9 Reductions

In Part I, we looked at a number of methods of establishing~xed-parameter tractability. Part II is devoted to problems where there is no known method which enables us to demonstrate fixed-parameter tractability. We are concerned with the situation where we can prove, or at least have good evidence to believe, that no such method exists. As we will see, for some combinatorial problems, we can actually prove intractability. However, as with classical NP and PSPACE "intractability," for most interesting natural problems we can only offer a completeness theory.

It is important to note that, although we share some methodology with classical NP-completeness, our theory is quite different. However, to motivate the method­ology, we will begin by reviewing classical NP-completeness. As we see it, the basic idea ~hind virtually all completeness results is the notion of a reduction. For an appropriate mode of computation, a reduction of a language L to a lan­guage L' is a method that computes L from L' in this mode. The best known classical examples are polynomial-time m-reductions (:::~) and polynomial time T -reductions (::::.) For languages Land L', we recall that L :::: L' iff there is a polynomial-time Oracle Turing Machine ct> such that for all x E E*, x E L iff ct>L' (x) = 1. Polynomial-time T-reductions are sometimes called Cook reduc­tions in honor of Stephen Cook who first studied them [131]. Polynomial-time m-reductions, or Karp reductions, were introduced by Richard Karp in his classic article [291]. They are often called polynomial-time transformations and are a special case of polynomial-time T -reductions. We recall that L :::~ L' iff there is a polynomial-time computable function f : E* ~ E*, such that for all x E E*, X E L iff f(x) E L'. Karp demonstrated the importance of polynomial time m­reductions by constructing transformations between many natural combinatorial problems in a wide variety of diverse areas, thereby revealing how widespread the NP-completeness phenomenon is.

R. G. Downey et al., Parameterized Complexity© Springer-Verlag New York, Inc. 1999

228 9. Reductions

Example 9.1 A polynomial-time m-reduction. Consider the following classical problems.

CNF SATISFIABILITY

Input: A formula X of propositional logic in conjunctive normal form (i.e., a collection of clauses).

Question:

3SAT

Input:

Question:

Is X satisfiable? That is, is there an assignment of the variables that makes X true?

A formula X of propositional logic in conjunctive normal form with maximum clause size 3. Is X satisfiable?

The classical result of Karp [291] is that CNF SATISFIABILITY ~~ 3SAT and hence CNF SATISFIABILITY =~ 3SAT. The transformation is the following. Let X be an instance of CNF SATISFIABILITY. We compute in polynomial time f (X) an instance of 3SAT such that X is satisfiable iff f (X) is satisfiable. Without loss of generality, we may assume that each clause of X has 2: 3 literals. Let U = {Ut, ... , urn} be a clause of X. We replace this clause by the following clauses:

where {Zt, ... , Zrn-3} are new variables, chosen for this clause. Let f (X) be the result obtained from X by performing this local replacement for each clause of X. Notice that the size of f(X) is bounded by 3mq, where m is the size of the largest clause and q is the number of clauses. Also, notice that any satisfying assignment for the collection U' must make U true, and hence X is satisfiable iff f(X) is satisfiable.

Example 9.2 A polynomial-time T -reduction. Nice examples of this sort of thing are often provided by showing that a search problem reduces to a decision problem. One such example is provided by

HAMILTON PATH

Input: Question:

A graph G. Does G have a path which travels through every vertex of G exactly once?

Although it is not a decision problem, we are often interested in the associated search problem: If G has a Hamilton path, find one or else return that G has no Hamilton path. Of course, if we can do the search problem in polynomial time, we can do the decision problem. The algorithm that reduces the search problem to the decision one is the following. Given an oracle Q for the decision problem and a graph G, first use Q to see if G has a Hamilton path. If the answer is no, then output that G has no Hamilton path. If Q tells us that G does have a Hamilton path, then choose any edge e of G and form the graph G' = G - e. Now ask Q if G' has a

9. Reductions 229

Hamilton path. If the answer is no, then we can conclude that every Hamilton path in G includes the edge e, and we should resolve never again to delete e. However, we can try again with a different edge e', forming G' = G - e'. It is obvious that if G is not simply a path (in which case we are done) then by repeatedly trying different edges we can eventually find an edge e such that for G' = G - e we will be informed by n that the answer for G' is yes. By making at most one query for each edge in the graph, we have thus narrowed our search to a sub graph G' of G with one fewer edge. Repeating this process, we will eventually identify a subgraph G' that is a Hamilton path in G, having made (in view of our resolutions along the way) at most one query to n for each edge in G.

