Parameterizing by the Number of Numbers
PParameterizing by the Number of Numbers (cid:63)
Michael R. Fellows , Serge Gaspers , and Frances A. Rosamond School of Engineering and IT, Charles Darwin University, NT 0909, Australia. { michael.fellows,frances.rosamond } @cdu.edu.au Institute of Information Systems, Vienna University of Technology, Vienna, Austria. [email protected]
Abstract.
The usefulness of parameterized algorithmics has often depended on whatNiedermeier has called “the art of problem parameterization”. In this paper we intro-duce and explore a novel but general form of parameterization: the number of numbers .Several classic numerical problems, such as
Subset Sum , Partition , 3-
Partition , Numerical 3-Dimensional Matching , and
Numerical Matching with TargetSums , have multisets of integers as input. We initiate the study of parameterizingthese problems by the number of distinct integers in the input. We rely on an FPTresult for
Integer Linear Programming Feasibility to show that all the above-mentioned problems are fixed-parameter tractable when parameterized in this way. Invarious applied settings, problem inputs often consist in part of multisets of integers ormultisets of weighted objects (such as edges in a graph, or jobs to be scheduled). Suchnumber-of-numbers parameterized problems often reduce to subproblems about tran-sition systems of various kinds, parameterized by the size of the system description.We consider several core problems of this kind relevant to number-of-numbers param-eterization. Our main hardness result considers the problem: given a non-deterministicMealy machine M (a finite state automaton outputting a letter on each transition), aninput word x , and a census requirement c for the output word specifying how manytimes each letter of the output alphabet should be written, decide whether there existsa computation of M reading x that outputs a word y that meets the requirement c .We show that this problem is hard for W [1]. If the question is whether there exists aninput word x such that a computation of M on x outputs a word that meets c , theproblem becomes fixed-parameter tractable. Parameterized complexity and algorithmics has been developing for more than twentyyears. Some important progress of the field has depended on what Niedermeier hascalled “the art of problem parameterization” (see Chapter 5 of his monograph [23]).For example, it was Cristina Bazgan who first suggested that the parameter mightbe k = 1 /(cid:15) in the study of the complexity of approximation, leading eventually tothe study of EPTASs [3].Here we explore, for the first time (to our knowledge), a parameterization thatseems widely relevant: the number of numbers . Many problems take as input informa-tion that consists (in part) of multisets of integers or multisets of weighted objects,such as weighted edges in a weighted graph, the time-requirements of jobs to be (cid:63) A preliminary version of this paper appeared in the proceedings of IPEC 2010 [7]. M.R.F. andF.A.R. acknowledge support from the Australian Research Council. S.G. acknowledges partialsupport from the European Research Council (COMPLEX REASON, 239962), from ConicytChile (Basal-CMM), and from the Australian Research Council. a r X i v : . [ c s . D S ] O c t Michael R. Fellows, Serge Gaspers, and Frances A. Rosamond scheduled, or the sequence of molecular weights of a spectrographic dataset. Our in-vestigations are of importance for problem input distributions where the number ofdistinct numerical values is small compared to the overall input size, and when themodeling of the problem allows rounding as a way to get to fewer distinct values.In classical complexity, this “parameterization” has been explored in distribution-sensitive algorithmics [29]. For example, while Ω ( n log n ) is a lower bound on sorting n values in the comparison model [18], a multiset of cardinality n and h distinctvalues can be sorted using O ( n log h ) comparisons [22].It is perhaps surprising that this parameterization in the sense of Niedermeier’s“art of problem parameterization” [23, 24] has not been considered before in parame-terized complexity, as it seems entirely well-motivated. While weighted combinatorialoptimization problems have generally strong claims to model realism, it is often thecase that, e.g., the jobs to be scheduled may be of certain standard sizes arising ina limited number of ways, or that the costs of the nodes in a network problem maydepend on the model and vendor of the device, of which there are a limited num-ber of possibilities. Many similar scenarios easily come to mind. A bounded numberof numbers may also arise naturally and implicitly in parameterized problems whennumbers are associated to other parameterized aspects of a problem, such as alphabetsize.As an initial foray, we first show that a number of classic NP-hard problemsabout multisets of integers, when parameterized in this way, become fixed-parametertractable. The proofs are easy, and the knowledgeable reader might anticipate themalmost as exercises today — they use the relatively deep result that Integer LinearProgramming , parameterized by the number of variables, is FPT. Until recently, asnoted in the 2006 monograph by Niedermeier [23], there were not so many interestingapplications of this fundamental result (see [1, 11, 12, 16] for some exceptions).At a deeper level of engagement with this parameterization, we describe someexamples of how number-of-numbers parameterized problems reduce to numericalproblems about Mealy machines, parameterized by the size of the description of themachine. We show that one basic problem about Mealy machines, parameterized inthis way, is FPT, and that another is W [1]-hard. Integer Linear Programming
In the
Integer Linear Programming Feasibility problem (
ILPF ), the input is an m × n matrix A of integers and an m -vector b ofintegers, the parameter is n , and the question is whether there exists an n -vector x ofintegers satisfying the m inequalities Ax ≤ b . ILPF , parameterized by the number ofvariables, was shown to be fixed-parameter tractable by Lenstra [20] and the runningtime has been improved by Kannan [17] and by Frank and Tardos [14].
