Standard local search algorithms for combinatorial optimization problems repeatedly apply small changes to a current solution to improve the problem's given objective function. In contrast, non-oblivious local search algorithms are guided by an auxiliary potential function, which is distinct from the problem's objective. In this thesis, we compare the standard and non-oblivious approaches for a variety of problems, and derive new, improved non-oblivious local search algorithms for several problems in the area of constrained linear and monotone submodular maximization. First, we give a new, randomized approximation algorithm for maximizing a monotone submodular function subject to a matroid constraint. Our algorithm's approximation ratio matches both the known hardness of approximation bounds for the problem and the performance of the recent ``continuous greedy'' algorithm. Unlike the continuous greedy algorithm, our algorithm is straightforward and combinatorial. In the case that the monotone submodular function is a coverage function, we can obtain a further simplified, deterministic algorithm with improved running time. Moving beyond the case of single matroid constraints, we then consider general classes of set systems that capture problems that can be approximated well. While previous such classes have focused primarily on greedy algorithms, we give a new class that captures problems amenable to optimization by local search algorithms. We show that several combinatorial optimization problems can be placed in this class, and give a non-oblivious local search algorithm that delivers improved approximations for a variety of specific problems. In contrast, we show that standard local search algorithms give no improvement over known approximation results for these problems, even when allowed to search larger neighborhoods than their non-oblivious counterparts. Finally, we expand on these results by considering standard local search algorithms for constraint satisfaction problems. We develop conditions under which the approximation ratio of standard local search remains limited even for super-polynomial or exponential local neighborhoods. In the special case of MaxCut, we further show that a variety of techniques including random or greedy initialization, large neighborhoods, and best-improvement pivot rules cannot improve the approximation performance of standard local search.

Standard local search algorithms for combinatorial optimization problems repeatedly apply small changes to a current solution to improve the problem's given objective function. In contrast, non-oblivious local search algorithms are guided by an auxiliary potential function, which is distinct from the problem's objective. In this thesis, we compare the standard and non-oblivious approaches for a variety of problems, and derive new, improved non-oblivious local search algorithms for several problems in the area of constrained linear and monotone submodular maximization. First, we give a new, randomized approximation algorithm for maximizing a monotone submodular function subject to a matroid constraint. Our algorithm's approximation ratio matches both the known hardness of approximation bounds for the problem and the performance of the recent ``continuous greedy'' algorithm. Unlike the continuous greedy algorithm, our algorithm is straightforward and combinatorial. In the case that the monotone submodular function is a coverage function, we can obtain a further simplified, deterministic algorithm with improved running time. Moving beyond the case of single matroid constraints, we then consider general classes of set systems that capture problems that can be approximated well. While previous such classes have focused primarily on greedy algorithms, we give a new class that captures problems amenable to optimization by local search algorithms. We show that several combinatorial optimization problems can be placed in this class, and give a non-oblivious local search algorithm that delivers improved approximations for a variety of specific problems. In contrast, we show that standard local search algorithms give no improvement over known approximation results for these problems, even when allowed to search larger neighborhoods than their non-oblivious counterparts. Finally, we expand on these results by considering standard local search algorithms for constraint satisfaction problems. We develop conditions under which the approximation ratio of standard local search remains limited even for super-polynomial or exponential local neighborhoods. In the special case of MaxCut, we further show that a variety of techniques including random or greedy initialization, large neighborhoods, and best-improvement pivot rules cannot improve the approximation performance of standard local search.

