arXiv subjects

Carlo Comin

Publications and source records attributed to Carlo Comin.

13 recordsLinked to original sources

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.

cs.CC↗

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

Energy Structure of Optimal Positional Strategies in Mean Payoff Games

This note studies structural aspects concerning Optimal Positional Strategies (OPSs) in Mean Payoff Games (MPGs), it is a contribution to understanding the relationship between OPSs in MPGs and Small Energy-Progress Measures (SEPMs) in reweighted Energy Games (EGs). Firstly, it is observed that the space of all OPSs, $\texttt{opt}_ΓΣ^M_0$, admits a unique complete decomposition in terms of so-called extremal-SEPM{s} in reweighted EG{s}; this points out what we called the "Energy-Lattice $\mathcal{X}^*_Γ$ of $\texttt{opt}_ΓΣ^M_0$". Secondly, 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 $\texttt{opt}_ΓΣ^M_0$. It is observed that the corresponding recursion tree defines an additional lattice $\mathcal{B}^*_Γ$, whose elements are certain subgames $Γ'\subseteq Γ$ that we call basic subgames. The extremal-SEPMs of a given \MPG $Γ$ coincide with the least-SEPMs of the basic subgames of $Γ$; so, $\mathcal{X}^*_Γ$ is the energy-lattice comprising all and only the least-SEPMs of the \emph{basic} subgames of $Γ$. The complexity of the proposed enumeration for both $\mathcal{B}^*_Γ$ and $\mathcal{X}^*_Γ$ is $O(|V|^3|E|W |\mathcal{B}^*_Γ|)$ total time and $O(|V||E|)+Θ\big(|E| \mathcal{B}^*_Γ|\big)$ working space. Finally, it is constructed an \MPG $Γ$ for which $|\mathcal{B}^*_Γ| > |\mathcal{X}^*_Γ|$, this proves that $\mathcal{B}^*_Γ$ and $\mathcal{X}^*_Γ$ are not isomorphic.

cs.GT↗

arXiv2016

Linear-Time Safe-Alternating DFS and SCCs

An alternating graph is a directed graph whose vertex set is partitioned into two classes, existential and universal. This forms the basic arena for a plethora of infinite duration two-player games where Player~$\square$ and~$\ocircle$ alternate in a turn-based sliding of a pebble along the arcs they control. We study alternating strongly-connectedness as a generalization of strongly-connectedness in directed graphs, aiming at providing a linear time decomposition and a sound structural graph characterization. For this a refined notion of alternating reachability is introduced: Player~$\square$ attempts to reach vertices without leaving a prescribed subset of the vertices, while Player~$\ocircle$ works against. This is named \emph{safe alternating reachability}. It is shown that every arena uniquely decomposes into safe alternating strongly-connected components where Player~$\square$ can visit each vertex within a given component infinitely often, without having to ever leave out the component itself. Our main result is a linear time algorithm for computing this alternating graph decomposition. Both the underlying graph structures and the algorithm generalize the classical decomposition of a directed graph into strongly-connected components. The algorithm builds on a linear time generalization of the depth-first search on alternation, taking inspiration from Tarjan 1972 machinery. Our theory has direct applications in solving well-known infinite duration pebble games faster. Dinneen and Khoussainov showed in 1999 that deciding a given Update Game costs $O(mn)$ time, where $n$ is the number of vertices and $m$ is that of arcs. We solve the task in $\Theta(m+n)$ linear~time. The complexity of Explicit McNaughton-M\"uller Games also improves from cubic to quadratic.

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

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.

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↗

arXiv2013

(Extended Version) Algebraic Characterization of the Class of Languages recognized by Measure Only Quantum Automata

We study a model of one-way quantum automaton where only measurement operations are allowed ($\mon$). We give an algebraic characterization of $\lmo(Σ)$, showing that the syntactic monoids of the languages in $\lmo(Σ)$ are exactly the $J$-trivial literally idempotent syntactic monoids, where $J$ is the Green's relation determined by two-sided ideals. We also prove that $\lmo(Σ)$ coincides with the literal variety of literally idempotent piecewise testable regular languages. This allows us to prove the existence of a polynomial time algorithm for deciding whether a regular language belongs to $\lmo(Σ)$ and to discuss definability issues in terms of the existential first-order logic $Σ_1[<]$ and the linear temporal logic without the next operator LTLWN.

cs.FL↗

arXiv2012

Algebraic Characterization of the Class of Languages recognized by Measure Only Quantum Automata

We study a model of one-way quantum automaton where only measurement operations are allowed (MOn-1qfa). We give an algebraic characterization of LMO, showing that the syntactic monoids of the languages in LMO are exactly the literal pseudovariety of J-trivial literally idempotent monoids, where J is the Green's relation determined by two-sided ideals. We also prove that LMO coincides with the literal variety of literally idempotent piecewise testable regular languages. This allows us to prove the existence of a polynomial time algorithm for deciding whether a regular language belongs to LMO.

cs.FL↗