arXiv subjects

Romeo Rizzi

Publications and source records attributed to Romeo Rizzi.

At least 37 records · Page 2Linked to original sources

arXiv2020

Safety in $s$-$t$ Paths, Trails and Walks

Given a directed graph $G$ and a pair of nodes $s$ and $t$, an \emph{$s$-$t$ bridge} of $G$ is an edge whose removal breaks all $s$-$t$ paths of $G$ (and thus appears in all $s$-$t$ paths). Computing all $s$-$t$ bridges of $G$ is a basic graph problem, solvable in linear time. In this paper, we consider a natural generalisation of this problem, with the notion of "safety" from bioinformatics. We say that a walk $W$ is \emph{safe} with respect to a set $\mathcal{W}$ of $s$-$t$ walks, if $W$ is a subwalk of all walks in $\mathcal{W}$. We start by considering the maximal safe walks when $\mathcal{W}$ consists of: all $s$-$t$ paths, all $s$-$t$ trails, or all $s$-$t$ walks of $G$. We show that the first two problems are immediate linear-time generalisations of finding all $s$-$t$ bridges, while the third problem is more involved. In particular, we show that there exists a compact representation computable in linear time, that allows outputting all maximal safe walks in time linear in their length. We further generalise these problems, by assuming that safety is defined only with respect to a subset of \emph{visible} edges. Here we prove a dichotomy between the $s$-$t$ paths and $s$-$t$ trails cases, and the $s$-$t$ walks case: the former two are NP-hard, while the latter is solvable with the same complexity as when all edges are visible. We also show that the same complexity results hold for the analogous generalisations of \emph{$s$-$t$ articulation points} (nodes appearing in all $s$-$t$ paths). We thus obtain the best possible results for natural "safety"-generalisations of these two fundamental graph problems. Moreover, our algorithms are simple and do not employ any complex data structures, making them ideal for use in practice.

cs.DS↗

arXiv2020

Computing all $s$-$t$ bridges and articulation points simplified

Given a directed graph $G$ and a pair of nodes $s$ and $t$, an $s$-$t$ bridge of $G$ is an edge whose removal breaks all $s$-$t$ paths of $G$. Similarly, an $s$-$t$ articulation point of $G$ is a node whose removal breaks all $s$-$t$ paths of $G$. Computing the sequence of all $s$-$t$ bridges of $G$ (as well as the $s$-$t$ articulation points) is a basic graph problem, solvable in linear time using the classical min-cut algorithm. When dealing with cuts of unit size ($s$-$t$ bridges) this algorithm can be simplified to a single graph traversal from $s$ to $t$ avoiding an arbitrary $s$-$t$ path, which is interrupted at the $s$-$t$ bridges. Further, the corresponding proof is also simplified making it independent of the theory of network flows.

cs.DS↗

arXiv2018

Instantaneous Reaction-Time in Dynamic-Consistency Checking of Conditional Simple Temporal Networks -- Extended version with an Improved Upper Bound --

CSTNs is a constraint-based graph-formalism for conditional temporal planning. In order to address the DC-Checking problem, in [Comin and Rizzi, TIME 2015] we introduced epsilon-DC (a refined, more realistic, notion of DC), and provided an algorithmic solution to it. The epsilon-DC notion is interesting per se, and the epsilon-DC-Checking algorithm in [Comin and Rizzi, TIME 2015] rests on the assumption that the reaction-time satisfies epsilon > 0; leaving unsolved the question of what happens when epsilon = 0. In this work, we introduce and study pi-DC, a sound notion of DC with an instantaneous reaction-time (i.e. one in which the planner can react to any observation at the same instant of time in which the observation is made). Firstly, we demonstrate by a counter-example that pi-DC is not equivalent to 0-DC, and that 0-DC is actually inadequate for modeling DC with an instantaneous reaction-time. This shows that the main results obtained in our previous work do not apply directly, as they were formulated, to the case of epsilon=0. Motivated by this observation, as a second contribution, our previous tools are extended in order to handle pi-DC, and the notion of ps-tree is introduced, also pointing out a relationship between pi-DC and HyTN-Consistency. Thirdly, a simple reduction from pi-DC-Checking to DC-Checking is identified. This allows us to design and to analyze the first sound-and-complete pi-DC-Checking procedure. Remarkably, the time complexity of the proposed algorithm remains (pseudo) singly-exponential in the number of propositional letters. Finally, it is observed that the technique can be leveraged to actually reduce from pi-DC to 1-DC, this allows us to further improve the exponents in the time complexity of pi-DC-Checking.