Multisets
Let A be a multiset. The cardinality of A , denoted | A | , is the total numberof elements in A , including repeated memberships. The variety of A , denoted || A || , isthe number of distinct elements in A . Element a has multiplicity m in A if it occurs m times in A . We denote the set of integers from 1 to n by [ n ] = { , . . . , n } . arameterizing by the Number of Numbers 3 Graphs
Let G = ( V, E ) be a graph, v ∈ V be a vertex of G , and S ⊆ V be a subsetof vertices of G . The subgraph of G induced on S is the graph G [ S ] = ( S, E ∩ { uv : u, v ∈ S } ). The set S is a clique of G if G [ S ] is complete , i.e. there is an edge betweenevery two distinct vertices of G [ S ]. The set S is an independent set of G if G [ S ] is empty , i.e. G [ S ] has no edge. The neighborhood of v is the set of vertices incident to v and denoted N ( v ). The degree of v is d ( v ) = | N ( v ) | . We also define N S ( v ) = N ( v ) ∩ S and d S ( v ) = | N S ( v ) | . Words
Let Σ be an alphabet . The elements of Σ are called letters , and a word x oflength n = | x | is a sequence of n letters. The symbol (cid:15) denotes the empty letter. Wedenote the concatenation of two words x , x ∈ Σ ∗ by x x . The i th power of a word x is denoted x i or ( x ) i and represents the word xx . . . x (cid:124) (cid:123)(cid:122) (cid:125) i times . Parameterized Complexity
We define the basic notions of Parameterized Complex-ity and refer to other sources [6, 13, 23] for an in-depth treatment. A parameterizedproblem is a set of pairs (
I, k ), the instances, where I is the main part and k isthe parameter. A parameterized problem is fixed-parameter tractable if there exist acomputable function f and an algorithm that solves any instance ( I, k ) of size n intime f ( k ) n O (1) . FPT denotes the class of all fixed-parameter tractable parameterizeddecision problems.Parameterized complexity offers a completeness theory that allows the accumula-tion of strong theoretical evidence that some parameterized problems are not fixed-parameter tractable. This theory is based on a hierarchy of complexity classesFPT ⊆ W[1] ⊆ W[2] ⊆ W[3] ⊆ · · · ⊆ XP . where all inclusions are believed to be strict. Each class W[ i ] contains all parameter-ized decision problems that can be reduced to a canonical parameterized satisfiabilityproblem P i under parameterized reductions . These are many-to-one reductions wherethe parameter for one problem maps into the parameter for the other. More specifi-cally, a parameterized problem L reduces to a parameterized problem L (cid:48) if there is amapping R from instances of L to instances of L (cid:48) such that1. ( I, k ) is a
Yes -instance of L if and only if ( I (cid:48) , k (cid:48) ) = R ( I, k ) is a
Yes -instanceof L (cid:48) ,2. there is a computable function g such that k (cid:48) ≤ g ( k ), and3. there is a computable function f such that R can be computed in time f ( k ) · n O (1) ,where n denotes the size of ( I, k ).A parameterized problem L is then in W[ i ], for i ∈ N , if it has a parameterizedreduction to the problem of deciding whether a Boolean decision circuit (a decisioncircuit is a circuit with exactly one output), with AND, OR, and NOT gates, ofconstant depth such that on each path from an input to the output, all but i gateshave a constant number of inputs, parameterized by the number of ones in a satisfyingassignment to the inputs of the circuit [6].A parameterized problem is in XP if there exist computable functions f and g and an algorithm that solves any instance ( I, k ) of size n in time f ( k ) n g ( k ) . Michael R. Fellows, Serge Gaspers, and Frances A. Rosamond
We start with two classic problems on multisets and show that they are fixed-para-meter tractable, parameterized by the number of numbers. variety - Subset Sum ( var - SubSum )Input: A multiset A of integers and an integer s .Parameter: k = || A || , the number of distinct integers in A .Question: Is there a multiset X ⊆ A such that (cid:80) a ∈ X a = s ? variety - Partition ( var - Part )Input: A multiset A of integers.Parameter: k = || A || .Question: Is there a multiset X ⊆ A such that (cid:80) a ∈ X a = (cid:80) b ∈ A \ X b ?The parameterizon of Subset Sum by | X | is W [1]-hard [9]. This hardness also holdsfor the parameterization of Partition by | X | as an easy reduction from SubsetSum adds the integer ( (cid:80) a ∈ A a ) − s to A if s ≤ ( (cid:80) a ∈ A a ) /
2, and if s > ( (cid:80) a ∈ A a ) / A \ X that sums to ( (cid:80) a ∈ A a ) − s and uses the previous construction.Our FPT results use a deep result of Lenstra, stating that Integer LinearProgramming Feasibility ( ILPF ), parameterized by the number of variables, isFPT. They are obtained by very natural formulations of the respective problems asinteger programs.
Theorem 1. var-
SubSum is fixed-parameter tractable.Proof.
Given an instance (
A, s ) for var - SubSum , with || A || = k , we create an equiv-alent instance of ILPF whose number of variables is upper bounded by a function of k . Let a , . . . , a k denote the distinct elements of A and let m , . . . , m k denote theirrespective multiplicities in A . The ILPF instance has the integer variables x , . . . , x k and the following inequalities and equalities. x i ≤ m i ∀ i ∈ [ k ] x i ≥ ∀ i ∈ [ k ] k (cid:88) i =1 x i · a i = s. For each i ∈ [ k ], the variable x i represents the number of times a i occurs in X , theset summing to s in a valid solution. Using standard techniques in mathematicalprogramming, these constraints can be transformed into the form Ax ≤ b . (cid:117)(cid:116) A very similar proof shows that var - Part is fixed-parameter tractable.
Theorem 2. var-
Part is fixed-parameter tractable. arameterizing by the Number of Numbers 5
Proof.
Given an instance A for var - Part , with || A || = k , we create an equivalentinstance of ILPF whose number n of variables is upper bounded by a function of k .Let a , . . . , a k denote the distinct elements of A and let m , . . . , m k denote theirrespective multiplicities in A . The ILPF instance has the integer variables x , . . . , x k and the following inequalities and equalities. x i ≤ m i ∀ i ∈ [ k ] x i ≥ ∀ i ∈ [ k ] k (cid:88) i =1 x i · a i = (cid:88) a ∈ A a/ . For each i ∈ [ k ], the variable x i represents the number of times a i occurs in X , suchthat (cid:80) a ∈ X a = (cid:80) b ∈ A \ X b = (cid:80) a ∈ A a/ Ax ≤ b . (cid:117)(cid:116) Using the
ILPF machinery, we show in this section that several other problems,which are often used in NP-hardness proofs, become fixed-parameter tractable whenparameterized by the number of numbers. variety - Numerical 3-Dimensional Matching ( var - Num3-DM )Input: Three multisets
A, B, C of n integers each and an integer s .Parameter: k = || A ∪ B ∪ C || .Question: Are there n triples S , . . . , S n , each containing one element fromeach of A, B, and C such that for every i ∈ [ n ], (cid:80) a ∈ S i a = s ? Theorem 3. var-
Num3-DM is fixed-parameter tractable.Proof.
