hardness and VNP -completeness of the latter. The lower bound under
ETH followssince (cid:107) λ (cid:107) = Θ( n /α ) .If Λ supports only sub-polynomial growth, the proof proceeds similarly. In this case, we can find asequence of partitions in which most rows have width , which allows us to peel a large amount of dominosfrom the left-most column. Proof of Theorem 2. If Λ supports polynomial growth, we use Theorem 1. Otherwise, let g ∈ ω (1) be agrowth supported by Λ . We reduce from ( h ) with h ( n ) = (cid:112) g ( n ) / : Let ( H, k ) be an instance for ( h ) with an n -vertex graph H and k ≤ h ( n ) . Using the growth condition on Λ and g ∈ O ( n . ) , wefind a partition λ ∈ Λ with b ( λ ) ≥ k and at least n + b ( λ ) boxes in the first column, which implies d ( λ ) ≥ n . With the bound on b ( λ ) , Lemma 18 yields that s ( λ ) ≥ k or w ( λ ) ≥ √ k − . In either case,24sing the second case of Lemmas 26 and 32, we compute a graph G on (cid:107) λ (cid:107) vertices and a number c ∈ Q suchthat H, k ) = c · [ x t ] imm λ ( G ) , where t = 2 k for the staircase-based reduction (Lemma 26) and t = 0 for the tetromino-based reduction (Lemma 32).Note that we can compute [ x t ] imm λ ( G ) via polynomial interpolation from the values imm λ ( G i ) for i ∈ { , . . . , (cid:107) λ (cid:107)} , where G i is the graph obtained from G by replacing each edge of weight x with an edge ofweight i . Overall, we obtain a polynomial-time Turing reduction from ( h ) to Imm(Λ) , which impliesby Lemma 4 that
Imm(Λ) / ∈ FP unless FPT = W [ ] . An analogous statement holds in the algebraic setting:As polynomial interpolation amounts to solving a system of linear equations, which can be performed withpolynomial-sized circuits, we obtain a parameterized c-reduction from the p-family ( h ) to the p-family Imm(Λ (cid:48) ) . We established a dichotomy for the complexity of immanants, concluding a sequence of partial results obtainedthroughout the last four decades [23, 9, 7, 27, 20].Let us note that immanants are not the only way of generalizing permanents and determinants into afamily of matrix forms. Other examples include the cover polynomials [14, 5] and the fermionants [27, 2],which are also sum-products over row-column permutations of a matrix. Unlike the immanant however, thesefamilies do not exhibit gradual progressions from easy to hard cases, and they feature no non-trivial easycases beside the determinant.Our result for immanants also prompts several interesting follow-up questions.
Modular immanants.
Writing λ (cid:48) for the transpose of a partition λ , it is known that χ λ and χ λ (cid:48) areequivalent modulo . This renders λ -immanants tractable over Z for partitions λ with constantly manyboxes outside of the first column or row . Are these the only tractable immanants over Z ? Which immanantsare tractable over Z p for odd primes p ? These questions lead into the tricky territory of representation theoryover fields of positive characteristic. Planar graphs.
The permanent is polynomial-time solvable for bi-adjacency matrices of planar bipartitegraphs. This is a consequence of the classical FKT algorithm [35], which expresses the number of perfectmatchings in a planar graph G as a determinant. The relevant matrix is derived from the adjacency matrixof G by flipping signs according to a Pfaffian orientation of G . Can this algorithmic idea be generalized tomore general immanants on bi-adjacency matrices of planar bipartite graphs? Which immanants remainhard on planar graphs? Removing weights.
Our proof establishes hardness for matrices with general entries from Z : Eventhough the source problems and are hard for unweighted graphs, the edge-gadget andinterpolation steps introduce non-unit weights. It may however still be possible to establish hardness for - matrices, as known for the permanent. While gadgets for simulating edge-weights are known, they change theformats of the relevant cycle covers, and it is more difficult to argue about character values on these formats. Acknowledgments
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