cs.DS↗

arXiv2018

On Restricted Disjunctive Temporal Problems: Faster Algorithms and Tractability Frontier

In 2005 Kumar studied the Restricted Disjunctive Temporal Problem (RDTP), a restricted but very expressive class of disjunctive temporal problems (DTPs). It was shown that that RDTPs are solvable in deterministic strongly-polynomial time by reducing them to the Connected Row-Convex (CRC) constraints problem; plus, Kumar devised a randomized algorithm whose expected running time is less than that of the deterministic one. Instead, the most general form of DTPs allows for multi-variable disjunctions of many interval constraints and it is NP-complete. This work offers a deeper comprehension on the tractability of RDTPs, leading to an elementary deterministic strongly-polynomial time algorithm for them, significantly improving the asymptotic running times of both the deterministic and randomized algorithms of Kumar. The result is obtained by reducing RDTPs to the Single-Source Shortest-Paths (SSSP) and the 2-SAT problem (jointly), instead of reducing to CRCs. In passing, we obtain a faster (quadratic-time) algorithm for RDTPs having only Type-1 and Type-2 constraints (and no Type-3 constraint). As a second main contribution, we study the tractability frontier of solving RDTPs by considering Hyper Temporal Networks (\HTNs), a strict generalization of \STNs grounded on hypergraphs: on one side, we prove that solving temporal problems having only Type-2 constraints and either only multi-tail or only multi-head hyperarc constraints lies in both NP and co-NP and it admits deterministic pseudo-polynomial time algorithms; on the other side, solving problems with Type-3 constraints and either only multi-tail or only multi-head hyperarc constraints turns strongly NP-complete.

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arXiv2018

Hardness of Covering Alignment: Phase Transition in Post-Sequence Genomics

Covering alignment problems arise from recent developments in genomics; so called pan-genome graphs are replacing reference genomes, and advances in haplotyping enable full content of diploid genomes to be used as basis of sequence analysis. In this paper, we show that the computational complexity will change for natural extensions of alignments to pan-genome representations and to diploid genomes. More broadly, our approach can also be seen as a minimal extension of sequence alignment to labelled directed acyclic graphs (labeled DAGs). Namely, we show that finding a \emph{covering alignment} of two labeled DAGs is NP-hard even on binary alphabets. A covering alignment asks for two paths $R_1$ (red) and $G_1$ (green) in DAG $D_1$ and two paths $R_2$ (red) and $G_2$ (green) in DAG $D_2$ that cover the nodes of the graphs and maximize the sum of the global alignment scores: $\mathsf{as}(\mathsf{sp}(R_1),\mathsf{sp}(R_2))+\mathsf{as}(\mathsf{sp}(G_1),\mathsf{sp}(G_2))$, where $\mathsf{sp}(P)$ is the concatenation of labels on the path $P$. Pair-wise alignment of haplotype sequences forming a diploid chromosome can be converted to a two-path coverable labelled DAG, and then the covering alignment models the similarity of two diploids over arbitrary recombinations. We also give a reduction to the other direction, to show that such a recombination-oblivious diploid alignment is NP-hard on alphabets of size $3$.

cs.CC↗

arXiv2018

Perfect phylogenies via branchings in acyclic digraphs and a generalization of Dilworth's theorem

