In the longest plane spanning tree problem, we are given a finite planar point set \(\mathcal{P}\) , and our task is to find a plane (i.e., noncrossing) spanning tree for \(\mathcal{P}\) with maximum total Euclidean edge length. Despite more than two decades of research, it remains open whether this problem is NP-hard. Thus, previous results have focused on polynomial-time algorithms that produce plane trees whose total edge length approximates \(\mathrm{OPT}\) , the maximum possible length. The approximate trees in these algorithms all have small unweighted diameter, typically two to four. It is natural to ask whether this is a common feature of longest plane spanning trees, or an artifact of the specific approximation algorithms. We provide three results to elucidate the interplay between the approximation guarantee and the unweighted diameter of the approximate trees. First, we describe a polynomial-time algorithm to construct a plane tree with diameter at most four and total edge length at least \(0.546\cdot\mathrm{OPT}\) . This constitutes a substantial improvement over the state of the art. Second, we show that a longest plane tree among those with diameter at most three can be found in polynomial time. Third, for any candidate diameter \(d\geq 3\) , we provide upper bounds on the approximation factor that can be achieved by a longest plane tree with diameter at most \( d \) (compared to a longest plane tree without constraints).

Moran Birth-death process is a standard stochastic process that is used to model natural selection in spatially structured populations. A newly occurring mutation that invades a population of residents can either fixate on the whole population or it can go extinct due to random drift. The duration of the process depends not only on the total population size n, but also on the spatial structure of the population. In this work, we consider the Moran process with a single type of individuals who invade and colonize an otherwise empty environment. Mathematically, this corresponds to the setting where the residents have zero reproduction rate, thus they never reproduce. The spatial structure is represented by a graph. We present two main contributions. First, in contrast to the Moran process in which residents do reproduce, we show that the colonization time is always at most a polynomial function of the population size n. Namely, we show that colonization always takes at most [Formula: see text] expected steps, and for each n, we identify the slowest graph where it takes exactly that many steps. Moreover, we establish a stronger bound of roughly [Formula: see text] steps for undirected graphs and an even stronger bound of roughly [Formula: see text] steps for so-called regular graphs. Second, we discuss various complications that one faces when attempting to measure fixation times and colonization times in spatially structured populations, and we propose to measure the real duration of the process, rather than counting the steps of the classic Moran process.

Computing the rate of evolution in spatially structured populations is difficult. A key quantity is the fixation time of a single mutant with relative reproduction rate r which invades a population of residents. We say that the fixation time is "fast" if it is at most a polynomial function in terms of the population size N. Here we study fixation times of advantageous mutants (r > 1) and neutral mutants (r = 1) on directed graphs, which are those graphs that have at least some one-way connections. We obtain three main results. First, we prove that for any directed graph the fixation time is fast, provided that r is sufficiently large. Second, we construct an efficient algorithm that gives an upper bound for the fixation time for any graph and any r ≥ 1. Third, we identify a broad class of directed graphs with fast fixation times for any r ≥ 1. This class includes previously studied amplifiers of selection, such as Superstars and Metafunnels. We also show that on some graphs the fixation time is not a monotonically declining function of r; in particular, neutral fixation can occur faster than fixation for small selective advantages.

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 bidding games, a class of two-player zero-sum graph games. The game proceeds as follows. Both players have bounded budgets. A token is placed on a vertex of a graph, in each turn the players simultaneously submit bids, and the higher bidder moves the token, where we break bidding ties in favor of Player 1. Player 1 wins the game iff the token visits a designated target vertex. We consider, for the first time, poorman discrete-bidding in which the granularity of the bids is restricted and the higher bid is paid to the bank. Previous work either did not impose granularity restrictions or considered Richman bidding (bids are paid to the opponent). While the latter mechanisms are technically more accessible, the former is more appealing from a practical standpoint. Our study focuses on threshold budgets, which is the necessary and sufficient initial budget required for Player 1 to ensure winning against a given Player 2 budget. We first show existence of thresholds. In DAGs, we show that threshold budgets can be approximated with error bounds by thresholds under continuous-bidding and that they exhibit a periodic behavior. We identify closed-form solutions in special cases. We implement and experiment with an algorithm to find threshold budgets.