Let (
A, B, C, s ) be an instance for var - Num3-DM , with k = || A || , k = || B || , k = || C || , and k = || A ∪ B ∪ C || . Let a , . . . , a k denote the distinct elements of A , b , . . . , b k denote the distinct elements of B , and c , . . . , c k denote the distinctelements of C . Also, let m ,a , . . . , m k ,a , m ,b , . . . , m k ,b , m ,c , . . . , m k ,c denote theirrespective multiplicities in A , B , and C . We create an instance of ILPF with at most k integer variables x i,j,(cid:96) , for i ∈ [ k ] , j ∈ [ k ] , (cid:96) ∈ [ k ]: x i,j,(cid:96) = 0 for each ( i, j, (cid:96) ) ∈ [ k ] × [ k ] × [ k ]such that a i + b j + c (cid:96) (cid:54) = s (cid:88) ( j,(cid:96) ) ∈ ([ k ] , [ k ]) x i,j,(cid:96) = m i,a ∀ i ∈ [ k ] (cid:88) ( i,(cid:96) ) ∈ ([ k ] , [ k ]) x i,j,(cid:96) = m j,b ∀ j ∈ [ k ] (cid:88) ( i,j ) ∈ ([ k ] , [ k ]) x i,j,(cid:96) = m (cid:96),c ∀ (cid:96) ∈ [ k ] Michael R. Fellows, Serge Gaspers, and Frances A. Rosamond
A variable x i,j,(cid:96) represents the number of times the elements a i ∈ A , b j ∈ B and c (cid:96) ∈ C are used together to form a triple summing to s . The first constraint makessure that such a triple is formed only if it sums to s . The remaining equalities makesure that each element of A ∪ B ∪ C appears in a triple. Thus n such triples areformed, all summing to s if the integer program is feasible. (cid:117)(cid:116) Note that the problem is also fixed-parameter tractable if parameterized by || A ∪ B || only: we face a No -instance if || C || > ||{ a + b : a ∈ A, b ∈ B }|| . A closely related, wellknown numerical problem, is the following. variety - Numerical Matching with Target Sums ( var - NMTS )Input: Three multisets
A, B, S of n integers each.Parameter: k = || A ∪ B ∪ S || .Question: Are there n triples C , . . . , C n ∈ A × B × S , such that the A -element and the B -element from each C i sum to its S -element? Corollary 1. var-
NMTS is fixed-parameter tractable.
By the previous discussion, the natural parameterization by || A ∪ B || is also fixed-parameter tractable. A straightforward adaptation of the proof of Theorem 3 showsthat variety - is fixed-parameter tractable. variety - ( var - )Input: A multiset A of 3 n integers.Parameter: k = || A || .Question: Are there n triples S , . . . , S n ⊆ A , all summing to the samenumber? Theorem 4. var- is fixed-parameter tractable.Proof.
Let A be an instance for var - , with || A || = k and | A | = 3 n . Let s = (cid:80) a ∈ A a/n . Let a , . . . , a k denote the distinct elements of A and let m , . . . , m k denote their multiplicities in A . We create an instance of ILPF with at most k integer variables x i,j,(cid:96) , for i, j, (cid:96) ∈ [ k ]: x i,j,(cid:96) = 0 for each i, j, (cid:96) ∈ [ k ]such that a i + a j + a (cid:96) (cid:54) = s (cid:88) j,(cid:96) ∈ [ k ] j,(cid:96) (cid:54) = i ( x i,j,(cid:96) + x j,i,(cid:96) + x j,(cid:96),i )+2 · (cid:88) j ∈ [ k ] j (cid:54) = i ( x i,i,j + x i,j,i + x j,i,i )+3 · x i,i,i = m i ∀ i ∈ [ k ]A variable x i,j,(cid:96) represents the number of times the elements a i , a j and a (cid:96) are usedtogether to form a triple summing to s . The first constraint makes sure that such a arameterizing by the Number of Numbers 7 triple is formed only if it sums to s . The second set of equalities make sure that eachelement of A appears in a triple. Thus n such triples are formed, all summing to s ifthe integer program is feasible. (cid:117)(cid:116) In this section, we explore how far we can generalize the rather simple FPT results ofthe previous two sections. To this end, we investigate the parameterized complexity oftwo problems about Mealy Machines. Both problems can be viewed as parameterizedproblems implicitly parameterized by the number of numbers, because in each casethe size of the alphabet is part of the parameterization, and each letter of the alphabetis associated with a census requirement. The richer structure of these problems meansthat a simple appeal to integer linear programming no longer suffices: one turns outto be FPT, and the other W[1]-hard. In Section 6, we show that other problemsparameterized by the number of numbers reduce to these two seemingly generalproblems of this kind.Mealy machines [21] are finite-state transducers, generating an output based ontheir current state and input. They have important applications in cryptanalysis [2],computational linguistics [27], and control and system theory [30]. A deterministicMealy machine is a dual-alphabet state transition system given by a 5-tuple M =( S, s , Γ, Σ, T ): – a finite set of states S , – a start state s ∈ S , – a finite set Γ , called the input alphabet, – a finite set Σ , called the output alphabet, and – a transition function T : S × Γ → S × Σ mapping pairs of a state and an inputletter to the corresponding next state and output letter.The alphabets Γ and Σ may contain the empty letter (cid:15) , as in [28]. This eases someof the description, but all our results also hold if we restrict (cid:15) / ∈ Γ ∪ Σ .In a non-deterministic Mealy machine , the only difference is that the transitionfunction is defined T : S × Γ → P ( S × Σ ) as for a given state and input letter, theremay be more than one possibility for the next state and output letter. (Here P ( X )denotes the powerset of a set X .)A census requirement c : Σ \ { (cid:15) } → N is a function assigning a non-negativeinteger to each letter of the output alphabet (except (cid:15) ). It is used to constrain howmany times each letter should appear in the output of a Mealy machine. A word x ∈ Σ ∗ meets the census requirement if every letter b ∈ Σ \ { (cid:15) } appears exactly c ( b )times in x .The notion of census requirement is related to Parikh images [25]. Let Σ \ { (cid:15) } = { b , . . . , b σ } . For x ∈ Σ ∗ , the Parikh image is Ψ ( x ) = ( c ( b ) , . . . , c ( b σ )), where c isthe census requirement such that x meets c . The Parikh image of a language L is Ψ ( L ) = { Ψ ( x ) : x ∈ L } . Parikh’s theorem [25] states that the Parikh image of acontext-free language is semilinear, i.e., that for every context-free language there isa regular language with the same Parikh image. Michael R. Fellows, Serge Gaspers, and Frances A. Rosamond
Our first problem about Mealy machines asks whether there exists an input wordand a computation of the Mealy machine such that the output word meets the censusrequirement. variety - Exists Word Mealy Machine ( var - EWMM )Input: A non-deterministic Mealy machine M = ( S, s , Γ, Σ, T ), and acensus requirement c : Σ \ { (cid:15) } → N .Parameter: | S | + | Γ | + | Σ | .Question: Does there exist a word x ∈ Γ ∗ for which a computation of M on input x generates an output y that meets c ?Our proof that var - EWMM is fixed-parameter tractable is inspired by the proof from[10] showing that
Bandwidth is fixed-parameter tractable when parameterized bythe maximum number of leaves in a spanning tree of the input graph. We need thefollowing definition and lemma from [10].In a digraph D , two directed walks ∆ and ∆ (cid:48) from a vertex s to a vertex t are arc-equivalent , if for every arc a of D , ∆ and ∆ (cid:48) pass through a the same number oftimes. Lemma 1 ([10]).