In this dissertation, we study multi-robot coordination in the context of multi-target tracking. Specifically, we are interested in the coordination achieved by means of submodular function optimization. Submodularity encodes the diminishing returns property that arises in multi-robot coordination. For example, the marginal gain of assigning an additional robot to track the same target diminishes as the number of robots assigned increases. The advantage of formulating coordination problems as submodular optimization is that a simple, greedy algorithm is guaranteed to give a good performance. However, often this comes at the expense of unrealistic models and assumptions. For example, the standard formulation does not take into account the fact that robots may fail, either randomly or due to adversarial attacks. When operating in uncertain conditions, we typically seek to optimize the expected performance. However, this does not give any flexibility for a user to seek conservative or aggressive behaviors from the team of robots. Furthermore, most coordination algorithms force robots to communicate at each time step, even though they may not need to. Our goal in this dissertation is to overcome these limitations by devising coordination algorithms that are parsimonious in communication, allow a user to manage the risk of the robot performance, and are resilient to worst-case robot failures and attacks. In the first part of this dissertation, we focus on designing parsimonious communication strategies for target tracking. Specifically, we investigate the problem of determining when to communicate and who to communicate with. When the robots use range sensors, the tracking performance is a function of the relative positions of the robots and the targets. We propose a self-triggered communication strategy in which a robot communicates its own position with its neighbors only when a certain set of conditions are violated. We prove that this strategy converges to the optimal robot positions for tracking a single target and in practice, reduces the number of communication messages by 30%. When tracking multiple targets, we can reduce the communication by forming subsets of robots and assigning one subset to track a target. We investigate a number of measures for tracking quality based on the observability matrix and show which ones are submodular and which ones are not. For non-submodular measures, we show a greedy algorithm gives a 1/(n+1) approximation, if we restrict the subset to n robots. In optimizing submodular functions, a common assumption is that the function value is deterministic, which may not hold in practice. For example, the sensor performance may depend on environmental conditions which are not known exactly. In the second part of the dissertation, we design an algorithm for stochastic submodular optimization. The standard formulation for stochastic optimization optimizes the expected performance. However, the expectation is a risk-neutral measure. Instead, we optimize the Conditional Value-at-Risk (CVaR), which allows the user the flexibility of choosing a risk level. We present an algorithm, based on the greedy algorithm, and prove that its performance has bounded suboptimality and improves with running time. We also present an online version of the algorithm to adapt to real-time scenarios. In the third part of this dissertation, we focus on scenarios where a set of robots may fail naturally or due to adversarial attacks. Our objective is to track as many targets as possible, a submodular measure, assuming worst-case robot failures. We present both centralized and distributed resilient tracking algorithms to cope with centralized and distributed communication settings. We prove these algorithms give a constant-factor approximation of the optimal in polynomial running time. / Doctor of Philosophy / Today, robotics and autonomous systems have been increasingly used in various areas such as manufacturing, military, agriculture, medical sciences, and environmental monitoring. However, most of these systems are fragile and vulnerable to adversarial attacks and uncertain environmental conditions. In most cases, even if a part of the system fails, the entire system performance can be significantly undermined. As robots start to coexist with humans, we need algorithms that can be trusted under real-world, not just ideal conditions. Thus, this dissertation focuses on enabling security, trustworthiness, and long-term autonomy in robotics and autonomous systems. In particular, we devise coordination algorithms that are resilient to attacks, trustworthy in the face of the uncertain conditions, and allow the long-term operation of multi-robot systems. We evaluate our algorithms through extensive simulations and proof-of-concept experiments. Generally speaking, multi-robot systems form the "physical" layer of Cyber-Physical Sytems (CPS), the Internet of Things (IoT), and Smart City. Thus, our research can find applications in the areas of connected and autonomous vehicles, intelligent transportation, communications and sensor networks, and environmental monitoring in smart cities.