Notice that the key difference between the T - and the m-reductions pointed out in the above examples is that in the m-reduction we compute f (x) and then ask a single question of n, namely if f (x) is in L', whereas in the T -reduction, many questions need to be asked of the oracle n to get the result. Also the reduction is adaptive in the sense that answers from earlier questions affect the next question. Even the two examples above suggest many natural questions. For instance, is the solution of the search problem from the decision problem intrinsically sequential? It has been shown that if we can solve the search problem from the decision problem without the use of sequential questions then many unlikely collapses occur for complexity theory. (See, e.g., [9] and [97].)

Reductions allow us to partially order languages in terms of their computational complexity. Two languages, Land L', are taken to have the same complexity from the point of view of the given reduction iff they are in the same degree; that is, there is a reduction from L to L' and vice versa. We will need to create reductions which express the fact that two languages have the same parameterized complexity. What is needed are reductions that ensure that if L' is computable in time f(k)n C "by the slice," then there is a parameterized algorithm for L running in time g(k)n C' "by the slice." After a moment's thought, we realize that we can achieve such reductions only if we allow each slice of L to reduce to a finite number of slices of L'. The easiest way to make such a reduction is to reduce, in parameterized polynomial time, the k-th slice of L to the k' -th slice of L'. This leads us to the working definition, Definition 9.3 below, of a parameterized reduction for combinatorial reductions. From the point of view of our hardness results, this is the most important definition in the book.

Definition 9.3 (Basic Working Definition) We say that L reduces to L' by a standard parameterized m-reduction if there are functions k 1-+ k' and k 1-+ k" from N to N, and a function (x, k) 1-+ x' from ~* x N to ~*, such that

(x, k) 1-+ x' is computable in time k" Ix IC , and

(x, k) E L iff (x', k') E L'.

We remark that usually one can take k' = k".

230 9. Reductions

Example 9.4 The following is a simple example of a standard parameterized reduction. It comes from an observation about a classical reduction of Karp. Define a formula (of propositional logic ) to be monotone if it contains no negation symbols, and antimonotone if it contains no negation symbols except for the literals which are all negated. Consider the following problem:

WEIGHTED CNF SATISFIABILITY

Input: Parameter: Question:

A propositional formula X in conjunctive normal form. A positive integer k. Does X have a satisfying assignment of Hamming weight k? (An assignment has weight k iff it has exactly k variables set to be true.)

WEIGHTED CNF SATISFIABILITY gives us a parameterized language which is a natural parameterized version of CNF SATISFIABILITY. We can similarly define a parameterized version of 3SAT, WEIGHTED 3SAT by considering weight k solu­tions of problems in 3CNF form. Finally, we can define WEIGHTED MONOTONE

CNF SATISFIABILITY and WEIGHTED ANTIMONOTONE CNF SATISFIABILITY by considering only monotone and antimonotone formulas.

We also need another problem.

WEIGHTED BINARY INTEGER PROGRAMMING

Instance: Parameter: Question:

A binary matrix A and a binary vector b. A positive integer k. Does A . x 2: b have a binary solution of weight k?

We have the following:

Theorem 9.S WEIGHTED MONOTONE CNF SATISFIABILITY reduces to WEIGH­

TED BINARY INTEGER PROGRAMMING via a standard reduction.

Proof. Let (X, k) be an instance of MONOTONE WEIGHTED CNF SAT. Let Cl , ... , Cp list the clauses of X and Xl, ... ,Xm list the variables. Let A be the matrix {ai.j : i = 1, ... p, j = 1, ... , m} with ai.j = 1 if Xj is present in Ci , and ai,j = 0 otherwise. Let b be the vector with 1 in the j -th position for j = 1, ... , p. It is easy to see that A . x 2: b has a solution of weight k iff X has a satisfying assignment of weight k (and the reasoning is reversible).

Actually, the situation in Theorem 9.5 is unusual. It is generally not the case that a classical reduction turns out to be also a parameterized reduction for natural parameterized versions of the problems. Classical reductions usually do not carry enough structure. This lack of structure in the classical reductions causes us to lose control of the parameter. For instance, consider the situation of Example 9.1 where CNF SATISFIABILITY is classically reduced to 3SAT. There, we replace a clause with m literals by m - 2, clauses each consisting of 3 literals. For each clause, we introduced m - 3 new variables. Now, suppose the original formula has a weight k satisfying assignment that makes exactly one variable u j in clause

9. Reductions 231

U of X true. Then, to make f(X) true, we would need to make Zj, ... , Zj-2 all true. So the weight of the satisfying assignment of f(X) would not only depend upon k but would depend on j, and hence the clause size of X. So this reduction from CNF SATISFIABILITY to 3SAT does not give a parameterized reduction from WEIGHTED CNF SATISFIABILITY to WEIGHTED 3SAT. It is not known if there is any such parameterized reduction. We will soon see that in fact, there is some evidence that there really is no such reduction, and hence from a parameterized point of view CNF SATISFIABILITY and 3SAT are not the same. We remark that if there are NP-complete problems that are not fixed parameter tractable, then it must be that some of the classical reductions do not lift to parameterized ones since we have seen that some NP-complete problems are fixed-parameter tractable.