Motivated by applications in cancer genomics and following the work of Hajirasouliha and Raphael (WABI 2014), Hujdurović et al. (IEEE TCBB, to appear) introduced the minimum conflict-free row split (MCRS) problem: split each row of a given binary matrix into a bitwise OR of a set of rows so that the resulting matrix corresponds to a perfect phylogeny and has the minimum possible number of rows among all matrices with this property. Hajirasouliha and Raphael also proposed the study of a similar problem, in which the task is to minimize the number of distinct rows of the resulting matrix. Hujdurović et al. proved that both problems are NP-hard, gave a related characterization of transitively orientable graphs, and proposed a polynomial-time heuristic algorithm for the MCRS problem based on coloring cocomparability graphs. We give new, more transparent formulations of the two problems, showing that the problems are equivalent to two optimization problems on branchings in a derived directed acyclic graph. Building on these formulations, we obtain new results on the two problems, including: (i) a strengthening of the heuristic by Hujdurović et al. via a new min-max result in digraphs generalizing Dilworth's theorem, which may be of independent interest, (ii) APX-hardness results for both problems, (iii) approximation algorithms, and (iv) exponential-time algorithms solving the two problems to optimality faster than the naïve brute-force approach. Our work relates to several well studied notions in combinatorial optimization: chain partitions in partially ordered sets, laminar hypergraphs, and (classical and weighted) colorings of graphs.

cs.DM↗

arXiv2017

Checking Dynamic Consistency of Conditional Hyper Temporal Networks via Mean Payoff Games (Hardness and (pseudo) Singly-Exponential Time Algorithm)

In this work we introduce the \emph{Conditional Hyper Temporal Network (CHyTN)} model, which is a natural extension and generalization of both the \CSTN and the \HTN model. Our contribution goes as follows. We show that deciding whether a given \CSTN or CHyTN is dynamically consistent is \coNP-hard. Then, we offer a proof that deciding whether a given CHyTN is dynamically consistent is \PSPACE-hard, provided that the input instances are allowed to include both multi-head and multi-tail hyperarcs. In light of this, we continue our study by focusing on CHyTNs that allow only multi-head or only multi-tail hyperarcs, and we offer the first deterministic (pseudo) singly-exponential time algorithm for the problem of checking the dynamic-consistency of such CHyTNs, also producing a dynamic execution strategy whenever the input CHyTN is dynamically consistent. Since \CSTN{s} are a special case of CHyTNs, this provides as a byproduct the first sound-and-complete (pseudo) singly-exponential time algorithm for checking dynamic-consistency in CSTNs. The proposed algorithm is based on a novel connection between CSTN{s}/CHyTN{s} and Mean Payoff Games. The presentation of the connection between \CSTN{s}/CHyTNs and \MPG{s} is mediated by the \HTN model. In order to analyze the algorithm, we introduce a refined notion of dynamic-consistency, named $ε$-dynamic-consistency, and present a sharp lower bounding analysis on the critical value of the reaction time $\hat{\varepsilon}$ where a \CSTN/CHyTN transits from being, to not being, dynamically consistent. The proof technique introduced in this analysis of $\hat{\varepsilon}$ is applicable more generally when dealing with linear difference constraints which include strict inequalities.

cs.CC↗

arXiv2017

Sorting With Forbidden Intermediates

A wide range of applications, most notably in comparative genomics, involve the computation of a shortest sorting sequence of operations for a given permutation, where the set of allowed operations is fixed beforehand. Such sequences are useful for instance when reconstructing potential scenarios of evolution between species, or when trying to assess their similarity. We revisit those problems by adding a new constraint on the sequences to be computed: they must \emph{avoid} a given set of \emph{forbidden intermediates}, which correspond to species that cannot exist because the mutations that would be involved in their creation are lethal. We initiate this study by focusing on the case where the only mutations that can occur are exchanges of any two elements in the permutations, and give a polynomial time algorithm for solving that problem when the permutation to sort is an involution.

