Perfect Code is W[1]-complete

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1 Information Processing Letters 81 (2002) Perfect Code is W[1]-complete Marco Cesati Department of Computer Science, Systems, and Industrial Engineering, University of Rome Tor Vergata, via di Tor Vergata 110, I Rome, Italy Received 3 October 2000; received in revised form 1 March 2001 Communicated by L.A. Hemaspaandra Abstract We show that the parameterized problem PERFECT CODE belongs to W[1]. This result closes an old open question, because it was often conjectured that PERFECT CODE could be a natural problem having complexity degree intermediate between W[1] and W[2]. This result also shows W[1]-membership of the parameterized problem WEIGHTED EXACT CNF SATISFIABILITY Elsevier Science B.V. All rights reserved. Keywords: Computational complexity; Parameterized computational complexity; Perfect Code; Weighted Exact Conjunctive Normal Form Satisfiability; W[1]-completeness 1. Introduction Parameterized Complexity [8] was introduced by Downey and Fellows about ten years ago. It is a powerful framework with which to address the different parameterized behavior of many computational problems. Almost all natural problems have instances consisting of at least two logical items; many NPcomplete problems admit efficient algorithms for small values of one item (the parameter). A parameterized problem is said to be fixed parameter tractable if there exists a decision algorithm whose running time on instance (x, k) is bounded by f(k) x α,wheref is an arbitrary function and α is a constant not depending on the parameter k. The class of fixed parameter tractable problems is denoted by FPT. In order to characterize those problems that do not seem to admit a fixed parameter efficient algorithm, Downey and Fellows defined a fixed parameter reduc- address: cesati@uniroma2.it (M. Cesati). tion and a hierarchy of classes W[1] W[2] including likely fixed parameter intractable problems. Each W-class is the closure under fixed parameter reductions with respect to a kernel problem, which is usually formulated in terms of special mixed-type Boolean circuits. Many natural parameterized problems have been proved to be complete for the first two levels W[1] and W[2]. Although W[1]-complete problems are not fixed parameter tractable (unless W[1] =FPT, which is very unlikely), they appear to be easier than W[2]-complete problems. Essentially, candidate solutions for W[1]-complete problems (like INDEPEN- DENT SET [6,8]) can be verified using constant-depth Boolean circuits having just one level of gates with unbounded fan-in, while solutions for W[2]-complete problems (like DOMINATING SET [5,8]) do not. The parameterized complexity of a problem complete for a W-class is precisely determined. However, many problems have been shown to be hard for some level and to belong to a higher level in the hierarchy /02/$ see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S (01)

2 164 M. Cesati / Information Processing Letters 81 (2002) Although we can safely assume that these problems are not fixed parameter tractable, it seems that we still lack some information about their complexity. Citing from Downey and Fellows book [8, p. 487]: [...] we find that there are many annoying gaps between hardness and membership results. For example, there is the PERFECT CODE problem that is hard for W[1] and a member of W[2]. It is very annoying not to know exactly where to put it. We conjecture that it either represents a natural degree intermediate between W[1] and W[2], or is complete for W[2]. In any case, it has resisted strenuous efforts to achieve a precise classification. This paper solves the question of the precise degree of parameterized complexity of the PERFECT CODE problem: we show that it belongs to W[1], and thus that it is W[1]-complete. As a corollary, the WEIGHTED EXACT CNF SATISFIABILITY problem, which is equivalent to PERFECT CODE, also belongs to W[1]. The structure of the paper is as follows. In Section 2 we introduce basic notions from Parameterized Complexity theory. In Section 3 we introduce the PERFECT CODE problem and we show that it belongs to W[1].In Section 4 we introduce the WEIGHTED EXACT CNF SATISFIABILITY problem. Finally, in Section 5 we discuss how the proof technique described in Section 3 could be used to establish other W-class membership results. 2. Parameterized computational complexity A parameterized problem is a set L Σ Σ, where Σ is a fixed alphabet. For convenience, we can always assume that L Σ N. 1 We say that a parameterized problem L is (uniformly) fixed parameter tractable if there exists a decision algorithm whose running time on instance (x, k) is bounded by f(k) x α,wheref is an arbitrary function and α is a constant not depending on the parameter k. The class of fixed parameter tractable problems is denoted by FPT. 1 In other words, it is always possible to establish a bijective mapping between the encoding of the parameter and the set of natural numbers N. Observe that the mapping function does not need to be polynomial. A parameterized problem A is (uniformly many: 1) reducible to a parameterized problem B if there is an algorithm Φ which transforms an instance (x, k) of A into an instance (x,k ) of B, and such that: (1) (x, k) A if and only if (x,k ) B; (2) k depends only on