Although Definition 9.3 is satisfactory as a working definition, for the general situation we need to be more precise. In Definition 2.4, we have seen that there are three basic flavors of fixed-parameter tractability, depending on the level of uniformity. Naturally, the three types of tractability will relativize to three flavors of reducibility. We get the following technical definitions. [We remark, how­ever, that if the reader is only interested in the concrete hardness results of Part II, then they can ignore the definitions to follow and take Definition 9.3 as the actual definition. We will not be needing the definitions below until Part III, where we look at the fine structure of problems under parameterized reducibilities. ]

Definition 9.6 (Uniform Fixed-Parameter Reducibility) Let A and B be param­eterized problems. We say that A is uniformly jixed-parameter-reducible to B if there is an oracle procedure 4>, a constant IX, and an arbitrary function f : N 1---+ N such that

(a) the running time of 4>(B; (x, k) is at most f(k)lxl a ,

(b) on input (x, k), 4> only asks oracle questions of B(f(k)), where

B(f(k)) = U Bj = {(x, j) : j :::: f(k)&(x, j) E B}, j:,,;J(k)

(c) 4>(B) = A.

If A is uniformly P-reducible to B, we write A ::::~ B. Where appropriate we may say that A ::::~ B via f. If the reduction is many: 1 (an m-reduction), we will write A ::::~ B.

Definition 9.7 (Strongly Uniform Reducibility) Let A and B be parameterized problems. We say that A is strongly uniformly jixed-parameter-reducible to B if A ::::~ B via f where f is recursive. We write A ::::1- B in this case.

Definition 9.8 (Nonuniform Reducibility) Let A and B be parameterized prob­lems. We say that A is nonuniformly jixed-parameter-reducible to B if there is a constant IX, a function f : N 1---+ N, and a collection of procedures {4>k : kEN} such that 4>k(B(f(k))) = Adoreachk E N, and the running time of 4>k is f(k)lxl a .

Here, we write A ::::} B.

232 9. Reductions

Note that the above are good definitions since whenever A ::s B with ::s any of the reducibilities, if B is fixed-parameter tractable, so too is A. Note also that the above definitions allow us to specify the notions of fixed-parameter tractability we had before. Now, nonunifonn fixed-parameter tractability corresponds to being ::Sr 0. We will henceforth write FPT(::s) as the fixed-parameter-tractable class corresponding to the reducibility ::s.

If A ::s B via a standard reduction in the sense of Definition 9.3, then A ::S::, B. Futhennore, in most cases, we will look at A ::s:;' B since the function k t-+ k' will be recursive. Finally, we remark that the standard definition works for natural languages since they all have the following property: for all k, Lk is "directly coded into" (or "trivially represented in") Lk+l. For instance, for any fonnula ({J

in variables XI, ... ,Xn , ({J has a weight k satisfying assignment iff ({J "Xn+1 has a weight k + 1 satisfying assignment. Thus, in a "natural" language L if L' ::s L, then L' reduces to a single slice of L because of this upward encoding of the slices.

Historical Notes

The notions of reducibilities and computability were developed in the 1930s in the setting of classical recursion theory by many authors, notably Post, Kleene, and Church. Of course, the notion of a reduction is in some sense much older: Consider the reduction that a ::s b iff a is, say, algebraic over b. The recognition of the difficulty of NP-complete problems has a very interesting history, and the reader is referred to the illuminating article of Hartrnanis [261]. The groundbreaking articles here are Karp [291], Cook [131], and Levin [330]. The notions of reducibilities we have used were developed by the authors in Downey-Fellows [165], [166], and [167]. We give fuller historical remarks on parameterized complexity at the end of the next chapter.

Exercises 9.0

9.0.1. Prove that INDEPENDENT SET ::s:;' MAXIMAL IRREDUNDANT SET via a standard reduction. These problems are defined as follows. (Bod­laender and Kratsch [88] have proven the apparently stronger result that CNF SATISFIABILITY ::S:;. MAXIMAL IRREDUNDANT SET. The Bodlaender-Kratsch result is apparently stronger because we will soon show that INDEPENDENT SET =:;' WEIGHTED 3SAT, and we believe that WEIGHTED 3SAT <:;' WEIGHTED CNF SATISFIABILITY.)