cs.DS↗

arXiv2017

Hyper Temporal Networks

Simple Temporal Networks (STNs) provide a powerful and general tool for representing conjunctions of maximum delay constraints over ordered pairs of temporal variables. In this paper we introduce Hyper Temporal Networks (HyTNs), a strict generalization of STNs, to overcome the limitation of considering only conjunctions of constraints but maintaining a practical efficiency in the consistency check of the instances. In a Hyper Temporal Network a single temporal hyperarc constraint may be defined as a set of two or more maximum delay constraints which is satisfied when at least one of these delay constraints is satisfied. HyTNs are meant as a light generalization of STNs offering an interesting compromise. On one side, there exist practical pseudo-polynomial time algorithms for checking consistency and computing feasible schedules for HyTNs. On the other side, HyTNs offer a more powerful model accommodating natural constraints that cannot be expressed by STNs like Trigger off exactly delta min before (after) the occurrence of the first (last) event in a set., which are used to represent synchronization events in some process aware information systems/workflow models proposed in the literature.

cs.DS↗

arXiv2016

Faster O(|V|^2|E|W)-Time Energy Algorithms for Optimal Strategy Synthesis in Mean Payoff Games

This study strengthens the links between Mean Payoff Games (\MPG{s}) and Energy Games (EG{s}). Firstly, we offer a faster $O(|V|^2|E|W)$ pseudo-polynomial time and $Θ(|V|+|E|)$ space deterministic algorithm for solving the Value Problem and Optimal Strategy Synthesis in \MPG{s}. This improves the best previously known estimates on the pseudo-polynomial time complexity to: \[ O(|E|\log |V|) + Θ\Big(\sum_{v\in V}\texttt{deg}_Γ(v)\cdot\ell_Γ(v)\Big) = O(|V|^2|E|W), \] where $\ell_Γ(v)$ counts the number of times that a certain energy-lifting operator $δ(\cdot, v)$ is applied to any $v\in V$, along a certain sequence of Value-Iterations on reweighted \EG{s}; and $\texttt{deg}_Γ(v)$ is the degree of $v$. This improves significantly over a previously known pseudo-polynomial time estimate, i.e. $Θ\big(|V|^2|E|W + \sum_{v\in V}\texttt{deg}_Γ(v)\cdot\ell_Γ(v)\big)$ \citep{CR15, CR16}, as the pseudo-polynomiality is now confined to depend solely on $\ell_Γ$. Secondly, we further explore on the relationship between Optimal Positional Strategies (OPSs) in \MPG{s} and Small Energy-Progress Measures (SEPMs) in reweighted \EG{s}. It is observed that the space of all OPSs, $\texttt{opt}_ΓΣ^M_0$, admits a unique complete decomposition in terms of extremal-SEPM{s} in reweighted EG{s}. This points out what we called the "Energy-Lattice $\mathcal{X}^*_Γ$ associated to $\texttt{opt}_ΓΣ^M_0$". Finally, it is offered a pseudo-polynomial total-time recursive procedure for enumerating (w/o repetitions) all the elements of $\mathcal{X}^*_Γ$, and for computing the corresponding partitioning of $\texttt{opt}_ΓΣ^M_0$.

