2.4 Computational Complexity
Our aim is to provide a deeper understanding of the computational complex-ity of someNP-hard problems. To this end, we employ classical complexity classes such asP(polynomial time) andNP(nondeterministic polynomial time) [GJ79] as well as the classLOGSNPof limited nondeterminism (pre-sumably lying between P and NP) [GLM96; PY96] and parameterized complexity classes such asFPT(fixed-parameter tractability),W[2](second level of the “weft hierarchy” of presumable parameterized intractability), andXP[DF13;FG06;Nie06].
To study computational complexity, we formally introduce the concept of decision problems as follows. LetΣbe some finite alphabet and letΣ∗denote the set of all finite words consisting of letters fromΣ. Adecision problem is a language L⊆Σ∗. Aninstanceof Lis an element of Σ∗. Intuitively, L encodes a set of instances which have some “common property”. For example, consider the toy problem ODDINTEGERwhich can be formalized by the language L={x∈{0, 1}∗|xends with 1}. Instead of this formal definition we use an input-question notation as follows.
ODDINTEGER
Input: An integerx(encoded in binary).
Question: Isxodd?
In general, the concrete encoding of a problem may have an influence on its computational complexity. However, for all problems considered in this thesis we can assume a standard encoding for example by bit strings.
Furthermore, we usually limit our computational complexity analysis to decision problems, but all our algorithms can be adapted to also construct the corresponding “solution”.
Finally, for an instancex∈Σ∗and a problemLwe sayxis ayes-instance ofLifx∈L; otherwise we sayxis ano-instanceofL.
Pvs. NP. In classical computational complexity theory, to distinguish between tractability and intractability one uses the complexity classesP andNP. Herein,Pcontains all problems that can be decided in polynomial time by a deterministic Turing machine andNPcontains all problems that can be decided in polynomial time by a non-deterministic Turing machine. It is widely believed thatNPcontains problems which are not inP. Intuitively,
2 Basic Concepts and Notation
NPcontains problems which are harder to decide than any problem inP. To identify the “hardest” problems inNP, we use the concept of polynomial-time many-one reductions as follows.
Definition 2.1. Let L and L0 be two decision problems. Then, a func-tion f:Σ∗→Σ∗is calledpolynomial-time many-one reduction(or simply polynomial-time reduction) fromLtoL0if for each instancex∈Σ∗
• x0:=f(x) can be computed in polynomial time, and
• x∈L ⇐⇒ x0∈L0.
This can be read as “L0is at least as hard asLup to polynomial factors”. A problemLisNP-hardif for each problemL0fromNPthere is a polynomial-time reduction to L. We say an NP-hard problem isNP-completeif it is also inNP. Hence, with respect to polynomial-time reductions,NP-complete problems are the “hardest problems inNP”.
LOGSNP. Papadimitriou and Yannakakis [PY96] introducedLOGSNPto precisely characterize the computational complexity of certain problems in NPthat are neither known to beNP-complete nor known to be solvable in polynomial time. LOGSNPis a subclass of problems inNPwhich can be decided in polynomial time with an initial phase ofO(log2N) nondetermin-istic steps, where N is the overall input size. LOGSNPdoes not include all problems decidable in polynomial time afterO(log2N) nondeterministic steps since it puts additional restrictions on the computation. We omit the formal definition ofLOGSNP, because it is quite technical and not needed in our proofs.
It is widely believed thatLOGSNPis properly intermediate betweenP andNP. Problems complete forLOGSNPunder polynomial-time reductions include RICHHYPERGRAPHCOVER(seeSubsection 6.2.2for the definition) and LOGADJUSTMENT[PY96]. In LOGADJUSTMENT, a boolean expression in conjunctive normal form withrvariables and a truth assignmentTare given, and the question is whether there is a satisfying truth assignment whose Hamming distance fromTis at most logr.
To identifyLOGSNP-complete problems we use polynomial-time reduc-tions from (showing hardness) or to (showing membership) knownLOGSNP -complete problems.
We mention in passing that alternative characterizations ofLOGSNP exist [Cai+97;FG06, Sec. 15.2]).
