Verifying Time Complexity of Deterministic Turing Machines

We show that, for all reasonable functions $T(n)=o(n\log n)$, we can algorithmically verify whether a given one-tape Turing machine runs in time at most $T(n)$. This is a tight bound on the order of growth for the function $T$ because we prove that, for $T(n)\geq(n+1)$ and $T(n)=\Omega(n\log n)$, there exists no algorithm that would verify whether a given one-tape Turing machine runs in time at most $T(n)$. We give results also for the case of multi-tape Turing machines. We show that we can verify whether a given multi-tape Turing machine runs in time at most $T(n)$ iff $T(n_0)< (n_0+1)$ for some $n_0\in\mathbb{N}$. We prove a very general undecidability result stating that, for any class of functions $\mathcal{F}$ that contains arbitrary large constants, we cannot verify whether a given Turing machine runs in time $T(n)$ for some $T\in\mathcal{F}$. In particular, we cannot verify whether a Turing machine runs in constant, polynomial or exponential time.Comment: 18 pages, 1 figur

Simple PTAS's for families of graphs excluding a minor

We show that very simple algorithms based on local search are polynomial-time approximation schemes for Maximum Independent Set, Minimum Vertex Cover and Minimum Dominating Set, when the input graphs have a fixed forbidden minor.Comment: To appear in Discrete Applied Mathematic

Preverjanje časovne zahtevnosti Turingovih strojev

The central problem in the dissertation is the following: "For a function ▫$T colon mathbb{N} rightarrow mathbb{R}_{,geq 0}$▫, how hard is it to verify whether a given Turing machine runs in time at most ▫$T(n)$▫? Is it even possible?" Our first main contibution is that, for all reasonable functions ▫$T(n) = operatorname{o}(nlog n)$▫, it is possible to verify with an algorithm whether a given one-tape Turing machine runs in time at most ▫$T(n)$▫. This is a tight bound on the order of growth for the function ▫$T$▫ because we prove that, for ▫$T(n) = Omega(nlog n)$▫ and ▫$T(n)geq n+1$▫, there exists no algorithm that would verify whether a given one-tape Turing machine runs in time at most ▫$T(n)$▫. As opposed to one-tape Turing machines, we show that we can verify with an algorithm whether a given multi-tape Turing machine runs in time at most ▫$T(n)$▫ if and only if ▫$T(n_0) < n_0+1$▫ for some ▫$n_0 in mathbb{N}$▫. Linear time bounds are the most natural algorithmically verifiable time bounds for one-tape Turing machines, because a one-tape Turing machine that runs in time ▫$operatorname{o}(nlog n)$▫ actually runs in linear time. This motivates our second main contibution which is the analysis of complexity of the following family of problems, parameterized by integers ▫$Cgeq 2$▫ and ▫$D geq 1$▫: "Does a given one-tape ▫$q$▫-state Turing machine run in time ▫$Cn+D$▫?" Assuming a fixed tape and input alphabet, we show that these problems are co-NP-complete and we provide good lower bounds. Specifically, these problems cannot be solved in ▫$operatorname{o}(q^{(C-1)/4})$▫ non-deterministic time by multi-tape Turing machines. We also show that the complements of these problems can be solved in ▫$operatorname{O}(q^{C+2})$▫ non-deterministic time and not in ▫$operatorname{o}(q^{(C-1)/4})$▫ non-deterministic time by multi-tape Turing machines. To prove the upper bound ▫$operatorname{O}(q^{C+2})$▫, we use the so-called compactness theorem which is our third main contribution. We need more notation to state it in full generality, but a simple corollary tells the following: To verify whether an input one-tape Turing machine runs in time ▫$Cn+D$▫, it is enough to verify this on a finite number of inputs. We argue that our main results are proved with techniques that relativize and that using only such techniques we cannot solve the P versus NP problem.Osrednji problem v disertaciji je sledeč: "Naj bo ▫$T colon mathbb{N} rightarrow mathbb{R}_{,geq 0}$▫ poljubna funkcija. Kako težko je preveriti, ali je časovna zahtevnost danega Turingovega stroja ▫$T(n)$▫? Je to sploh mogoče preveriti?" Naš prvi večji prispevek pove, da je za vse "normalne" funkcije ▫$T(n)=operatorname{o}(nlog n)$▫ možno z algoritmom preveriti, ali je časovna zahtevnost danega enotračnega Turingovega stroja ▫$T(n)$▫. Meja ▫$operatorname{o}(nlog n)$▫ je tesna, saj za ▫$T(n) = Omega(nlog n)$▫ in ▫$T(n)geq n+1$▫ ni mogoče z algoritmom preveriti, ali je časovna zahtevnost danega enotračnega Turingovega stroja ▫$T(n)$▫. Pri večtračnih Turingovih strojih je rezultat enostavnejši. Zanje namreč velja, da je časovno zahtevnost ▫$T(n)$▫ moč z algoritmom preveriti natanko tedaj, ko velja ▫$T(n_0) < n_0+1$▫ za neki ▫$n_0 in mathbb{N}$▫. Znano je, da je vsak enotračni Turingov stroj časovne zahtevnosti ▫$operatorname{o}(nlog n)$▫ tudi linearne časovne zahtevnosti. Posledično je linearna časovna zahtevnost najbolj naravna časovna zahtevnost, ki jo lahko z algoritmom preverimo pri enotračnih Turingovih strojih. V disertaciji se zato ukvarjamo tudi z naslednjimi problemi, ki so parametrizirani z naravnima številoma ▫$C geq 2$▫ in ▫$D geq 1$▫: "Ali je dani enotračni Turingov stroj s ▫$q$▫ stanji časovne zahtevnosti ▫$Cn+D$▫?" Pri analizi teh problemov, kar je naš drugi večji prispevek, predpostavljamo fiksno vhodno in tračno abecedo. Ti problemi so co-NP-polni in zanje lahko dokažemo dobre spodnje meje računske zahtevnosti. Ni jih namreč mogoče rešiti v času ▫$operatorname{o}(q^{(C-1)/4})$▫ z nedeterminističnimi večtračnimi Turingovimi stroji. Še več, komplementi teh problemov so rešljivi z večtračnimi nedeterminističnimi Turingovimi stroji v času ▫$operatorname{O}(q^{C+2})$▫, ne pa v času ▫$operatorname{o}(q^{(C-1)/4})$▫. Pri dokazu zgornje meje ▫$operatorname{O}(q^{C+2})$▫ uporabimo tako imenovani izrek o kompaktnosti, naš tretji večji prispevek. Potrebovali bi več notacije, da bi ga na tem mestu navedli, zato povejmo le njegovo posledico: Da bi preverili, ali dani enotračni Turingov stroj teče v času ▫$Cn+D$▫, je dovolj preveriti čas izvajanja Turingovega stroja le na končno mnogo vhodih. Glavni prispevki te disertacije so dokazani s tehnikami, ki relativizirajo. Dokažemo tudi znano dejstvo, da s takimi tehnikami ni mogoče rešiti slavnega problema ▫$rm{Pstackrel{?}{=}NP}$▫