Cdmtcs Research Report Series Parameterized Circuit Complexity and the W

A parameterized problem hL; ki belongs to W [t] if there exists k computed from k such that hL; ki reduces to the weight-k satis ability problem for weft-t circuits. We relate the fundamental question of whether the W [t] hierarchy is proper to parameterized problems for constant-depth circuits. We de ne classes G[t] as the analogues of AC depth-t for parameterized problems, and N [t] by weight-k existential quanti cation on G[t], by analogy with NP = 9 P. We prove that for each t, W [t] equals the closure under xed-parameter reductions of N [t]. Then we prove, using Sipser's results on the AC depth-t hierarchy, that both the G[t] and the N [t] hierarchies are proper. If this separation holds up under parameterized reductions, then the W [t] hierarchy is proper. We also investigate the hierarchy H[t] de ned by alternating quanti cation over G[t]. By trading weft for quanti ers we show thatH[t] coincides withH[1]. We also consider the complexity of unique solutions, and show a randomized reduction from W [t] to Unique W [t]. 1 Parameterized Problems and the W Hierarchy Many important and familiar problems have the general form Instance: An object x, a number k 1. Question: Does x have some property k that depends on k? For example, the NP-complete Clique problem asks: given an undirected graph G and natural number k, does G have a clique of size k? The Vertex Cover and Dominating Set problems ask whether G has a vertex cover, respectively dominating set, of size k. Here k is called the parameter . Formally, a parameterized language is a subset of N. A parameterized language A is said to be xed-parameter tractable, and to belong to the class FPT, if there is a polynomial p, a function f : N! N, and a Turing machine M such that on any input (x; k), M decides whether (x; k) 2 A within f(k) p(jxj) steps. A is in strongly uniform FPT if the function f is computable. Note that if M runs in time polynomial in the length of (x; k) then it meets this condition with f computable. Examples of problems in FPT for which the only f are uncomputable are given in [DF93], while [DF95c] describes natural problems in FPT for which the only known f are not known to be computable. The best known method for solving the parameterized Clique problem is the algorithm of Nesetril and Poljak [NP85] that runs in time O(n 2+ 3 ), where 2+ represents the exponent on the time for multiplying two n n matrices (best known is 2:376 : : :, see [CW90]). For Dominating Set we know of nothing better than the trivialO(n)-time algorithm that tries all vertex subsets of size k. Vertex Cover, however, belongs to FPT, via a depthrst search algorithm that runs in time 2 O(n) (see [DF95c]). Quite a few other NP-complete problems, with natural parameter k, are in FPT via algorithms of time f(k) O(n) through f(k) O(n), while many others treated in [DF95a] seem to be hard in the manner of Clique and Dominating Set. The established way in complexity theory of comparing the hardness of problems is by formulating appropriate notions of reducibility and completeness. Here the former is provided by De nition 1.1. A parameterized language A FPT-many-one reduces to a parameterized language B, written A fpt m B, if there are a polynomial q, functions f; g : N ! N, and a Turing machine T such that on any input (x; k), T runs for f(k) q(jxj) steps and outputs (x; g(k)) such that (x; k) 2 A () (x; g(k)) 2 B. The reduction is strongly uniform if f is computable. Then (strongly uniform) FPT is closed downward under (strongly uniform) FPT reductions. Note that g is computable, and the parameter k = g(k) in the reduction does not depend on x. For the completeness notion, Downey and Fellows [DF95a] de ned a natural hierarchy of classes of parametrized languages FPT W [1] W [2] W [3] : : : W [poly ]; (1) and showed that the parameterized version of Clique is complete forW [1] under FPT reductions, while that of Dominating Set is complete for W [2]. This gives a sense in which Dominating Set is apparently harder than Clique. The formal de nition of the W hierarchy is deferred to the next section, but the main idea can be seen by examining the logical de nitions of Clique and Dominating Set. For each k, the language of graphs with a clique of size k is de ned by