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1 Final Notes on K-Path 2 Parameterized Complexity 6.S078 Fine-Grained Algorithms and Complexity MIT Lecture 19: Parameterized Complexity November 9, 2020 1 Final Notes on k-Path Here are the fastest known k-PATH algorithms: •[Wil09]: k-PATH (on directed or undirected graphs) is in randomized O?(2k) time. •[Tsu19]: k-PATH (on directed or undirected graphs) in deterministic O?(2:56k) time. •[BHKK17]: Undirected k-PATH is in randomized O?(1:66k) time. •[BDH18]: For graphs with at most p distinct k-paths, can find one in O?(p2 · 2k) deterministic time. These algorithms also use very different techniques from the ones we showed you! Check the references if you’re interested. In general, one reference for finding the fastest algorithm for various parametrized problems is http: //fpt.wikidot.com/fpt-races, but note it’s not always up to date (for deterministic k-PATH for example). The gap between the randomized and deterministic algorithms provokes the question: is there an inherent gap between randomized and deterministic algorithms? The answer is “probably not”, given the very recent work [CT20] (which builds on a lot of prior work). Lijie and Roei show that, under plausible hypotheses, that every randomized algorithm running in T (n) time can be deterministically simulated in O(nT (n)1+") time, for every desired " > 0. 2 Parameterized Complexity We now turn to developing a theoretical understanding of what parametric problems do not have FPT algorithms, i.e., the complexity theory of parametric problems. In these notes, we’ll only cover a few of the known results on this topic, picking results that are closest to other material from this class. The textbook (https://www.mimuw.edu.pl/ ˜malcin/book/parameterized-algorithms.pdf) covers some more results in detail. Recall the class FPT consists of the parametric problems solvable in time f (k) nc, for some constant c and some computable f. This “extended” notion of tractability not only includes polynomial-time solvable problems but also appropriate parameterizations of some NP-hard problems. We have discussed various techniques for deriving FPT algorithms for problems, but so far we haven’t said much about limitations on solving problems in an FPT way. 2.1 The class XP What should be considered intractable in the parameterized world? Generally speaking, intractable problems should be those for which the running time dependence on the parameter is huge, such as nf(k) where f(k) ≥ k. When the exponent of the polynomial depends on the parameter, this is a much worse dependence: such a bound would be intractable for large n and even moderately sized k. This motivates the following class: Definition 2.1 The class XP consists of parameteric problems solvable in nf(k) time, where f is a computable func- tion. 1 On the positive side, any problem L 2 XP is indeed solvable in polynomial time, for every fixed value of k. To formalize this, let L be a parametric problem, and for any constant k 2 N we define the k-slice of L to be Lk = f(x; k) j (x; k) 2 Lg : That is, Lk only includes the instances of L with parameter equal to exactly k. Proposition 2.1 If L 2 XP, then for all k 2 N, Lk 2 P. f(k) Proof. For every k 2 N, LK is solvable in n time, where f(K) is some fixed constant. Let A be an algorithm f(k) f(k) solving L in at most n time. Then the following algorithm B solves LK in n time (which is polynomial for constant K): B(x): Run A(x; K) and output the answer. On the negative side, the degree of the polynomial in the running time for solving LK can obviously increase with K, hence we may consider the algorithm A for L to be inefficient. Exercise: Is the converse of the proposition true? (For all parametric L, if for all k 2 N, Lk 2 P , then is L 2 XP?) Prove your answer. 