Any directed walk ∆ through a finite digraph D on n vertices froma vertex s to a vertex t of D is arc-equivalent to a directed walk ∆ (cid:48) from s to t , where ∆ (cid:48) has the form:(1) ∆ (cid:48) consists of an underlying directed walk ρ from s to t of length at most n ,(2) together with some number of short loops , where each such short loop l beginsand ends at a vertex of ρ , and has length at most n . The algorithm will first subdivide state transitions in order to make the underlyingdirected graph simple. As suggested by Lemma 1, the algorithm goes over all possiblechoices for selecting an underlying directed walk ρ starting from s . For every shortloop starting and ending at a vertex from ρ , the algorithm associates an integervariable representing the number of times this short loop is executed while movingalong ρ . Again by Integer Linear Programming Feasibility , it can be checkedwhether there is a set of integers, representing the number of executions of the shortloops, such that the number of times each output letter is written is compatible withthe census requirement.
Theorem 5. var-
EWMM is fixed-parameter tractable.Proof.
Let ( M (cid:48) = ( S (cid:48) , s (cid:48) , Γ (cid:48) , Σ (cid:48) , T (cid:48) ) , c ) be an instance for var - EWMM with k = | S (cid:48) | + | Γ (cid:48) | + | Σ (cid:48) | . As M (cid:48) might have multiple transitions from one state to another,we first subdivide each transition in order to obtain a simple digraph underlyingthe Mealy machine (so we can use Lemma 1): create a new non-deterministic Mealymachine M = ( S, s , Γ, Σ, T ) such that, initially, S = S (cid:48) , s = s (cid:48) , Γ = Γ (cid:48) ∪ { (cid:15) } , and Σ = Σ (cid:48) ∪ { (cid:15) } ; for each transition t of T (cid:48) from a couple ( s i , (cid:104) i (cid:105) ) to a couple ( s o , (cid:104) o (cid:105) ),add a new state s t to S and add the transition from ( s i , (cid:104) i (cid:105) ) to ( s t , (cid:104) o (cid:105) ) and thetransition from ( s t , (cid:15) ) to ( s o , (cid:15) ) to T . Clearly, there is at most one transition betweenevery two states in M . arameterizing by the Number of Numbers 9 Our algorithm goes over all transition walks in M of length at most | S | that startfrom s . There are at most | S | ( | S | ) such transition walks and each such transitionwalk has at most | S | | S | short loops, as they have length at most | S | by Lemma 1. Let P = ( s , s , . . . , s | P | ) be such a transition walk and L = ( (cid:96) , (cid:96) , . . . , (cid:96) | L | ) be its shortloops. It remains to check whether there exists a set of integers X = { x , x , . . . , x | L | } such that a word output by a computation of M moving from s to s | P | along thewalk P , and executing x i times each short loop (cid:96) i , 1 ≤ i ≤ | L | , meets the censusrequirement. Note that if one such word meets the census requirement, then all suchwords meet the census requirement, as it does not matter in which order the shortloops are executed. We verify whether such a set X exists by ILPF .Let Σ \ { (cid:15) } = {(cid:104) (cid:96), (cid:105) , (cid:104) (cid:96), (cid:105) , . . . , (cid:104) (cid:96), σ (cid:105)} . Define m ( i, j ), for 1 ≤ i ≤ | L | , 1 ≤ j ≤ σ ,to denote the number of times that M writes the letter (cid:104) (cid:96), j (cid:105) when it executes theloop (cid:96) i once. Define m ( j ), for 1 ≤ j ≤ σ , to be the number of times that M writesthe letter (cid:104) (cid:96), j (cid:105) when it transitions from s to s | P | along the walk P . Then, we onlyneed to verify that there exist integers x , x , . . . , x | L | such that m ( j ) + | L | (cid:88) i =0 x i · m ( i, j ) = c ( (cid:104) (cid:96), j (cid:105) ) , ∀ j ∈ [ σ ] . By construction, | S | ≤ | S (cid:48) | + | T (cid:48) | ≤ | S (cid:48) | + | S (cid:48) | · | Γ (cid:48) | · | Σ (cid:48) | ≤ k + k . As the number ofinteger variables of this program is at most | L | ≤ | S | | S | ≤ ( k + k ) k + k , and the numberof transition walks that the algorithm considers is at most | S | ( | S | ) ≤ ( k + k ) k +2 k + k , var - EWMM is fixed-parameter tractable. (cid:117)(cid:116)
We note that the proof in [10] concerned a special case of a deterministic Mealymachine where the input and output alphabet are the same, and all transitions thatread a letter (cid:104) (cid:96) (cid:105) also write (cid:104) (cid:96) (cid:105) .In our second Mealy machine problem, the question is whether, for a given inputword, there is a computation of the Mealy machine which outputs a word that meetsthe census requirement. variety - Given Word Mealy Machine ( var - GWMM )Input: A non-deterministic Mealy machine M = ( S, s , Γ, Σ, T ), a word x ∈ Γ ∗ , and a census requirement c : Σ \ { (cid:15) } → N .Parameter: | S | + | Γ | + | Σ | Question: Is there a computation of M on input x generating an output y that meets c ?By dynamic programming we show that two restrictions of this problem are in XP.In the first one, the census requirement is encoded in unary. This restriction of theproblem seems lenient, especially when one is actually interested in finding the outputword, as the census function acts then as a placeholder for the produced word. Theorem 6. var-
