arXiv subjects
Publications and source records attributed to Marzio De Biasi.
At first glance, one-state Turing machines are very weak: the halting problem for them is decidable, and, without memory, they cannot even accept a simple one element language such as $L = \{ 1 \}$ . Nevertheless it has been showed that a one-state Turing machine can accept non regular languages. We extend such result and prove that they can also recognize non context-free languages, so for some tasks they are more powerful than pushdown automata.
In the \probrFix problem, we are given a graph $G$, a (non-proper) vertex-coloring $c : V(G) \to [r]$, and a positive integer $k$. The goal is to decide whether a proper $r$-coloring $c'$ is obtainable from $c$ by recoloring at most $k$ vertices of $G$. Recently, Junosza-Szaniawski, Liedloff, and Rz{ą}{ż}ewski [SOFSEM 2015] asked whether the problem has a polynomial kernel parameterized by the number of recolorings $k$. In a full version of the manuscript, the authors together with Garnero and Montealegre, answered the question in the negative: for every $r \geq 3$, the problem \probrFix does not admit a polynomial kernel unless $\NP \subseteq \coNP / \poly$. Independently of their work, we give an alternative proof of the theorem. Furthermore, we study the complexity of \probrFixSwap, where the only difference from \probrFix is that instead of $k$ recolorings we have a budget of $k$ color swaps. We show that for every $r \geq 3$, the problem \probrFixSwap is $\W[1]$-hard whereas \probrFix is known to be FPT. Moreover, when $r$ is part of the input, we observe both \probFix and \probFixSwap are $\W[1]$-hard parameterized by treewidth. We also study promise variants of the problems, where we are guaranteed that a proper $r$-coloring $c'$ is indeed obtainable from $c$ by some finite number of swaps. For instance, we prove that for $r=3$, the problems \probrFixPromise and \probrFixSwapPromise are $\NP$-hard for planar graphs. As a consequence of our reduction, the problems cannot be solved in $2^{o(\sqrt{n})}$ time unless the Exponential Time Hypothesis (ETH) fails.
We prove that the problem of reconstructing a permutation $π_1,\dotsc,π_n$ of the integers $[1\dotso n]$ given the absolute differences $|π_{i+1}-π_i|$, $i = 1,\dotsc,n-1$ is NP-complete. As an intermediate step we first prove the NP-completeness of the decision version of a new puzzle game that we call Crazy Frog Puzzle. The permutation reconstruction from differences is one of the simplest combinatorial problems that have been proved to be computationally intractable.
A two-way deterministic finite state automaton with one counter (2D1CA) is a fundamental computational model that has been examined in many different aspects since sixties, but we know little about its power in the case of unary languages. Up to our knowledge, the only known unary nonregular languages recognized by 2D1CAs are those formed by strings having exponential length, where the exponents form some trivial unary regular language. In this paper, we present some non-trivial subsets of these languages. By using the input head as a second counter, we present simulations of two-way deterministic finite automata with linearly bounded counters and linear--space Turing machines. We also show how a fixed-size quantum register can help to simplify some of these languages. Finally, we compare unary 2D1CAs with two--counter machines and provide some insights about the limits of their computational power.
The problem of deciding if a Traveling Salesman Problem (TSP) tour is minimal was proved to be coNP-complete by Papadimitriou and Steiglitz. We give an alternative proof based on a polynomial time reduction from 3SAT. Like the original proof, our reduction also shows that given a graph $G$ and an Hamiltonian path of $G$, it is NP-complete to check if $G$ contains an Hamiltonian cycle (Restricted Hamiltonian Cycle problem).
Data replication technologies enable efficient and highly-available data access, thus gaining more and more interests in both the academia and the industry. However, data replication introduces the problem of data consistency. Modern commercial data replication systems often provide weak consistency for high availability under certain failure scenarios. An important weak consistency is Pipelined-RAM (PRAM) consistency. It allows different processes to hold different views of data. To determine whether a data replication system indeed provides PRAM consistency, we study the problem of Verifying PRAM Consistency over read/write traces (or VPC, for short). We first identify four variants of VPC according to a) whether there are Multiple shared variables (or one Single variable), and b) whether write operations can assign Duplicate values (or only Unique values) for each shared variable; the four variants are labeled VPC-SU, VPC-MU, VPC-SD, and VPC-MD. Second, we present a simple VPC-MU algorithm, called RW-CLOSURE. It constructs an operation graph $\mathcal{G}$ by iteratively adding edges according to three rules. Its time complexity is $O(n^5)$, where n is the number of operations in the trace. Third, we present an improved VPC-MU algorithm, called READ-CENTRIC, with time complexity $O(n^4)$. Basically it attempts to construct the operation graph $\mathcal{G}$ in an incremental and efficient way. Its correctness is based on that of RW-CLOSURE. Finally, we prove that VPC-SD (so is VPC-MD) is $\sf{NP}$-complete by reducing the strongly $\sf{NP}$-complete problem 3-PARTITION to it.