Maria Paola Bianchi

Online Coloring of Bipartite Graphs with and without Advice

In the online version of the well-known graph coloring problem, the vertices appear one after the other together with the edges to the already known vertices and have to be irrevocably colored immediately after their appearance. We consider this problem on bipartite, i.e., two-colorable graphs. We prove that at least ⌊1.13746⋅log2(n)−0.49887⌋ colors are necessary for any deterministic online algorithm to be able to color any given bipartite graph on n vertices, thus improving on the previously known lower bound of ⌊log2n⌋+1 for sufficiently large n.Recently, the advice complexity was introduced as a method for a fine-grained analysis of the hardness of online problems. We apply this method to the online coloring problem and prove (almost) tight linear upper and lower bounds on the advice complexity of coloring a bipartite graph online optimally or using 3 colors. Moreover, we prove that

Behaviours of Unary Quantum Automata

O(\sqrt{n})

On the advice complexity of the online L ( 2 , 1 ) -coloring problem on paths and cycles

advice bits are sufficient for coloring any bipartite graph on n vertices with at most ⌈log2n⌉ colors.

On the Advice Complexity of the Online L(2,1)-Coloring Problem on Paths and Cycles

We study the stochastic events induced by MM-qfas working on unary alphabets. We give two algorithms for unary MM-qfas: the first computes the dimension of the ergodic and transient components of the non halting subspace, while the second tests whether the induced event is d-periodic. These algorithms run in polynomial time whenever the MM-qfa given in input has complex amplitudes with rational components. We also characterize the recognition power of unary MM-qfas, by proving that any unary regular language can be accepted by a MM-qfa with constant cut point and isolation. Yet, the amount of states of the resulting MM-qfa is linear in the size of the corresponding minimal dfa. We also single out families of unary regular languages for which the size of the accepting MM-qfas can be exponentially decreased.

Complexity of Promise Problems on Classical and Quantum Automata

In an L ( 2 , 1 ) -coloring of a graph, the vertices are colored with colors from an ordered set such that neighboring vertices get colors that have distance at least 2 and vertices at distance 2 in the graph get different colors. We consider the problem of finding an L ( 2 , 1 ) -coloring using a minimum range of colors in an online setting where the vertices arrive in consecutive time steps together with information about their neighbors and vertices at distance 2 among the previously revealed vertices. For this, we restrict our attention to paths and cycles.Offline, paths can easily be colored within the range { 0 , ? , 4 } of colors. We prove that, considering deterministic algorithms in an online setting, the range { 0 , ? , 6 } is necessary and sufficient while a simple greedy strategy needs range { 0 , ? , 7 } .Advice complexity is a recently developed framework to measure the complexity of online problems. The idea is to measure how many bits of advice about the yet unknown parts of the input an online algorithm needs to compute a solution of a certain quality. We show a sharp threshold on the advice complexity of the online L ( 2 , 1 ) -coloring problem on paths and cycles. While achieving color range { 0 , ? , 6 } does not need any advice, improving over this requires a number of advice bits that is linear in the size of the input. Thus, the L ( 2 , 1 ) -coloring problem is the first known example of an online problem for which sublinear advice does not help.We further use our advice complexity results to prove that no randomized online algorithm can achieve a better expected competitive ratio than 5 4 ( 1 - ? ) , for any ? 0 .

On the Size of Unary Probabilistic and Nondeterministic Automata

In an L(2,1)-coloring of a graph, the vertices are colored with colors from an ordered set such that neighboring vertices get colors that have distance at least 2 and vertices at distance 2 in the graph get different colors. We consider the problem of finding an L(2,1)-coloring using a minimum range of colors in an online setting where the vertices arrive in consecutive time steps together with information about their neighbors and vertices at distance two among the previously revealed vertices. For this, we restrict our attention to paths and cycles.

Size lower bounds for quantum automata

We consider the promise problem \(A^{N,r_1,r_2}\) on a unary alphabet \({\left\{ \sigma \right\} }\) studied by Gruska et al. in [21]. This problem is formally defined as the pair \(A^{N,r_1,r_2}=(A^{N,r_1}_{yes},A^{N,r_2}_{no})\), with \(0\le r_1\ne r_2<N\), \(A^{N,r_1}_{yes}={\left\{ \sigma ^n \ \mid \ n\equiv r_1 \mod N\right\} }\) and \(A^{N,r_2}_{no}={\left\{ \sigma ^n \ \mid \ n \equiv r_2 \mod N\right\} }\). There, it is shown that a measure-once one-way quantum automaton can solve exactly \(A^{N,r_1,r_2}\) with only \(3\) basis states, while any one-way deterministic finite automaton requires \(d\) states, \(d\) being the smallest integer such that \(d\mid N\) and \(d \not \mid (r_2-r_1) \mod N\). Here, we introduce the promise problem \({\textsc {Diof}}^{\,{a},N}_{r_1,r_2}\) as an extension of \(A^{N,r_1,r_2}\) to general alphabets. Even for this problem, we show the same descriptional superiority of the quantum paradigm over one-way deterministic automata. Moreover, we prove that even by adding features to classical automata, namely nondeterminism, probabilism, two-way motion, we cannot obtain automata for \(A^{N,r_1,r_2}\) and \({\textsc {Diof}}^{\,{a},N}_{r_1,r_2}\) smaller than one-way deterministic.

Size Lower Bounds for Quantum Automata

We investigate and compare the descriptional power of unary probabilistic and nondeterministic automata (pfas and nfas, respectively). We show the existence of a family of languages hard for pfas in the following sense: For any positive integer d, there exists a unary d-cyclic language such that any pfa accepting it requires d states, as the smallest deterministic automaton. On the other hand, we prove that there exist infinitely many languages having pfas which from one side do not match a known optimal state lower bound and, on the other side, they are smaller than nfas which, in turn, are smaller than deterministic automata.

On the Size of Two-Way Reasonable Automata for the Liveness Problem

We analyze the descriptional power of Quantum Finite Automata with control language.We show an exponential size cost conversion from QFAs with control language to DFAs.We simulate Latvian QFAs by QFAs with control languages.We obtain size lower bounds for several models of QFAs. We compare the descriptional power of quantum finite automata with control language (qfcs) and deterministic finite automata (dfas). By suitably adapting Rabins technique, we show how to convert any given qfc to an equivalent dfa, incurring in an at most exponential size increase. This enables us to state a lower bound on the size of qfcs, which is logarithmic in the size of equivalent minimal dfas. In turn, this result yields analogous size lower bounds for several models of quantum finite automata in the literature.

On inverse operations and their descriptional complexity

We compare the descriptional power of quantum finite automata with control language (qfcs) and deterministic finite automata (dfas). By suitably adapting Rabin’s technique, we show how to convert any given qfc to an equivalent dfa, incurring in an at most exponential size increase. This enables us to state a lower bound on the size of qfcs, which is logarithmic in the size of equivalent minimal dfas. In turn, this result yields analogous size lower bounds for several models of quantum finite automata in the literature.