k (i.e., there exists a function g such that k = g(k)); (3) Φ s running time is bounded by f(k) x α,for some arbitrary function f and constant α. Consider Boolean circuits which have two kinds of gates: small gates with bounded fan-in (usually, at most 2), and large gates with unrestricted fan-in. The depth of a circuit C is defined to be the maximum number of small and large gates on an input output path in C. Theweft of C is the maximum number of large gates on an input output path in C.Theweight of a Boolean vector x is the number of 1 s in the vector. A decision circuit is a Boolean circuit having just one output line. A decision circuit is said to accept every input vector that forces the value 1 on the output line. Let F be a family of decision circuits, possibly having several circuits with a given number of input lines. Let L F ={(C, k): C F accepts an input vector of weight k}. A parameterized problem L belongs to W[t] if there is a constant h such that L reduces to the parameterized problem L F(t,h),whereF(t,h) is the family of decision circuits of weft at most t and depth at most h. Thus, there is a hierarchy of W classes: FPT W[1] W[2] W[t]. From these definitions, if a parameterized problem A reduces to B, andb W[t], thena W[t] as well. Many parameterized problems have been proved to be complete for W[1] and W[2], thatis,thefirsttwo levels of the W-hierarchy. As an example, INDEPEN- DENT SET is W[1]-complete [6]. Informally, W[1]- membership means that there exists an algorithm that, given a graph G and a parameter k, produces in time bounded by f(k) G α a constant-depth decision circuit C having just one large gate on every input output path, and such that C accepts an input vector x of weight k if and only if x encodes an independent set of G. W[1]-hardness implies that all parameterized problems in W[1] reduce to INDEPENDENT SET;since there are many problems in W[1] for which no FPT

3 M. Cesati / Information Processing Letters 81 (2002) algorithm is known, it is very unlikely that INDEPEN- DENT SET is fixed parameter tractable The PERFECT CODE problem A perfect code in a graph G = (V, E) is a subset of vertices V such that for each vertex v V,the subset V includes exactly one element of the closed neighborhood N[v] of v, that is, exactly one element among v and all vertices adjacent to v. Formally: PERFECT CODE Instance: AgraphG = (V, E); a positive integer k N. Question: Does G have a k-element perfect code? A perfect code is a set of vertices V V with the property that for each vertex v V there is precisely one vertex in N[v] V. Downey and Fellows proved that the PERFECT CODE problem is W[1]-hard by means of a reduction from INDEPENDENT SET [6]. Although PERFECT CODE may be easily placed in W[2] (see [6]), until now there was no evidence that it belongs to W[1]. Downey and Fellows conjectured that the problem could be of difficulty intermediate between W[1] and W[2], and thus not W[1]-complete [8, pp. 277, 458, 487]. The following theorem is the main result of this paper. Theorem 1. The PERFECT CODE problem belongs to W[1]. Proof. We show a parameterized reduction from PER- FECT CODE to the following problem: SHORT NONDETERMINISTIC TURING MACHINE COMPUTATION Instance: A single-tape, single-head nondeterministic Turing machine M; awordx on the alphabet of M; a positive integer k N. 2 Moreover, Downey and Fellows proved that if INDEPENDENT SET is fixed parameter tractable, then 3-CNF SATISFIABILITY belongs to DTIME(2 o(n) ),wheren denotes the number of variables in the Boolean formula [8]. Question: Is there a computation of M on input x that reaches a final accepting state in at most k steps? Since SHORT NONDETERMINISTIC TURING MA- CHINE COMPUTATION is W[1]-complete [9,2], thereduction proves that PERFECT CODE belongs to W[1]. Given a graph G = (V, E) with n vertices and a positive integer k, let us construct a nondeterministic Turing machine T = (Σ,Q,,,q A ),whereσ includes the alphabet symbols Σ ={ } {σ v : v V } {s i : i = 1,...,n}, Q contains the internal states Q ={q A,q R } {q i : i = 0,...,k} { q l v,qv r : v V } { qj s : j = 1,...,n}, is the blank symbol, and q A is the final accepting state. (Notice that both the alphabet size Σ and the state set size Q depend on n, hence the Turing machine behavior cannot be predicted with an FPTalgorithm unless W[1]=FPT [4].) When the Turing machine starts, the internal state is q 0 and all tape cells contain the blank symbol ( ). The machine operates in three phases. Phase 1: Guess k vertices. The Turing machine chooses nondeterministically k vertices of G writing the corresponding symbols onto the tape. This is achieved by including the following instructions in the transition table :,q i,σ v,q i+1, +1 ( ) i {0,...,k 2}, v V,,q k 1,σ v,q k, 0 ( v V). (Each instruction specifies, in order, the symbol scanned under the head, the internal state of the machine, the new symbol written by the head, the new internal state, and the movement of the head: 1 forleft,+1 for right, and 0 for no move.) Phase 2: Check that the k vertices are perfect. The Turing machine scans the guessed vertices and rejects as soon as it finds two vertices that violate one of the following conditions: For every pair of guessed vertices x and y, x and y are different. For every pair of guessed vertices x and y, x and y are not adjacent.