GWMM is in XP if c is encoded in unary.Proof. Let | Σ \ { (cid:15) }| = σ and Σ \ { (cid:15) } = { b , . . . , b σ } . Our dynamic programmingalgorithm computes the entries of a boolean table A . The table A has an entry A [ s, c , . . . , c σ , i, p ] for each state s ∈ S , each c j ∈ { , . . . , c ( b j ) } , j ∈ [ σ ], each index i ∈{ , . . . , | x |} , and each integer p ∈ P = { , . . . , | S | − } . The entry A [ s, c , . . . , c σ , i, p ]is set to true if there exists a computation of M reading the first i letters of x ,outputting a word y in which the letter b j occurs c j times, for each j ∈ [ σ ], followedby p transitions that read (cid:15) and write (cid:15) , and ending up in state s , and to false otherwise.Set A [ s, c , . . . , c σ , ,
0] to true if s = s and c = . . . = c σ = 0, and to false otherwise. Compute the values of the table by increasing values of (cid:80) σi =1 c i , index i ,propagation integer p , and state number s : A [ s, c , . . . , c σ , i, p ] = (cid:95) s (cid:48)∈ S,bj ∈ Σ \{ (cid:15) } ,p (cid:48)∈ P :( s,bj ) ∈ T ( s (cid:48) ,x [ i ]) A [ s (cid:48) , c , . . . , c j − , c j − , c j +1 , . . . , c σ , i − , p (cid:48) ] ∨ (cid:95) s (cid:48)∈ S,p (cid:48)∈ P :( s,(cid:15) ) ∈ T ( s (cid:48) ,x [ i ]) A [ s (cid:48) , c , . . . , c σ , i − , p (cid:48) ] ∨ (cid:95) s (cid:48)∈ S,bj ∈ Σ \{ (cid:15) } ,p (cid:48)∈ P :( s,bj ) ∈ T ( s (cid:48) ,(cid:15) ) A [ s (cid:48) , c , . . . , c j − , c j − , c j +1 , . . . , c σ , i, p (cid:48) ] ∨ (cid:95) s (cid:48) ∈ S :( s,(cid:15) ) ∈ T ( s (cid:48) ,(cid:15) ) A [ s (cid:48) , c , . . . , c σ , i, p − x -computation of M generating a word y that meets thecensus requirement if and only if (cid:87) s ∈ S,p ∈ P A [ s, c ( b ) , . . . , c ( b σ ) , | x | , p ] is true . Denoteby n is the length of the description of an input instance. The table has | S | · | x | · | P | · Π σj =1 c ( b j ) ≤ | S | · n σ +1 entries, and each entry can be computed in time O ( | S | · σ ).The running time of the algorithm is thus upper bounded by O ( n σ +1 · k ), where k is the parameter. (cid:117)(cid:116) For the version where c is encoded in binary, a restriction on the input alphabet givesan XP algorithm as well. Corollary 2. var-
GWMM is in XP if (cid:15) / ∈ Γ .Proof. If (cid:80) b ∈ Σ \{ (cid:15) } c ( b ) > | x | , then return false , as M cannot output more than | x | letters. Otherwise, run the algorithm described in the proof of Theorem 6. Its runningtime is O ( k · | x | · Π σj =1 c ( b j )) = O ( n σ +1 · k ). (cid:117)(cid:116) Note that the XP-results also hold if the parameter is only | Σ | .To show that var - GWMM is W [1]-hard, we reduce from the MulticoloredClique problem, which is W [1]-hard [26, 8]. arameterizing by the Number of Numbers 11 Multicolored Clique ( MCC )Input: An integer k and a connected undirected graph G = ( V (1) ∪ V (2) . . . ∪ V ( k ) , E ) such that for every i ∈ [ k ], the vertices of V ( i ) induce an independent set in G .Parameter: k .Question: Is there a clique of size k in G ?Clearly, a solution to this problem has one vertex from each color.Our parameterized reduction encodes G in the input word x of the Mealy machine M , and the description of M depends only on k . The Mealy machine is divided into k parts, one for each color class V ( i ), with 1 ≤ i ≤ k . Its i th part is responsible forselecting a vertex v i from V ( i ) and edges v i v j for every v j ∈ V ( j ), with 1 ≤ j (cid:54) = i ≤ k .All consistency issues and communication is done via the census requirement. Withinpart i , we need to make sure that the selected edges are all incident to the selectedvertex v i . This is achieved by making M output p times each letter (cid:104) l , i, j (cid:105) , with1 ≤ j (cid:54) = i ≤ k , if it selects the p th vertex in V ( i ). The census requirement for (cid:104) l , i, j (cid:105) is | V ( i ) | + 1, meaning that (cid:104) l , i, j (cid:105) needs to be output | V ( i ) | + 1 − p times later. Toselect an edge v i v j , the machine M will be constrained to select this edge among theedges incident to v i . To achieve this, edges from V ( i ) to V ( j ) appear in x grouped bythe vertex from V ( i ) on which they are incident. After each group of edges incidenton one vertex from V ( i ), there is a special state where (cid:104) l , i, j (cid:105) is output if and onlyif the edge towards V ( j ) has already been selected. As (cid:104) l , i, j (cid:105) needs to be outputexactly | V ( i ) | + 1 − p times, we force in this way that an edge is selected which isincident on v i . This enforces that all edges selected in the i th part are incident on thesame vertex. It remains to make sure that distinct parts i and j select the same edgebetween V ( i ) and V ( j ). This is again achieved by a census requirement where a partof the census of letter (cid:104) l , ¯ e , i, j (cid:105) is output in the i th part and the remaining part inthe j th part of M . Theorem 7. var-
GWMM is W [1] -hard.Proof. Let ( k, G = ( V (1) ∪ V (2) . . . ∪ V ( k ) , E )) be an instance of MCC . Suppose V ( i ) = { v i, , v i, , . . . , v i, | V ( i ) | } is the vertex set of color i , for each color class i ∈ [ k ], E = { e , e , . . . , e | E | } , and E ( i, j ) = { e ( i, j, , e ( i, j, , . . . , e ( i, j, | E ( i, j ) | ) } is thesubset of edges with one vertex in color class i and the other in color class j , for i, j ∈ [ k ]. Moreover, suppose E ( i, j ) follows the same order as E , that is if e p = e ( i, j, p (cid:48) ), e q = e ( i, j, q (cid:48) ), and p ≤ q , then p (cid:48) ≤ q (cid:48) . For a vertex v i,p and two integers j ∈ [ k ] \ { i } and q ∈ [ d V ( j ) ( v i,p ) + 1], we define gap ( v i,p , j, q ) = t − s , where e ( i, j, t ) is the q th edgein E ( i, j ) incident to v i,p (respectively, t = | E ( i, j ) | if q = d V ( j ) ( v i,p ) + 1) and e ( i, j, s )is the ( q − th edge in E ( i, j ) incident to v i,p (respectively, s = 0 if q = 1).We construct an instance ( M = ( S, s , Γ, Σ, T ) , x, c ) for var - GWMM as follows. M ’s input alphabet, Γ , is {(cid:104) i (cid:105) , (cid:104) i, j (cid:105) , (cid:104) ¯ e , i, j (cid:105) , (cid:104) e , i, j (cid:105) : i, j ∈ [ k ] , i (cid:54) = j } . M ’s outputalphabet, Σ , is { (cid:15) } ∪ {(cid:104) l , i, j (cid:105) , (cid:104) l , ¯ e , i, j (cid:105) : i, j ∈ [ k ] , i (cid:54) = j } . The word x is defined x := x x . . . x k x i := x i, x i, . . . x i,i − x i,i +1 x i,i +2 . . . x i,k (cid:104) i (cid:105) ∀ i ∈ [ k ] x i, := ( (cid:104) i, (cid:105)(cid:104) i, (cid:105) . . . (cid:104) i, i − (cid:105)(cid:104) i, i + 1 (cid:105)(cid:104) i, i + 2 (cid:105) . . . (cid:104) i, k (cid:105) ) | V ( i ) | ∀ i ∈ [ k ] x i,j := (cid:104) i, j (cid:105) x i,j, (cid:104) i, j (cid:105) x i,j, . . . (cid:104) i, j (cid:105) x i,j, | V ( i ) | (cid:104) i, j (cid:105) ∀ i, j ∈ [ k ] , i (cid:54) = jx i,j,p := (cid:104) ¯ e , i, j (cid:105) gap ( v i,p ,j, (cid:104) e , i, j (cid:105)(cid:104) ¯ e , i, j (cid:105) gap ( v i,p ,j, (cid:104) e , i, j (cid:105) . . . (cid:104) ¯ e , i, j (cid:105) gap ( v i,p ,j,d V ( j ) ( v i,p )) (cid:104) e , i, j (cid:105)(cid:104) ¯ e , i, j (cid:105) gap ( v i,p ,j,d V ( j ) ( v i,p )+1) . The census requirement c is, for every i, j ∈ [ k ] , i (cid:54) = j , c ( (cid:104) l , i, j (cid:105) ) := | V ( i ) | + 1 c ( (cid:104) l , ¯ e , i, j (cid:105) ) := | E ( i, j ) | . On reading a subword x i , the Mealy machine will select a vertex v i,p in V ( i ) and oneedge incident to v i,p for each color class j ∈ [ k ] \ { i } . The vertex v i,p is selected inthe subword x i, of x i . Next, for each j ∈ [ k ] \ { i } , a vertex in V ( i ) and a vertex in V ( j ) are selected in the subword x i,j . The census requirement for (cid:104) l , i, j (cid:105) makes surethat the vertex from V ( i ) is v i,p . The subword x i,j,p ensures that v i,p and the vertexthat is selected from V ( j ) are joined by an edge. Finally, the census requirement for (cid:104) l , ¯ e , i, j (cid:105) is responsible for the inter-partition communication and makes sure thatthe edge selected in x i,j is equal to the edge selected in x j,i .The Mealy machine M consists of k parts. The i th part of M is depicted in Fig. 1.Its initial state is s v, . There is a transition from the last state of each part, s (4) e,i,k , tothe first state of the following part, s v,i +1 (from the k th part, there is a transition to afinal state): it reads the letter (cid:104) i (cid:105) and writes the letter (cid:15) . We set (cid:104) l (cid:48) , ¯ e , i, j (cid:105) = (cid:104) l , ¯ e , j, i (cid:105) for all i (cid:54) = j ∈ [ k ]. Note that, in the description of M , the letter (cid:104) l , i, j (cid:105) can onlybe output on reading (cid:104) i, j (cid:105) , and (cid:104) l , ¯ e , i, j (cid:105) can only be output on reading (cid:104) ¯ e , i, j (cid:105) or (cid:104) ¯ e , j, i (cid:105) .Let us first verify that the parameter for var - GWMM is a function of k , andthat there exists a function f such that the size of the instance for var - GWMM is f ( k ) · n O (1) , where n is the number of vertices of G . We have | Γ | = k + 3 · k · ( k − | Σ | = 1 + 2 · k · ( k − | S | = 1 + k · (2 + 4 · ( k − var - GWMM is thus bounded by a function of k . The length of x is O ( k · n ). Now, we show that( M, x, c ) is a
Yes -instance for var - GWMM if and only if (
G, k ) is a
Yes -instancefor
MCC .First, suppose ( M = ( S, s , Γ, Σ, T ) , x, c ) is a Yes -instance for var - GWMM .We say that M selects a vertex v i,p if it makes a transition from state s v,i to state s (cid:48) v,i reading (cid:104) i, k (cid:105) (respectively (cid:104) i, k − (cid:105) if i = k ) for the p th time. In other words,in the i th part of M , it reads p · ( k − − x i, , staying in state s v,i andoutputs the letter (cid:104) l , i, r (cid:105) for each letter (cid:104) i, r (cid:105) it reads; then it transitions to state s (cid:48) v,i on reading (cid:104) i, k (cid:105) (respectively (cid:104) k, k − (cid:105) if i = k ) and outputs (cid:104) l , i, k (cid:105) (respectively (cid:104) l , k, k − (cid:105) ); in the state s (cid:48) v,i it outputs the empty letter for each letter (cid:104) i, r (cid:105) it reads. arameterizing by the Number of Numbers 13 s v,i s v,i s (1) e,i, s (2) e,i, s (3) e,i, s (4) e,i, s (1) e,i, s (2) e,i, s (3) e,i, s (4) e,i, s (1) e,i,k s (2) e,i,k s (3) e,i,k s (4) e,i,k h i − i ,(cid:15) h i, i , h l ,i, ih i, i , h l ,i, i ... h i,k i , h l ,i,k i h i,k i , h l ,i,k i h i, i ,(cid:15) h i, i ,(cid:15)... h i,k i ,(cid:15) h i, i ,(cid:15) h i, i ,(cid:15) h e ,i, i ,(cid:15) h ¯ e ,i, i ,(cid:15) h ¯ e ,i, i , h l , ¯ e,i, i h e ,i, i ,(cid:15) h ¯ e ,i, i , h l , ¯ e ,i, i h e ,i, i ,(cid:15) h e ,i, i ,(cid:15) h ¯ e ,i, i , h l , ¯ e ,i, ih i, i , h l ,i, ih i, i , h l ,i, ih e ,i, i ,(cid:15) h ¯ e ,i, i ,(cid:15) h i, i ,(cid:15) h i, i ,(cid:15) h e ,i, i ,(cid:15) h ¯ e ,i, i ,(cid:15) h ¯ e ,i, i , h l , ¯ e ,i, ih e ,i, i ,(cid:15) h ¯ e ,i, i , h l , ¯ e ,i, ih e ,i, i ,(cid:15) h e ,i, i ,(cid:15) h ¯ e ,i, i , h l , ¯ e ,i, ih i, i , h l ,i, i h i, i , h l ,i, ih e ,i, i ,(cid:15) h ¯ e ,i, i ,(cid:15) h i,k i ,(cid:15) h e ,i,k i ,(cid:15) h ¯ e ,i,k i ,(cid:15) h ¯ e ,i,k i , h l , ¯ e ,i,k i h e ,i,k i ,(cid:15) h ¯ e ,i,k i , h l , ¯ e ,i,k i h e ,i,k i ,(cid:15) h e ,i,k i ,(cid:15) h ¯ e ,i,k i , h l , ¯ e ,i,k ih i,k i , h l ,i,k ih i,k i , h l ,i,k ih e ,i,k i ,(cid:15) h ¯ e ,i,k i ,(cid:15) h i i ,(cid:15) Fig. 1.
