We consider a class of optimization problems defined by a system of linear equations with min and max operators. This class of optimization problems has been studied under restrictive conditions, such as, (C1) the halting or stability condition; (C2) the non-negative coefficients condition; (C3) the sum upto 1 condition; and (C4) the only min or only max operator condition. Several seminal results in the literature focus on special cases. For example, turn-based stochastic games correspond to conditions C2 and C3; and Markov decision process to conditions C2, C3, and C4. However, the systematic computational complexity study of all the cases has not been explored, which we address in this work. Some highlights of our results are: with conditions C2 and C4, and with conditions C3 and C4, the problem is NP-complete, whereas with condition C1 only, the problem is in UP intersects coUP. Finally, we establish the computational complexity of the decision problem of checking the respective conditions.

Liquid democracy is a transitive vote delegation mechanism over voting graphs. It enables each voter to delegate their vote(s) to another better-informed voter, with the goal of collectively making a better decision. The question of whether liquid democracy outperforms direct voting has been previously studied in the context of local delegation mechanisms (where voters can only delegate to someone in their neighbourhood) and binary decision problems. It has previously been shown that it is impossible for local delegation mechanisms to outperform direct voting in general graphs. This raises the question: for which classes of graphs do local delegation mechanisms yield good results?

Abstract Two standard models for probabilistic systems are Markov chains (MCs) and Markov decision processes (MDPs). Classic objectives for such probabilistic models for control and planning problems are reachability and stochastic shortest path. The widely studied algorithmic approach for these problems is the Value Iteration (VI) algorithm which iteratively applies local updates called Bellman updates. There are many practical approaches for VI in the literature but they all require exponentially many Bellman updates for MCs in the worst case. A preprocessing step is an algorithm that is discrete, graph-theoretical, and requires linear space. An important open question is whether, after a polynomial-time preprocessing, VI can be achieved with sub-exponentially many Bellman updates. In this work, we present a new approach for VI based on guessing values. Our theoretical contributions are twofold. First, for MCs, we present an almost-linear-time preprocessing algorithm after which, along with guessing values, VI requires only subexponentially many Bellman updates. Second, we present an improved analysis of the speed of convergence of VI for MDPs. Finally, we present a practical algorithm for MDPs based on our new approach. Experimental results show that our approach provides a considerable improvement over existing VI-based approaches on several benchmark examples from the literature.

International audience

Reinforcement Learning (RL) from temporal logical specifications is a fundamental problem in sequential decision making. One of the basic and core such specification is the reachability specification that requires a target set to be eventually visited. Despite strong empirical results for RL from such specifications, the theoretical guarantees are bleak, including the impossibility of Probably Approximately Correct (PAC) guarantee for reachability specifications. Given the impossibility result, in this work we consider the problem of RL from reachability specifications along with the information of expected conditional distance (ECD). We present (a) lower bound results which establish the necessity of ECD information for PAC guarantees and (b) an algorithm that establishes PAC-guarantees given the ECD information. To the best of our knowledge, this is the first RL from reachability specifications that does not make any assumptions on the underlying environment to learn policies.

Turn-based discounted-sum games are two-player zero-sum games played on finite directed graphs. The vertices of the graph are partitioned between player 1 and player 2. Plays are infinite walks on the graph where the next vertex is decided by a player that owns the current vertex. Each edge is assigned an integer weight and the payoff of a play is the discounted-sum of the weights of the play. The goal of player 1 is to maximize the discounted-sum payoff against the adversarial player 2. These games lie in NP ∩ coNP and are among the rare combinatorial problems that belong to this complexity class and the existence of a polynomial-time algorithm is a major open question. Since breaking the general exponential barrier has been a challenging problem, faster parameterized algorithms have been considered. If the discount factor is expressed in unary, then discounted-sum games can be solved in polynomial time. However, if the discount factor is arbitrary (or expressed in binary), but the weights are in unary, none of the existing approaches yield a sub-exponential bound. Our main result is a new analysis technique for a classical algorithm (namely, the strategy iteration algorithm) that present a new runtime bound which is [EQUATION] for game graphs with n vertices and absolute weights of at most W. In particular, our result yields a deterministic sub-exponential bound for games with weights that are constant or represented in unary.