4 166 M. Cesati / Information Processing Letters 81 (2002) For every pair of guessed vertices x and y, thereis no vertex z V that is adjacent to both x and y. Moreover, in this phase each symbol σ v is replaced by the symbol s m,wherem represents the size of the neighborhood N[v] of v. This is achieved by including the following instructions into the transition table : σv,q k,σ v,q l v, 1 ( v V), σw,q l v,σ w,q, 1 ( v,w V), where q R if v = w, q R if (v, w) E, q = q R if z V such that (v, z) E and (w, z) E, otherwise, q l v,q l v,,q r v, +1 ( v V), σw,q r v,σ w,q r v, +1 ( v,w V,v w), σv,q r v,s m,q k, 1 ( v V), where m = N[v],,q k,,q k, +1. Phase 3: Taking the sum. The tape now contains exactly k symbols s i ; each of them represents the neighborhood size of a guessed vertex. The Turing machine must accept if and only if the sum of all neighborhood sizes is equal to n. In fact, the checks in Phase 2 grant that no vertex in G belongs to the neighborhood of two different guessed vertices. In other words, the guessed vertices cover non-overlapping subsets of V.Therefore, the sum cannot be greater than n; moreover,if the sum is equal to n, all vertices in G are dominated by the guessed k-element subset of V. The following instructions in computethe sum: si,q k,s i,q s i, +1 ( i {1,...,n} ) sj,q s i,s j,q s t, +1 ( i, j {1,...,n},i+ j n ), where t = i + j,,q s n,,q A, 0,,q s j,,q R, 0 ( j {1,...,n 1} ). It is straightforward to verify that the Turing machine T includes (5/2)n 2 +(k +7/2)n+1instructions, that it can be derived in polynomial time in the size of G, and that it accepts in k 2 + 4k + 2 steps if and only if there exists a perfect code of size k in G. 4. WEIGHTED EXACT CNF SATISFIABILITY The theoretical framework of parameterized computational complexity offers several results that are the analogs of fundamental results in classical computational complexity. In particular, the parameterized analog of the Cook s Theorem is based on the following problem: WEIGHTED t-normalized SATISFIABILITY Instance: A Boolean t-normalized expression E; a positive integer k N. Question: Does E have a satisfying truth assignment of Hamming weight k? Given t 2, E is t-normalized if it is of the form product-of-sums-of-products... of literals with t 1 alternations (that is, t occurrences of products and sums ). Thus, a 2-normalized expression is in conjunctive normal form. The Hamming weight of a truth assignment is the number of variables set to true. Downey and Fellows established [5] that WEIGHTED t-normalized SATISFIABILITY is W[t]-complete, for any t 2. The case t = 1 must be handled separately. In [6] Downey and Fellows proved that the class W[1] can be characterized as the closure under fixed parameter reductions of the WEIGHTED q-cnf SATISFIABILITY problem: the instance is restricted to Boolean expressions in conjunctive normal form and each clause has no more than q literals, where q is a fixed constant. We can now establish another analog of the Cook s Theorem for the class W[1]. In fact, PERFECT CODE is equivalent to the following parameterized version of the SATISFIABILITY problem: WEIGHTED EXACT CNF SATISFIABILITY Instance: A Boolean expression E in conjunctive normal form; a positive integer k N.