(k- )INDEPENDENT SET

Input: A graph G. Parameter: A positive integer k. Question: Does G have a set of k vectors {XI, ..• , Xl} such that

for all i i= j, Xi is not adjacent to Xi?

9. Reductions 233

(k-)MAXIMAL IRREDUNDANT SET

Input: A graph G. Parameter: A positive integer k. Question: Does G have a set X of k vertices such that for each

member x of X, there is ayE V(G) such that either y = x or (x, y) is an edge, and y is not adjacent to nor a member of S, and, furthermore, S is maximal with this property?

[Hint. Given a graph G and a k, we construct a graph H and a k' so that G has a k-element independent set iff H has a maximal irredundant set of size k'. H is constructed as follows. Construct k blocks consisting of k + 1 columns, each consisting of n = I G I points. So column i in block j consists of n points x(j, i, t) for t = 1, ... , n. The idea is that a block represents a choice for a vertex in the independent set. Now, within a block, join all vertices except those with the same last coordinate. (We refer to this as a row.) The first column is not connected to anything else and is called the dummy column. Now, for each block i pick a column in each block i' =1= i to identify with block i, and similarly identify a corresponding column i" in block i. Do this in such a way that for each pair (i, j), there is exactly one column ci(i, j) identified with exactly one column Cj (i, j) in block j, and these columns are used for no other identifications, for any other pair except (i, j). Let (q, r) be some such pair of columns. Now, connect each xCi, q, t) to x(j, r, s) iff t and s are adjacent vertices in G.

Now, let k' = k(k + 1). First, if G has an independent set of size k, let {tl, ... tk-d be this independent set. The relevant irredundant set for H is obtained by taking the union of vertices of the ti-th row of block i, for i = 1, ... , k. They are each their own private neighbors, as the corresponding set in G is independent, and it is maximal by the fact that the blocks are almost cliques. Conversely, take S to be a size k' maximal irredundant set in H. Prove that it must be of the form above and, hence, corresponds to an independent set in G of size k. There are two cases depending on whether some block B

has more than k vertices of S. Remember that a maximal irredundant set must also be a dominating set.]

9.0.2. Prove that WEIGHTED ANTIMONOTONE 2SAT (see Example 9.4) reduces to COLORED AUTOMORPlllSM (below) by a standard reduction.

COLORED AUTOMORPmSM

Instance: Parameter: Question:

A 2-colored (bipartite) graph G. A positive integer k. Is there an automorphism preserving colors moving exactly k blue vertices?

[Hint: Let X be an instance of WEIGHTED ANTIMONOTONE 2-CNFSAT with parameter k. Let C I , ... Cm be the clauses of X and U\, •.• , Un the vari­ables which will all appear in the clauses as negated since X is antimonotone. The graph G(X) is constructed as follows:

234 9. Reductions

For each U;, we have two blue nodes u(i) and u(i). For each clause Cj , we have three red nodes C(j, k) : k = 1,2,3. For each i, append to each u(i) and u(i) a rigid tree T; so that additionally for all i oj: j, T; is not isomorphic to 1j. All the nodes of the T; are red. (This will force u(i) to go to u(i) if it moves at all.) Finally, we need some further edges. Let Cj = {up, Uq1. Add the edges u(p)C(j, I), u(q)C(j, I), u(p)C(j, 2), u(q)C(j, 2), and u(p)C(j, 3), u(q)C(j,3).

Prove that X has a weight k satisfying assignment iff G(X) has an automorphism that moves exactly 2k blue vertices.]

9.0.3. (The Advice View.) In Part I, Chapter 5, we saw that an alternative view of parameterized tractability is provided by considering FPT to be the class of parameterized problems that are solvable in time c . na

by an oracle Turing machine <l> with a finite piece of advice f(k). We can define A -:::'advice B via f : N 1-+ ~* if there exists an oracle Turing machine <l> running in time na such that

(x, k) E A iff <l>B(f[k»E9f(k) «(x, k)) = 1.

Prove that A -:::.'T B iff A -:::'advice B.

9.0.4. (i) Prove that there exists a language L E FPT(-:::.':,.) such that L ~ FPT(-:::'T)'

(ii) Prove that there exists a recursive language L' E FPT(-:::.':r,) with L' ~ FPT(-:::''T).

[Hint: (i) Construct a language L = UsLs in stages to diagonalize against all possible ::::~ reductions. The reader should see Theorem 19.21 in Chapter 19.]