The i th part of the Mealy machine M . It does not have the states s (1) e,i,i , s (2) e,i,i , s (3) e,i,i , and s (4) e,i,i ;there is instead a transition from s (4) e,i,i − to s (1) e,i,i +1 reading (cid:104) i − (cid:105) and writing (cid:15) , and there is atransition from s (4) e,k,k − to the last state reading (cid:104) k (cid:105) and writing (cid:15) . There are no transitions startingat the last state. (Drawing all this would have cluttered the figure too much.) We say that M selects an edge e ( i, j, q ) if it makes a transition from state s (2) e,i,j to state s (3) e,i,j after having read the letter (cid:104) ¯ e , i, j (cid:105) of x i,j,p exactly q times, where v i,p is the vertex of color i that e ( i, j, q ) is incident on. In other words, in the i th part of M , it transitions from the state s (1) e,i,j to the state s (2) e,i,j on reading the first letter of x i,j,p (if it did this transition any later, the census requirement of (cid:104) l , ¯ e , i, j (cid:105) could notbe met, as shown in the proof of Claim 2 below); then it stays in the state s (2) e,i,j untilit has read q times the letter (cid:104) ¯ e , i, j (cid:105) of x i,j,p ; then it transitions to the state s (3) e,i,j onreading (cid:104) e , i, j (cid:105) ; it stays in this state and outputs (cid:104) l (cid:48) , ¯ e , i, j (cid:105) for each letter (cid:104) ¯ e , i, j (cid:105) itreads until transitioning to the state s (4) e,i,j on reading the letter following x i,j,p . The following claims ensure that the edge-selection and the vertex-selection arecompatible, i.e., that exactly one edge is selected from color i to color j , and thatthis edge is incident on the selected vertex of color i . Claim 1.
Let i be a color and let v i,p be the vertex selected in the i th part of M . Inits i th part, M selects one edge incident to v i,p and to a vertex of color j , for each j ∈ [ k ] \ { i } .Proof. After M has selected v i,p , it has output p times each of the letters (cid:104) l , i, (cid:105) , (cid:104) l , i, (cid:105) , . . . , (cid:104) l , i, i − (cid:105) , (cid:104) l , i, i + 1 (cid:105) , (cid:104) l , i, i + 2 (cid:105) , . . . , (cid:104) l , i, k (cid:105) . For each j ∈ [ k ] \ { i } , theonly other transitions that output (cid:104) l , i, j (cid:105) are the transition from s (3) e,i,j to s (4) e,i,j and atransition that loops on s (4) e,i,j . To meet the census requirement of | V ( i ) | + 1 for (cid:104) l , i, j (cid:105) , M selects an edge while reading x i,j,p . This edge is incident on v i,p by construction. (cid:117)(cid:116) The following claim makes sure that the edge selected from color i to color j is thesame as the edge selected from color j to color i . Claim 2.
Suppose M selects the edge e ( i, j, q ) in its i th part. Then, M selects theedge e ( j, i, q ) in its j th part.Proof. Before M selects e ( i, j, q ), it has output q (cid:48) ≤ q times the letter (cid:104) l , ¯ e , i, j (cid:105) . Onselecting e ( i, j, q ) it transitions to the state s (3) e,i,j , and after the selection it outputs (cid:104) l (cid:48) , ¯ e, i, j (cid:105) for every letter (cid:104) ¯ e , i, j (cid:105) of x i,j,p it reads. As it reads( d V ( j ) ( v i,p )+1 (cid:88) r =1 gap ( v i,p , j, r )) − q = | E ( i, j ) | − q times the letter (cid:104) ¯ e , i, j (cid:105) of x i,j,p after it has selected e ( i, j, q ), the Mealy machineoutputs | E ( i, j ) | − q times the letter (cid:104) l (cid:48) , ¯ e, i, j (cid:105) in its i th part.The only other transition where it outputs (cid:104) l , ¯ e, i, j (cid:105) = (cid:104) l (cid:48) , ¯ e, j, i (cid:105) is the transitionin the j th part of M looping on s (3) e,j,i that reads (cid:104) ¯ e , j, i (cid:105) and outputs (cid:104) l (cid:48) , ¯ e , j, i (cid:105) . To meetthe census requirement for (cid:104) l , ¯ e, i, j (cid:105) , this transition must be used exactly | E ( i, j ) | − q (cid:48) times.The only other transitions where it outputs (cid:104) l (cid:48) , ¯ e, i, j (cid:105) = (cid:104) l , ¯ e, j, i (cid:105) are two transi-tions in the j th part of M : the transition from s (1) e,j,i to s (2) e,j,i and the transition loopingon s (2) e,j,i , both reading (cid:104) ¯ e , j, i (cid:105) and writing (cid:104) l , ¯ e, j, i (cid:105) . These transitions can be used atmost q (cid:48) times as the transition of the previous paragraph is used | E ( i, j ) | − q (cid:48) times.These transitions have to be used at least q times to meet the census requirement for (cid:104) l (cid:48) , ¯ e , i, j (cid:105) . Thus, these transitions are used exactly q times and q = q (cid:48) .Finally, the transition from s (2) e,j,i to s (3) e,j,i happens after having read q times theletter (cid:104) ¯ e , j, i (cid:105) of some vertex x j,i,p (cid:48) , p (cid:48) ∈ [ | V ( j ) | ], which means that M selects the edge e ( j, i, q ) in its j th part. (cid:117)(cid:116) By Claims 1 and 2, the k vertices that are selected by M form a multicolored clique.Thus, ( k, G = ( V (1) ∪ V (2) . . . ∪ V ( k ) , E )) is a Yes -instance for
MCC . arameterizing by the Number of Numbers 15 Now, suppose that ( k, G = ( V (1) ∪ V (2) . . . ∪ V ( k ) , E )) is a Yes -instance for