5 M. Cesati / Information Processing Letters 81 (2002) Question: Is there a truth assignment of Hamming weight k to the variables of E that makes exactly one literal in each clause of E true? Downey and Fellows showed that PERFECT CODE reduces to WEIGHTED EXACT CNF SATISFIABILI- TY, and vice versa [6,8]. As an immediate consequence of Theorem 1 we have the following: Corollary 1. The WEIGHTED EXACT CNF SATISFI- ABILITY problem is W[1]-complete. 5. Conclusions In this paper we have shown that PERFECT CODE belongs to W[1]; as an easy corollary, we also established a new analog of the Cook s Theorem for the class W[1]. In general, it seems fruitful to think in terms of Turing machine computations when trying to establish membership in W[1]. In order to prove that a parameterized problem A belongs to W[1], wealways pick (more or less randomly) a problem B W[1] and we try to show that A reduces to B. However,it could be very difficult to devise such a reduction, because problems A and B can be very different; thus, even if the reduction exists, it could be very complicated. This is quite different from what we do in classical computational complexity, where usually we just show that a suitable model of Turing machine (for instance, a polynomial-time nondeterministic Turing machine) is able to decide an instance of our problem. Apparently, the proof technique of Theorem 1 is the usual way to show membership in a W-class: a reduction from our candidate problem to a target problem already placed in W[1]. However, we recognize an important difference: the target problem is formulated in terms of a general-purpose Turing machine computation: to show W[1]-membership of PERFECT CODE we devised a single-tape, single-head nondeterministic Turing machine that guesses and verifies a candidate perfect code with a short computation. This is exactly the same as what we would have done to show that a problem belongs to NP: devise a nondeterministic Turing machine that guesses and verifies a candidate solution in polynomial time. To be fair, other reductions to the SHORT NON- DETERMINISTIC TURING MACHINE COMPUTATION problem have already been used (see for instance [2]). However, we suggest that devising a reduction to this particular problem should be the very first technique to consider when trying to show membership in W[1]. In fact, we believe that if the reduction exists, then it is usually simple, thanks to the computational power offered by the nondeterministic Turing machine model. For example, a simple reduction for the INDEPEN- DENT SET problem consists of a single-tape nondeterministic Turing machine that guesses a k-element subset of vertices and then checks that there is no edge between the guessed vertices. (The instructions of such Turing machine are a subset of the instructions of the Turing machine shown in this paper.) Thinking in terms of Turing machines may also be useful when establishing W[2]-membership. In fact, the SHORT NONDETERMINISTIC TURING MACHINE COMPUTATION problem extended to multi-tape machines is W[2]-complete [3,4,7]. Therefore, in order to prove that some problem belongs to W[2], we can simply try to solve it with a multi-tape nondeterministic Turing machine. This technique allowed us to close a long-standing open question by establishing W[2]-membership for a parameterized version of the STEINER TREE problem [3,1]. References [1] H.L. Bodlaender, D. Kratsch, A note on fixed parameter intractability of some domination-related problems, Private communication, April [2] L. Cai, J. Chen, R.G. Downey, M.R. Fellows, On the parameterized complexity of short computation and factorization, Arch. Math. Logic 36 (4 5) (1997) [3] M. Cesati, The Turing way to the parameterized intractability, in: Workshop on Parameterized Complexity, Chennai, India, December 7 9, [4] M. Cesati, M. Di Ianni, Computation models for parameterized complexity, Math. Logic Quarterly 43 (1997) [5] R.G. Downey, M.R. Fellows, Fixed-parameter tractability and completeness. I. Basic results, SIAM J. Comput. 24 (4) (1995) [6] R.G. Downey, M.R. Fellows, Fixed-parameter tractability and completeness. II. On completeness for W[1], Theoret. Comput. Sci. 141 (1 2) (1995) [7] R.G. Downey, M.R. Fellows, Threshold dominating set and an improved characterization of W[2], Theoret. Comput. Sci. 209 (1 2) (1998)

6 168 M. Cesati / Information Processing Letters 81 (2002) [8] R.G. Downey, M.R. Fellows, Parameterized Complexity, 1st edn., Monographs in Computer Science, Springer, New York, [9] R.G. Downey, M.R. Fellows, B. Kapron, M.T. Hallett, H.T. Wareham, Parameterized complexity of some problems in logic and linguistics (extended abstract), in: Proc. 2nd Workshop on Structural Complexity and Recursion-theoretic Methods in Logic Programming, 1993, Lecture Notes in Comput. Sci., Vol. 813, Springer, Berlin, 1994, pp