cs.DS↗

arXiv2016

The Complexity of Simulation and Matrix Multiplication

Computing the simulation preorder of a given Kripke structure (i.e., a directed graph with $n$ labeled vertices) has crucial applications in model checking of temporal logic. It amounts to solving a specific two-players reachability game, called simulation game. We offer the first conditional lower bounds for this problem, and we relate its complexity (for computation, verification, and certification) to some variants of $n\times n$ matrix multiplication. We show that any $O(n^α)$-time algorithm for simulation games, even restricting to acyclic games/structures, can be used to compute $n\times n$ boolean matrix multiplication (BMM) in $O(n^α)$ time. This is the first evidence that improving the existing $O(n^{3})$-time solutions may be difficult, without resorting to fast matrix multiplication. In the acyclic case, we match this lower bound presenting the first subcubic algorithm, based on fast BMM, and running in $n^{ω+o(1)}$ time (where $ω<2.376$ is the exponent of matrix multiplication). For both acyclic and cyclic structures, we point out the existence of natural and canonical $O(n^{2})$-size certificates, that can be verified in truly subcubic time. In the acyclic case, $O(n^{2})$ time is sufficient, employing standard matrix product verification. In the cyclic case, a $\max$-semi-boolean matrix multiplication (MSBMM) is used, i.e., a matrix multiplication on the semi-ring $(\max,\times)$ where one matrix contains only $0$'s and $1$'s. This MSBMM is computable (hence verifiable) in truly subcubic $n^{(3+ω)/2+o(1)}$ time by reduction to $(\max,\min)$-multiplication. Finally, we show a reduction from MSBMM to cyclic simulation games which implies a separation between the cyclic and the acyclic cases, unless MSBMM can be verified in $n^{ω+o(1)}$ time.

cs.CC↗

arXiv2016

Dynamic Controllability of Conditional Simple Temporal Networks is PSPACE-complete

Even after the proposal of various solution algorithms, the precise computational complexity of checking whether a Conditional Temporal Network is Dynamically Controllable had still remained widely open. This issue gets settled in this paper which provides constructions, algorithms, and bridging lemmas and arguments to formally prove that: (1) the problem is PSPACE-hard, and (2) the problem lies in PSPACE.

cs.DS↗

arXiv2016

Decomposing Cubic Graphs into Connected Subgraphs of Size Three

Let $S=\{K_{1,3},K_3,P_4\}$ be the set of connected graphs of size 3. We study the problem of partitioning the edge set of a graph $G$ into graphs taken from any non-empty $S'\subseteq S$. The problem is known to be NP-complete for any possible choice of $S'$ in general graphs. In this paper, we assume that the input graph is cubic, and study the computational complexity of the problem of partitioning its edge set for any choice of $S'$. We identify all polynomial and NP-complete problems in that setting, and give graph-theoretic characterisations of $S'$-decomposable cubic graphs in some cases.

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arXiv2016

An Improved Pseudo-Polynomial Upper Bound for the Value Problem and Optimal Strategy Synthesis in Mean Payoff Games

In this work we offer an $O(|V|^2 |E|\, W)$ pseudo-polynomial time deterministic algorithm for solving the Value Problem and Optimal Strategy Synthesis in Mean Payoff Games. This improves by a factor $\log(|V|\, W)$ the best previously known pseudo-polynomial time upper bound due to Brim,~\etal The improvement hinges on a suitable characterization of values, and a description of optimal positional strategies, in terms of reweighted Energy Games and Small Energy-Progress Measures.

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arXiv2015

Decoding Hidden Markov Models Faster Than Viterbi Via Online Matrix-Vector (max, +)-Multiplication

In this paper, we present a novel algorithm for the maximum a posteriori decoding (MAPD) of time-homogeneous Hidden Markov Models (HMM), improving the worst-case running time of the classical Viterbi algorithm by a logarithmic factor. In our approach, we interpret the Viterbi algorithm as a repeated computation of matrix-vector $(\max, +)$-multiplications. On time-homogeneous HMMs, this computation is online: a matrix, known in advance, has to be multiplied with several vectors revealed one at a time. Our main contribution is an algorithm solving this version of matrix-vector $(\max,+)$-multiplication in subquadratic time, by performing a polynomial preprocessing of the matrix. Employing this fast multiplication algorithm, we solve the MAPD problem in $O(mn^2/ \log n)$ time for any time-homogeneous HMM of size $n$ and observation sequence of length $m$, with an extra polynomial preprocessing cost negligible for $m > n$. To the best of our knowledge, this is the first algorithm for the MAPD problem requiring subquadratic time per observation, under the only assumption -- usually verified in practice -- that the transition probability matrix does not change with time.