MCC .Let { v ,p , v ,p , . . . , v k,p k } be a multicolored clique in G . We will construct a word y meeting c such that a computation of M on input x generates y . For two adjacentvertices v i,p i and v j,p j , define edge ( v i,p i , v j,p j ) = t such that e ( i, j, t ) = v i,p i v j,p j . Theword y is y y . . . y k , where y i , for i ∈ [ k ], is( (cid:104) l , i, (cid:105)(cid:104) l , i, (cid:105) . . . (cid:104) l , i, i − (cid:105)(cid:104) l , i, i + 1 (cid:105)(cid:104) l , i, i + 2 (cid:105) . . . (cid:104) l , i, k (cid:105) ) p i y i, y i, . . . y i,i − y i,i +1 y i,i +2 . . . y i,k and y i,j , for i (cid:54) = j ∈ [ k ], is (cid:104) l , ¯ e, i, j (cid:105) edge ( v i,pi ,v j,pj ) (cid:104) l (cid:48) , ¯ e , i, j (cid:105) | E ( i,j ) |− edge ( v i,pi ,v j,pj ) (cid:104) l , i, j (cid:105) | V ( i ) |− p i +1 . We note that y meets the census requirement c . Moreover, the computation of M oninput x , which selects (as defined in the first part of the proof) exactly the verticesand edges of the multicolored clique { v ,p , v ,p , . . . , v k,p k } , outputs y . Thus ( M =( S, s , Γ, Σ, T ) , x, c ) is a Yes -instance for var - GWMM . (cid:117)(cid:116) The theorem holds if we restrict (cid:15) / ∈ Γ ∪ Σ . Indeed, (cid:15) / ∈ Γ in the target instance,and one can add a new letter e to Σ , which replaces (cid:15) and has census requirement c ( e ) = | x | − (cid:80) i,j ∈ [ k ] ,i (cid:54) = j ( c ( (cid:104) l , i, j (cid:105) ) + c ( (cid:104) l , ¯ e , i, j (cid:105) ). This instance is equivalent since themodified M outputs one letter for each letter in x . In this section we sketch two examples that illustrate how number-of-numbers pa-rameterized problems may reduce to census problems about Mealy machines, param-eterized by the size of the machine. For another application, see [10].
Example 1: Heat-Sensitive Scheduling.
In a recent paper Chrobak et al. [5]introduced a model for the issue of temperature-aware task scheduling for micropro-cessor systems. The motivation is that different jobs with the same time requirementsmay generate different heat loads, and it may be important to schedule the jobs sothat some temperature threshold is not breached.In the model, the input consists of a set of jobs that are all assumed to be ofunit length, with each job assigned a numerical heat level. If at time t the processortemperature is T t , and if the next job that is scheduled has heat level H , then theprocessor temperature at time t + 1 is T t +1 = ( T t + H ) / t + 1 (that is, idle time isscheduled), in which case H = 0 in the above calculation of the updated temperature.The relevant decision problem is whether all of the jobs can be scheduled, meet-ing a specified deadline, in such a way that a given temperature threshold is neverexceeded. This problem has been shown to be NP-hard [5] by a reduction from . An image instance of the reduction, however, involves arbitrarily many distinct heat levels asymptotically close to H = 2, for a tempera-ture threshold of 1.In the spirit of the “deconstruction of hardness proofs” advocated by Komusiewiczet al. [19] (see also [4, 24]), one might regard this problem as ripe for parameterizationby the number of numbers, for example (scaling appropriately), a model based on 2 k equally-spaced heat levels and a temperature threshold of k . Furthermore, if the heatlevels of the jobs are only roughly classified in this way, it also makes sense to treatthe temperature transition model similarly, as: T t +1 = (cid:100) ( T t + H ) / (cid:101) The input to the problem can now be viewed equivalently as a census of howmany jobs there are for each of the 2 k + 1 heat levels, with the available potentialunits of idle time allowed to meet the deadline treated as “jobs” for which H = 0.Because of the ceiling function modeling the temperature transition, the problem nowimmediately reduces to var - EWMM , for a machine on k + 1 states (that representthe temperature of the processor) and an alphabet of size at most 2 k +1. By Theorem5, the problem is fixed-parameter tractable. Example 2: A Problem in Computational Chemistry.
The parameterizedproblem of
Weighted Splits Reconstruction for Paths that arises in com-putational chemistry [15] reduces to a special case of var - GWMM . The input tothe problem is obtained from time-series spectrographic data concerning molecularweights. The problem as defined in [15] is equivalent to the following two-processorscheduling problem. The input consists of – a sequence x of positive integer time gaps taken from a set of positive integers Γ ,and – a census requirement c on a set of positive integers Σ of job lengths .The question is whether there is a “winning play” for the following one-person two-processor scheduling game. At each step, first, Nature plays the next positive integer“gap” of the sequence of time gaps x — this establishes the next immediate deadline .Second, the Player responds by scheduling on one of the two processors, a job thatbegins at the last stop-time on that processor, and ends at the immediate deadline.The
Player wins if there is a sequence of plays (against x ) that meets the censusrequirement c on job lengths. Fig. 2 illustrates such a game. Processor 1 4 3 3Processor 2 5 3 1 5 x = 4 1 2 1 1 1 4 Fig. 2.
A winning game for the census: 1 (1), 3 (3), 4 (1), 5 (2)
This problem easily reduces to a special case of var - GWMM . Whether this specialcase is also W [1]-hard remains open. arameterizing by the Number of Numbers 17 The practical world of computing is full of computational problems where inputsare “weighted” in a realistic model — weighted graphs provide a simple examplerelevant to many applications. Here we have begun to explore parameterizing on the numbers of numbers as a way of mitigating computational complexity for problemsthat are numerically structured. One might view some of the impulse here as movingapproximation issues into the modeling , as illustrated by Example 1 in Section 6. Webelieve this line of attack may be widely applicable.To date, there has been little attention to parameterized complexity issues inthe context of cryptography, control theory, and other numerically structered areasof application. Number of numbers parameterization may provide some inroads intothese underdeveloped areas.Our main FPT result, Theorem 5, has a poor worst-case running-time guarantee.Can this be improved – at least in important special cases?
Acknowledgment.
We thank Iyad Kanj for stimulating conversations about thiswork.
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