2.4 Computational Complexity
Fixed-Parameter Tractability andXP. The concept of parameterized complexity was pioneered by Downey and Fellows [DF13] (see also further textbooks [FG06;Nie06]). The fundamental goal is to find out whether the seemingly unavoidable combinatorial explosion occurring in algorithms to solveNP-hard problems can be confined to certain problem-specific parame-ters. If such a parameter assumes only small values in applications, then an algorithm with a running time that is exponential exclusively with respect to this parameter may be efficient. Formally, aparameterized problemis a languageL⊆Σ∗×Σ∗and the second component is called parameter. For the sake of convenience we assume that a parameter is an non-negative integer and acombined parameter(which is a vector of parameters) can be simply seen as the sum of its components.
Definition 2.2. A parameterized problemLisfixed-parameter tractableif for every instance (x,p) there is a deterministic algorithm which decides whether (x,p)∈Linf(p)·|x|O(1)time wherefis a computable function solely depending onp. Equivalently, we sayLis contained in the parameterized complexity classFPT.
If the problem can only be solved in polynomial running time where the de-gree of the polynomial depends onp(such as|x|f(p)), then, for parameterp, the problem is said to lie in the—strictly larger [DF13]—parameterized complexity classXP. Note that containment inXPensures polynomial-time solvability for a constant parameterpwhereasFPTadditionally ensures that the degree of the corresponding polynomial is independent of the pa-rameterp.
Kernelization. A common way of showing fixed-parameter tractability is through kernelization. Akernelization algorithmtakes as input a problem instance xtogether with a parameter pand transforms it in polynomial time into an instancex0with parameterp0such that (x,p) is a yes-instance if and only if (x0,p0) is a yes-instance and there is a function f such that p0≤f(p) and|x0| ≤f(p). The function f measures the size of the(problem) kernel(x0,p0). A problem kernel is said to be apolynomial kernelif f is polynomially bounded. Note that it is well-known that a decidable problem is fixed-parameter tractable with respect to a parameter if and only if it admits a problem kernel [Cai+97]. The corresponding kernels, however, may have exponential size and it is of particular interest to determine which
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problems, with respect to which parameter(s), allow for polynomial-size problem kernels [Bod09;GN07;Kra14].
Bodlaender, Thomassé, and Yeo [BTY11] introduced a refined concept of reductions that allows to transfer “non-existence results” for polynomial-size problem kernels to other problems. It is defined as follows.
Definition 2.3. Let L andL0 be two parameterized problems. Then, a function f :Σ∗×Σ∗→Σ∗×Σ∗ is called polynomial time and parameter transformationfromLtoL0if there is some polynomialgsuch that for each instance (x,p)∈Σ∗×Σ∗
• (x0,p0) :=f(x,p) can be computed polynomial time,
• (x,p)∈L ⇐⇒(x0,p0)∈L0, and
• p0≤g(p).
The difference between a polynomial time and parameter transformation and a classical many-one reduction is that the parameter in the instance one reduces to has to be bounded by some polynomial solely depending on the parameter in the problem instance one reduces from.
Bodlaender, Thomassé, and Yeo [BTY11] showed that, if there are two parameterized problemsLandL0such that the unparameterized versions ofLandL0are NP-complete and there is a polynomial time and parameter transformation fromLtoL0, then a polynomial problem kernel forLimplies a polynomial problem kernel forL0.
Parameterized Intractability. Downey and Fellows [DF13] introduced a framework ofparameterized intractability. Herein, the central tool is the W-hierarchy consisting of the following classes and interrelations:
FPT⊆W[1]⊆W[2]⊆ · · · ⊆W[t]⊆. . .⊆XP.
To showW[t]-hardness for any positive integer t, we use the concept of parameterized reductionwhich is defined as follows.
Definition 2.4. Let L andL0 be two parameterized problems. Then, a functionf:Σ∗×Σ∗→Σ∗×Σ∗is calledparameterized reductionfromLtoL0 if there are two computable functionsg1andg2such that for each instance (x,p)∈Σ∗×Σ∗