This work is inspired by robotic search applications where a robot or team of robots is equipped with sensors and tasked to autonomously acquire as much information as possible from a region of interest. To accomplish this task, robots must plan paths through the region of interest that maximize the effectiveness of the sensors they carry. Receding horizon path planning is a popular approach to addressing the computationally expensive task of planning long paths because it allows robotic agents with limited computational resources to iteratively construct a long path by solving for an optimal short path, traversing a portion of the short path, and repeating the process until a receding horizon path of the desired length has been constructed. However, receding horizon paths do not retain the optimality properties of the short paths from which they are constructed and may perform quite poorly in the context of achieving the robotic search objective. The primary contributions of this work address the worst-case performance of receding horizon paths by developing methods of using terminal rewards in the construction of receding horizon paths. We prove that the proposed methods of constructing receding horizon paths provide theoretical worst-case performance guarantees. Our result can be interpreted as ensuring that the receding horizon path performs no worse in expectation than a given sub-optimal search path. This result is especially practical for subsea applications where, due to use of side-scan sonar in search applications, search paths typically consist of parallel straight lines. Thus for subsea search applications, our approach ensures that expected performance is no worse than the usual subsea search path, and it might be much better. The methods proposed in this work provide desirable lower-bound guarantees for a single robot as well as teams of robots. Significantly, we demonstrate that existing planning algorithms may be easily adapted to use our proposed methods. We present our theoretical guarantees in the context of subsea search applications and demonstrate the utility of our proposed methods through simulation experiments and field trials using real autonomous underwater vehicles (AUVs). We show that our worst-case guarantees may be achieved despite non-idealities such as sub-optimal short-paths used to construct the longer receding horizon path and unreliable communication in multi-agent planning. In addition to theoretical guarantees, An important contribution of this work is to describe specific implementation solutions needed to integrate and implement these ideas for real-time operation on AUVs. / Doctor of Philosophy / This work is inspired by robotic search applications where a robot or team of robots is equipped with sensors and tasked to autonomously acquire as much information as possible from a region of interest. To accomplish this task, robots must plan paths through the region of interest that maximize the effectiveness of the sensors they carry. Receding horizon path planning is a popular approach to addressing the computationally expensive task of planning long paths because it allows robotic agents with limited computational resources to iteratively construct a long path by solving for an optimal short path, traversing a portion of the short path, and repeating the process until a receding horizon path of the desired length has been constructed. However, receding horizon paths do not retain the optimality properties of the short paths from which they are constructed and may perform quite poorly in the context of achieving the robotic search objective. The primary contributions of this work address the worst-case performance of receding horizon paths by developing methods of using terminal rewards in the construction of receding horizon paths. The methods proposed in this work provide desirable lower-bound guarantees for a single robot as well as teams of robots. We present our theoretical guarantees in the context of subsea search applications and demonstrate the utility of our proposed methods through simulation experiments and field trials using real autonomous underwater vehicles (AUVs). In addition to theoretical guarantees, An important contribution of this work is to describe specific implementation solutions needed to integrate and implement these ideas for real-time operation on AUVs.

<p>With an increasing number of users spending time on social media platforms and engaging with family, friends, and influencers within communities of interest (such as in fashion, cooking, gaming, etc.), there are significant opportunities for marketing firms to leverage word-of-mouth advertising on these platforms. In particular, marketing firms can select sets of influencers within relevant communities to sponsor, namely by providing free product samples to those influencers so that so they will discuss and promote the product on their social media accounts.</p><p>The question of which set of influencers to sponsor is known as <b>influence maximization</b> (IM) formally defined as follows: "if we can try to convince a subset of individuals in a social network to adopt a new product or innovation, and the goal is to trigger a large cascade of further adoptions, which set of individuals should we target?'' Under standard diffusion models, this optimization problem is known to be NP-hard. This problem has been widely studied in the literature and several approaches for solving it have been proposed. Some approaches provide near-optimal solutions but are costly in terms of runtime. On the other hand, some approaches are faster but heuristics, i.e., do not have approximation guarantees.</p><p>In this dissertation, we study the influence maximization problem extensively. We provide efficient algorithms for solving the original problem and its important generalizations. Furthermore, we provide theoretical guarantees and experimental evaluations to support the claims made in this dissertation.</p><p>We first study the original IM problem referred to as the discrete influence maximization (DIM) problem where the marketer can either provide a free sample to an influencer or not, i.e., they cannot give fractional discounts like 10% off, etc. As already mentioned the existing solution methods (for instance, the simulation-based greedy algorithm) provide near-optimal solutions that are costly in terms of runtime and the approaches that are faster do not have approximation guarantees. Motivated by the idea of addressing this trade-off between accuracy and runtime, we propose a community-aware divide-and-conquer framework to provide a time-efficient solution to the DIM problem. The proposed framework outperforms the standard methods in terms of runtime and the heuristic methods in terms of influence.