Minimal Reachability is Hard To Approximate

In this note, we consider the problem of choosing which nodes of a linear dynamical system should be actuated so that the state transfer from the system's initial condition to a given final state is possible. Assuming a standard complexity hypothesis, we show that this problem cannot be efficiently solved or approximated in polynomial, or even quasi-polynomial, time

Orienting Graphs to Optimize Reachability

The paper focuses on two problems: (i) how to orient the edges of an undirected graph in order to maximize the number of ordered vertex pairs (x,y) such that there is a directed path from x to y, and (ii) how to orient the edges so as to minimize the number of such pairs. The paper describes a quadratic-time algorithm for the first problem, and a proof that the second problem is NP-hard to approximate within some constant 1+epsilon > 1. The latter proof also shows that the second problem is equivalent to ``comparability graph completion''; neither problem was previously known to be NP-hard

Strengthening Model Checking Techniques with Inductive Invariants

This paper describes optimized techniques to efficiently compute and reap benefits from inductive invariants within SAT-based model checking. We address sequential circuit verification, and we consider both equivalences and implications between pairs of nodes in the logic networks. First, we present a very efficient dynamic procedure, based on equivalence classes and incremental SAT, specifically oriented to reduce the set of checked invariants. Then, we show how to effectively integrate the computation of inductive invariants within state-of-the-art SAT-based model checking procedures. Experiments (on more than 600 designs) show the robustness of our approach on verification instances on which stand-alone techniques fai

Reachability and Shortest Paths in the Broadcast CONGEST Model

In this paper we study the time complexity of the single-source reachability problem and the single-source shortest path problem for directed unweighted graphs in the Broadcast CONGEST model. We focus on the case where the diameter D of the underlying network is constant. We show that for the case where D = 1 there is, quite surprisingly, a very simple algorithm that solves the reachability problem in 1(!) round. In contrast, for networks with D = 2, we show that any distributed algorithm (possibly randomized) for this problem requires Omega(sqrt{n/ log{n}}) rounds. Our results therefore completely resolve (up to a small polylog factor) the complexity of the single-source reachability problem for a wide range of diameters. Furthermore, we show that when D = 1, it is even possible to get an almost 3 - approximation for the all-pairs shortest path problem (for directed unweighted graphs) in just 2 rounds. We also prove a stronger lower bound of Omega(sqrt{n}) for the single-source shortest path problem for unweighted directed graphs that holds even when the diameter of the underlying network is 2. As far as we know this is the first lower bound that achieves Omega(sqrt{n}) for this problem

A recursive paradigm for aligning observed behavior of large structured process models

The alignment of observed and modeled behavior is a crucial problem in process mining, since it opens the door for conformance checking and enhancement of process models. The state of the art techniques for the computation of alignments rely on a full exploration of the combination of the model state space and the observed behavior (an event log), which hampers their applicability for large instances. This paper presents a fresh view to the alignment problem: the computation of alignments is casted as the resolution of Integer Linear Programming models, where the user can decide the granularity of the alignment steps. Moreover, a novel recursive strategy is used to split the problem into small pieces, exponentially reducing the complexity of the ILP models to be solved. The contributions of this paper represent a promising alternative to fight the inherent complexity of computing alignments for large instances.Peer ReviewedPostprint (author's final draft

Zero-Reachability in Probabilistic Multi-Counter Automata

We study the qualitative and quantitative zero-reachability problem in probabilistic multi-counter systems. We identify the undecidable variants of the problems, and then we concentrate on the remaining two cases. In the first case, when we are interested in the probability of all runs that visit zero in some counter, we show that the qualitative zero-reachability is decidable in time which is polynomial in the size of a given pMC and doubly exponential in the number of counters. Further, we show that the probability of all zero-reaching runs can be effectively approximated up to an arbitrarily small given error epsilon > 0 in time which is polynomial in log(epsilon), exponential in the size of a given pMC, and doubly exponential in the number of counters. In the second case, we are interested in the probability of all runs that visit zero in some counter different from the last counter. Here we show that the qualitative zero-reachability is decidable and SquareRootSum-hard, and the probability of all zero-reaching runs can be effectively approximated up to an arbitrarily small given error epsilon > 0 (these result applies to pMC satisfying a suitable technical condition that can be verified in polynomial time). The proof techniques invented in the second case allow to construct counterexamples for some classical results about ergodicity in stochastic Petri nets.Comment: 20 page

Algorithmic Verification of Continuous and Hybrid Systems

We provide a tutorial introduction to reachability computation, a class of computational techniques that exports verification technology toward continuous and hybrid systems. For open under-determined systems, this technique can sometimes replace an infinite number of simulations.Comment: In Proceedings INFINITY 2013, arXiv:1402.661