Given the increasingly stringent requirements on the performance and efficiency of communication networks, over the last years, great efforts have been made to render networks more flexible and programmable. In particular, modern networks support a flexible rerouting of flows, e.g., depending on the dynamically changing traffic or network conditions. However, the underlying algorithmic problems are still not well-understood today.In this paper, we revisit the k-Network Flow Update problem that asks for a schedule to reroute k unsplittable flows from their current paths to the given new paths, in a congestion-free manner in a capacitated network. We show that the problem is already NP-hard for three acyclic flows on simple directed graphs. Our main contribution is an efficient algorithm for sparse networks; specifically the algorithm is fixed parameter tractable in the number of flows and the treewidth of a graph that is the union of all flows. Our results also settle the open complexity question in the literature.

Populations evolve by accumulating advantageous mutations. Every population has some spatial structure that can be modeled by an underlying network. The network then influences the probability that new advantageous mutations fixate. Amplifiers of selection are networks that increase the fixation probability of advantageous mutants, as compared to the unstructured fully-connected network. Whether or not a network is an amplifier depends on the choice of the random process that governs the evolutionary dynamics. Two popular choices are Moran process with Birth-death updating and Moran process with death-Birth updating. Interestingly, while some networks are amplifiers under Birth-death updating and other networks are amplifiers under death-Birth updating, so far no spatial structures have been found that function as an amplifier under both types of updating simultaneously. In this work, we identify networks that act as amplifiers of selection under both versions of the Moran process. The amplifiers are robust, modular, and increase fixation probability for any mutant fitness advantage in a range r ∈ (1, 1.2). To complement this positive result, we also prove that for certain quantities closely related to fixation probability, it is impossible to improve them simultaneously for both versions of the Moran process. Together, our results highlight how the two versions of the Moran process differ and what they have in common.

We consider a natural problem dealing with weighted packet selection across a rechargeable link, which e.g., finds applications in cryptocurrency networks. The capacity of a link (u,v) is determined by how much nodes u and v allocate for this link. Specifically, the input is a finite ordered sequence of packets that arrive in both directions along a link. Given (u,v) and a packet of weight x going from u to v, node u can either accept or reject the packet. If u accepts the packet, the capacity on link (u,v) decreases by x. Correspondingly, v's capacity on (u,v) increases by x. If a node rejects the packet, this will entail a cost affinely linear in the weight of the packet. A link is “rechargeable” in the sense that the total capacity of the link has to remain constant, but the allocation of capacity at the ends of the link can depend arbitrarily on the nodes' decisions. The goal is to minimise the sum of the capacity injected into the link and the cost of rejecting packets. We show that the problem is NP-hard, but can be approximated efficiently with a ratio of (1+ε)⋅(1+3) for some arbitrary ε>0.

Turn-based stochastic games (aka simple stochastic games) are two-player zero-sum games played on directed graphs with probabilistic transitions. The goal of player-max is to maximize the probability to reach a target state against the adversarial player-min. These games lie in NP ∩ coNP and are among the rare combinatorial problems that belong to this complexity class for which the existence of polynomial-time algorithm is a major open question. While randomized sub-exponential time algorithm exists, all known deterministic algorithms require exponential time in the worst-case. An important open question has been whether faster algorithms can be obtained parametrized by the treewidth of the game graph. Even deterministic sub-exponential time algorithm for constant treewidth turn-based stochastic games has remain elusive. In this work our main result is a deterministic algorithm to solve turn-based stochastic games that, given a game with n states, treewidth at most t, and the bit-complexity of the probabilistic transition function log D, has running time O ((tn2 log D)t log n). In particular, our algorithm is quasi-polynomial time for games with constant or poly-logarithmic treewidth.

Game of Life is a simple and elegant model to study dynamical system over networks. The model consists of a graph where every vertex has one of two types, namely, dead or alive. A configuration is a mapping of the vertices to the types. An update rule describes how the type of a vertex is updated given the types of its neighbors. In every round, all vertices are updated synchronously, which leads to a configuration update. While in general, Game of Life allows a broad range of update rules, we focus on two simple families of update rules, namely, underpopulation and overpopulation, that model several interesting dynamics studied in the literature. In both settings, a dead vertex requires at least a desired number of live neighbors to become alive. For underpopulation (resp., overpopulation), a live vertex requires at least (resp. at most) a desired number of live neighbors to remain alive. We study the basic computation problems, e.g., configuration reachability, for these two families of rules. For underpopulation rules, we show that these problems can be solved in polynomial time, whereas for overpopulation rules they are PSPACE-complete.