The Moran process is a classic stochastic process that models invasion dynamics on graphs. A single mutant (e.g., a new opinion, strain, social trait etc.) invades a population of residents spread over the nodes of a graph. The mutant fitness advantage δ>=0 determines how aggressively mutants propagate to their neighbors. The quantity of interest is the fixation probability, i.e., the probability that the initial mutant eventually takes over the whole population. However, in realistic settings, the invading mutant has an advantage only in certain locations. E.g., the ability to metabolize a certain sugar is an advantageous trait to bacteria only when the sugar is actually present in their surroundings. In this paper we introduce the positional Moran process, a natural generalization in which the mutant fitness advantage is only realized on specific nodes called active nodes, and study the problem of fixation maximization: given a budget k, choose a set of k active nodes that maximize the fixation probability of the invading mutant. We show that the problem is NP-hard, while the optimization function is not submodular, thus indicating strong computational hardness. We focus on two natural limits. In the limit of δ to infinity (strong selection), although the problem remains NP-hard, the optimization function becomes submodular and thus admits a constant-factor approximation using a simple greedy algorithm. In the limit of δ to 0 (weak selection), we show that we can obtain a tight approximation in O(n^{2×ω}) time, where ω is the matrix-multiplication exponent. An experimental evaluation of the new algorithms along with some proposed heuristics corroborates our results.

The voter process is a classic stochastic process that models the invasion of a mutant trait A (e.g., a new opinion, belief, legend, genetic mutation, magnetic spin) in a population of agents (e.g., people, genes, particles) who share a resident trait B, spread over the nodes of a graph. An agent may adopt the trait of one of its neighbors at any time, while the invasion bias r quantifies the stochastic preference towards (r>1) or against (r<1) adopting A over B. Success is measured in terms of the fixation probability, i.e., the probability that eventually all agents have adopted the mutant trait A. In this paper we study the problem of fixation probability maximization under this model: given a budget k, find a set of k agents to initiate the invasion that maximizes the fixation probability. We show that the problem is NP-hard for both regimes r>1 and r<1, while the latter case is also inapproximable within any multiplicative factor that is independent of r. On the positive side, we show that when r>1, the optimization function is submodular and thus can be greedily approximated within a factor 1-1/e. An experimental evaluation of some proposed heuristics corroborates our results.

In this paper we introduce and study all-pay bidding games, a class of two player, zero-sum games on graphs. The game proceeds as follows. We place a token on some vertex in the graph and assign budgets to the two players. Each turn, each player submits a sealed legal bid (non-negative and below their remaining budget), which is deducted from their budget and the highest bidder moves the token onto an adjacent vertex. The game ends once a sink is reached, and Player 1 pays Player 2 the outcome that is associated with the sink. The players attempt to maximize their expected outcome. Our games model settings where effort (of no inherent value) needs to be invested in an ongoing and stateful manner. On the negative side, we show that even in simple games on DAGs, optimal strategies may require a distribution over bids with infinite support. A central quantity in bidding games is the ratio of the players budgets. On the positive side, we show a simple FPTAS for DAGs, that, for each budget ratio, outputs an approximation for the optimal strategy for that ratio. We also implement it, show that it performs well, and suggests interesting properties of these games. Then, given an outcome c, we show an algorithm for finding the necessary and sufficient initial ratio for guaranteeing outcome c with probability 1 and a strategy ensuring such. Finally, while the general case has not previously been studied, solving the specific game in which Player 1 wins iff he wins the first two auctions, has been long stated as an open question, which we solve.

The fixation probability of a single mutant invading a population of residents is among the most widely-studied quantities in evolutionary dynamics. Amplifiers of natural selection are population structures that increase the fixation probability of advantageous mutants, compared to well-mixed populations. Extensive studies have shown that many amplifiers exist for the Birth-death Moran process, some of them substantially increasing the fixation probability or even guaranteeing fixation in the limit of large population size. On the other hand, no amplifiers are known for the death-Birth Moran process, and computer-assisted exhaustive searches have failed to discover amplification. In this work we resolve this disparity, by showing that any amplification under death-Birth updating is necessarily bounded and transient. Our boundedness result states that even if a population structure does amplify selection, the resulting fixation probability is close to that of the well-mixed population. Our transience result states that for any population structure there exists a threshold r⋆ such that the population structure ceases to amplify selection if the mutant fitness advantage r is larger than r⋆. Finally, we also extend the above results to δ-death-Birth updating, which is a combination of Birth-death and death-Birth updating. On the positive side, we identify population structures that maintain amplification for a wide range of values r and δ. These results demonstrate that amplification of natural selection depends on the specific mechanisms of the evolutionary process.

This paper deals with the multiple-rule problem which arises when several decision rules (of different classes) match ("fire" for) an input to-be-classified (unseen) object. The paper focuses on formal aspects and theoretical methodology for the above problem. The general definitions of the notions of a Designer, Learner and Classifier are presented in a formal matter, including parameters that are usually attached to the above concepts such as rule consistency, completeness, quality, matching rate, etc. We thus provide the minimum-requirement definitions as necessary conditions for these concepts. Any designer (decision-system builder) of a new multiple-rule system may start with these minimum requirements. We only expect that the Classifier makes its decisions according to its decision scheme induced as a knowledge base (theory, model, concept description). Also, two case studies are discussed. We conclude with a general flow chart for a decision-system builder. He/she can just pursue it and select parameters of a Learner and Classifier, following the minimum requirements provided.