2.2 A Lower Bound for XP So, is XP really an “intractable” complexity class? The following theorem confirms this intuition, in some sense: Theorem 2.1 FPT 6= XP. This separation may be viewed as an analogue of the time hierarchy theorem from complexity theory, with FPT ≈ P and XP ≈ EXP. Proof. We give an explicit problem L 2 XP n FPT. Define SIM = (M; x; k) jTuring machine M accepts input x in at most (jMj + jxj)k+1 steps : (Note: You don’t actually need to know what a Turing machine is here, to get this argument. If it bothers you, substitute “algorithm” for “Turing machine” in the below.) First, observe that SIM 2 XP: note the input length n is O(jMj + jxj + jkj). using a universal Turing machine we can simulate any given Turing machine M on an input x with polynomial overhead, so L is solvable in nO(k) time. Suppose SIM 2 FPT. We will derive a contradiction. This implies that SIM has a g(k) · nc algorithm, for some c c computable g and constant c ≥ 1. Then the (c + 1)-th slice of SIM, SIMc+1, is decidable in g(c + 1) · n ≤ O(n ) time. Let Ac+1 be a Turing machine for SIMc+1 running in this time. Now let B be any Turing machine running in nc+1 time. We can now simulate B on n-bit inputs in only O(nc) time, as follows: C: On input x, run Ac+1 on (B; x) and output its answer. c c On input x, C runs in O(jBj + jxj) ≤ O(jxj ) time, by assumption on Ac+1. But we have given a way to simulate every Turing machine running in nc+1, that runs in only O(nc) time. Essentially we have derived that TIME nc+1 ⊆ TIME (nc) ; which contradicts the time hierarchy theorem from complexity theory (recall the time hierarchy theorem says that c c+1 TIME (n ) ( TIME n ). 2 2.3 The W-hierarchy There are many problems contained in XP that are not believed to be in FPT. In fact there is an infinite hierarchy of problems between FPT and XP (sort of like the polynomial hierarchy, but not quite). The most widely-studied such hierarchy is the so-called W -hierarchy. The W has traditionally stood for a circuit parameter called Weft but for our purposes, we will think of the W as standing for Weight – this is the Weight-hierarchy. The W -hierarchy is defined with respect to various weighted satisfiability problems. n Definition 2.2 Let x 2 f0; 1g . The weight of x is the number of bits set to 1 (that is, its L1 norm). A parameterized weighted satisfiability problem has the following high-level form: Given a logical expression E encoding a Boolean function, is there a satisfying assignment to E with weight equal to k? The W -hierarchy is obtained by considering parameterized weighted SAT problems over different classes of logical expressions. The canonical problem for the W -hierarchy is the Weighted Depth-t SAT problem. Let t ≥ 1 be a fixed constant. Weighted Depth-t SAT (a.k.a. W-Depth-t-SAT): given an AND, OR, and NOT circuit with t layers of unbounded fan-in gates, followed by a bottom layer (closest to the inputs) with fan-in at most two, is there an input assignment x of weight k that satisfies C? In circuit complexity language, Weighted Depth-t SAT is asking if a given “AC0 circuit of depth t + 1” with bottom fan-in two has a SAT assignment with exactly k variables set to true. Intuitively, this problem should get harder as the depth of the circuit increases. (Don’t worry if you are uncomfortable with circuits; we’ll only really deal with CNFs in these lecture notes!) To understand the true difficulty of these SAT problems, we would like to set up a complexity class where these weighted SAT problems are “complete” for those classes, in the sense that if the weighted SAT problem is FPT, then all problems in the class are also FPT. To have a notion of “completeness” for a complexity class, we first need a notion of reducibility. 2.4 Reductions What makes NP-completeness so powerful is that an NP-complete problem can be used to express any other NP prob- lem Π, with only a polynomial-time reduction from Π. Fine-grained hardness is powerful because its reductions from A to B “preserve exponents”. We want reductions between parametric problems that “preserve” their parametrized tractability. The generic form of FPT reducibility, defined below, fits the bill: Definition 2.3 We say that there is an FPT-reduction from A to B, which we denote A ≤fpt B if there is a computable function g : N!N and an oracle Turing Machine M B (with oracle access to B) such that: • For all (x; k), M B (x; k) accepts if and only if (x; k) 2 A. • The running time of M B is FPT. • For every query (y; k0) to B, we have k0 ≤ g (k). The last item says that all queries to B by M B(x; k) must preserve the original parameter k. All parameters appearing in all queries must solely be a function of the original parameter k in the input to A. 3 Our definition of FPT reduction is more general than normally considered: typically in the literature, people have considered many-one reductions from A to B (where there is at most one oracle query).
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