cs.LG↗

arXiv2015

Pattern matching in $(213,231)$-avoiding permutations

Given permutations $σ\in S_k$ and $π\in S_n$ with $k<n$, the \emph{pattern matching} problem is to decide whether $π$ matches $σ$ as an order-isomorphic subsequence. We give a linear-time algorithm in case both $π$ and $σ$ avoid the two size-$3$ permutations $213$ and $231$. For the special case where only $σ$ avoids $213$ and $231$, we present a $O(max(kn^2,n^2\log(\log(n)))$ time algorithm. We extend our research to bivincular patterns that avoid $213$ and $231$ and present a $O(kn^4)$ time algorithm. Finally we look at the related problem of the longest subsequence which avoids $213$ and $231$.

cs.DS↗

arXiv2015

An Improved Upper Bound on Maximal Clique Listing via Rectangular Fast Matrix Multiplication

The first output-sensitive algorithm for the Maximal Clique Listing problem was given by Tsukiyama et.al. in 1977. As any algorithm falling within the Reverse Search paradigm, it performs a DFS visit of a directed tree (the RS-tree) having the objects to be listed (i.e. maximal cliques) as its nodes. In a recursive implementation, the RS-tree corresponds to the recursion tree of the algorithm. The time delay is given by the cost of generating the next child of a node, and Tsukiyama showed it is $O(mn)$. In 2004, Makino and Uno sharpened the time delay to $O(n^ω)$ by generating all the children of a node in one single shot performed by computing a \emph{square} fast matrix multiplication. In this paper, we further improve the asymptotics for the exploration of the same RS-tree by grouping the offsprings' computation even further. Our idea is to rely on rectangular fast matrix multiplication in order to compute all children of $n^2$ nodes in one shot. According to the current upper bounds on fast matrix multiplication, with this the time delay improves from $O(n^{2.3728639})$ to $O(n^{2.093362})$.

cs.DS↗

arXiv2015

Dynamic Consistency of Conditional Simple Temporal Networks via Mean Payoff Games: a Singly-Exponential Time DC-Checking

Conditional Simple Temporal Network (CSTN) is a constraint-based graph-formalism for conditional temporal planning. It offers a more flexible formalism than the equivalent CSTP model of Tsamardinos, Vidal and Pollack, from which it was derived mainly as a sound formalization. Three notions of consistency arise for CSTNs and CSTPs: weak, strong, and dynamic. Dynamic consistency is the most interesting notion, but it is also the most challenging and it was conjectured to be hard to assess. Tsamardinos, Vidal and Pollack gave a doubly-exponential time algorithm for deciding whether a CSTN is dynamically-consistent and to produce, in the positive case, a dynamic execution strategy of exponential size. In the present work we offer a proof that deciding whether a CSTN is dynamically-consistent is coNP-hard and provide the first singly-exponential time algorithm for this problem, also producing a dynamic execution strategy whenever the input CSTN is dynamically-consistent. The algorithm is based on a novel connection with Mean Payoff Games, a family of two-player combinatorial games on graphs well known for having applications in model-checking and formal verification. The presentation of such connection is mediated by the Hyper Temporal Network model, a tractable generalization of Simple Temporal Networks whose consistency checking is equivalent to determining Mean Payoff Games. In order to analyze the algorithm we introduce a refined notion of dynamic-consistency, named ε-dynamic-consistency, and present a sharp lower bounding analysis on the critical value of the reaction time \hat{\varepsilon} where the CSTN transits from being, to not being, dynamically-consistent. The proof technique introduced in this analysis of \hat{\varepsilon} is applicable more in general when dealing with linear difference constraints which include strict inequalities.

cs.DS↗