</p><p>We next study a natural extension of the DIM problem referred to as the fractional influence maximization (FIM) problem where the marketer may offer fractional discounts (as opposed to either providing a free sample to an influencer or not in the DIM problem) to the influencers. Clearly, the FIM problem provides more flexibility to the marketer in allocating the available budget among different influencers. The existing solution methods propose to use a continuous extension of the simulation-based greedy approximation algorithm for solving the DIM problem. This continuous extension suggests greedily building the solution for the given fractional budget by taking small steps through the interior of the feasible region. On the contrary, we first characterize the solution to the FIM problem in terms of the solution to the DIM problem. We then use this characterization to propose an efficient greedy approximation algorithm that only iterates through the corners of the feasible region. This leads to huge savings in terms of runtime compared to the existing methods that suggest iterating through the interior of the feasible region. Furthermore, we provide an approximation guarantee for the proposed greedy algorithm to solve the FIM problem.</p><p>Finally, we study another extension of the DIM problem referred to as the online discrete influence maximization (ODIM) problem, where the marketer provides free samples not just once but repeatedly over a given time horizon and the goal is to maximize the cumulative influence over time while receiving instantaneous feedback. The existing solution methods are based on semi-bandit instantaneous feedback where the knowledge of some intermediate aspects of how the influence propagates in the social network is assumed or observed. For instance, which specific individuals became influenced at the intermediate steps during the propagation? However, for social networks with user privacy, this information is not available. Hence, we consider the ODIM problem with full-bandit feedback where no knowledge of the underlying social network or diffusion process is assumed. We note that the ODIM problem is an instance of the stochastic combinatorial multi-armed bandit (CMAB) problem with submodular rewards. To solve the ODIM problem, we provide an efficient algorithm that outperforms the existing methods in terms of influence, and time and space complexities.</p><p>Furthermore, we point out the connections of influence maximization with a related problem of disease outbreak prevention and a more general problem of submodular maximization. The methods proposed in this dissertation can also be used to solve those problems.</p>

Recently, Udwani defined a new class of set functions under monotonicity and subadditivity, called submodular order functions, which is a subfamily of submodular functions. Informally, the submodular order function admits a very limited form of submodularity which is defined over a specific permutation of the ground set. His work pointed out the intriguing connection between streaming submodular maximization and submodular order maximization. Inspired by a 0.25-approximation streaming algorithm for maximizing a monotone submodular function subject to a matroid constraint, Udwani gave a 0.25-approximation algorithm for submodular order functions maximization subject to a matroid constraint. Based on the above results, we would like to explore further in which cases it is feasible to generalize from streaming submodular maximization algorithms to submodular order maximization algorithms. As a more general constraint than matroid, p-matchoid is a collection of p matroids with each matroid defined on some subsets of the ground set. Related work gave a 1/4p-approximation streaming algorithm for monotone submodular functions maximization under a p-matchoid constraint. Inspired by the above algorithms and the intriguing connection, we used some techniques to try to generalize several streaming algorithms for submodular functions to the offline algorithms for submodular order functions, including interleaved partitions and incremental values. Assuming that the objective function f is subadditive and non-negative, we gave a 1/4p-approximation algorithm for monotone submodular order maximization to a p-matchoid constraint. In addition, we summarize the failures of other cases. / Nyligen definierade Udwani en ny klass av mängdfunktioner under monotonicitet och subadditivitet, som kallas submodulära ordningsfunktioner och som är en underfamilj av submodulära funktioner. Informellt sett medger den submodulära ordningsfunktionen en mycket begränsad form av submodularitet som är definierad över en specifik permutation av grundmängden. Hans arbete pekade på det spännande sambandet mellan strömmande submodulär maximering och submodulär ordermaximering. Inspirerad av en strömningsalgoritm med 0.25-approximation för maximering av en monoton submodulär funktion som är föremål för en matroidbegränsning, gav Udwani en algoritm med 0.25-approximation för maximering av submodulära ordningsfunktioner som är föremål för en matroidbegränsning. Baserat på ovanstående resultat skulle vi vilja utforska ytterligare i vilka fall det är möjligt att generalisera från algoritmer för strömning av submodulära maximeringsfunktioner till algoritmer för maximering av submodulära orderfunktioner. Som en mer allmän begränsning än matroid är p-matchoid en samling av p matroider där varje matroid definieras på vissa delmängder av grundmängden. Relaterade arbeten gav en strömmingsalgoritm med 1/4p-tillnärmning för monoton submodulär funktionsmaximering under en p-matchoid-begränsning. Inspirerade av ovanstående algoritmer och det spännande sambandet använde vi vissa tekniker för att försöka generalisera flera strömningsalgoritmer för submodulära funktioner till offline-algoritmer för submodulära ordningsfunktioner, inklusive interleaved partitions och inkrementella värden. Under förutsättning att målfunktionen f är subadditiv och icke-negativ gav vi en algoritm för 1/4p-tillnärmning för monoton submodulär ordermaximering till ett p-matchoid-begränsningsvillkor. Dessutom sammanfattar vi misslyckanden i andra fall.