Convergence and approximation in potential games

We initiate the study of the computational complexity of the covering radius problem for point lattices, and approximation versions of the problem for both lattices and linear codes. We also investigate the computational complexity of the shortest linearly independent vectors problem, and its relation to the covering radius problem for lattices. For the covering radius on n-dimensional lattices, we show that the problem can be approximated within any constant factor /spl gamma/(n) > 1 in random exponential time 2/sup O(n)/, it is in AM for /spl gamma/(n) = 2, in coAM for /spl gamma/(n) = /spl radic/(n log n), and in NP /spl cap/ coNP for /spl gamma/(n) = /spl radic/n. For the covering radius on n-dimensional linear codes, we show that the problem can be solved in deterministic polynomial time for approximation factor /spl gamma/(n) = log n, but cannot be solved in polynomial time for some /spl gamma/(n) = /spl Omega/(log log n) unless NP can be simulated in deterministic n/sup O(log log log n)/ time. Moreover, we prove that the problem is NP-hard for every constant approximation factor, it is /spl Pi//sub 2/-hard for some constant approximation factor, and it is in AM for approximation factor 2. So, it is unlikely to be /spl Pi//sub 2/-hard for approximation factors larger than 2. This is a natural hardness of approximation result in the polynomial hierarchy. For the shortest independent vectors problem, we give a coAM protocol achieving approximation factor /spl gamma/(n) = /spl radic/(n/log n), solving an open problem of Blomer and Seifert (1999), and prove that the problem is also in coNP for /spl gamma/(n) = /spl radic/n. Both results are obtained by giving a gap-preserving nondeterministic polynomial time reduction to the closest vector problem.

We prove that the distance-r dominating set, distance-r connected dominating set, distance-r vertex cover, and distance-r connected vertex cover problems admit constant factor approximations in the CONGEST model of distributed computing in a constant number of rounds on classes of sparse high-girth graphs. In this paper, sparse means bounded expansion, and high-girth means girth at least 4r+2. Our algorithm is quite simple; however, the proof of its approximation guarantee is non-trivial. To complement the algorithmic results, we show tightness of our approximation by providing a loosely matching lower bound on rings.Our result is the first to show the existence of constant-factor approximations in a constant number of rounds in non-trivial classes of graphs for distance-r covering problems.

Minimizing the total length has been one of the main optimality criteria for triangulations and other kinds of partitions. Indeed, the minimum weight triangulation or MWT (i.e. a triangulation of minimum total edge length) has frequently been referred to as the optimal This triangulation has some good properties [1] and is e.g. useful in numerical approximation of bivariate data [23]. Its complexity is one of the most intriguing problems in computational geometry. No proof has yet been presented showing that it is NP-hard. On the other hand, no polynomial time heuristic for it was shown to guarantee even a constant approximation factor. However, very recently we proved that what we called the quasi-greedy triangulation has length within a constant factor from the minimum [13]. A standard implementation of that heuristic using known results would yield a quadratic time algorithm. In this paper, we improve on this by describing an O(n log n) algorithm for quasi-greedy triangulation. Moreover, if the Delaunay triangulation is given, our algorithm runs in linear time. The minimum weight convex partition (MWCP) problem has been shown to be NP-complete [7, 17]. By a MWCP we mean a set of non-crossing diagonals (i.e. without Steiner points) of minimum total length which includes the convex hull and partitions it into empty convex polygons. In [13] it was shown that, given the quasi-greedy triangulation, a convex partition of length within a constant factor from the MWCP can be computed in linear time. Thus from the results in this paper we also obtain a fast heuristic for approximating the MWCP. The quasi-greedy triangulation is very closely related to the greedy triangulation (both are defined in the next section). In fact, we will describe an algorithm that, given the Delannay triangulation, computes the greedy triangulation in linear time. This algorithm behaves in such a way that it is very easily adapted to yield an algorithm for the quasi-greedy triangulation. Consequently, we also present the first O(n log n) algorithm for greedy triangulation. (By using a substantially different approach, an attempt to develop an O(n log n) algorithm for greedy triangulation was done in [21]. Even if that approach could be corrected

Coevolutionary algorithms are a family of black-box optimisation algorithms with many applications in game theory. We study a coevolutionary algorithm on an important class of games in game theory: potential games. In these games, a real-valued function defined over the entire strategy space encapsulates the strategic choices of all players collectively. We present the first theoretical analysis of a coevolutionary algorithm on potential games, showing a runtime guarantee that holds for all exact potential games, some weighted and ordinal potential games, and certain non-potential games. Using this result, we show a polynomial runtime on singleton congestion games. Furthermore, we show that there exist games for which coevolutionary algorithms find Nash equilibria exponentially faster than best or better response dynamics, and games for which coevolutionary algorithms find better Nash equilibria as well. Finally, we conduct experimental evaluations showing that our algorithm can outperform widely used algorithms, such as better response on random instances of singleton congestion games, as well as fictitious play, counterfactual regret minimisation (CFR), and external sampling CFR on dynamic routing games.

The framework of budget-feasible mechanism design studies procurement auctions where the auctioneer (buyer) aims to maximize his valuation function subject to a hard budget constraint. We study the problem of designing truthful mechanisms that have good approximation guarantees and never pay the participating agents (sellers) more than the budget. We focus on the case of general (non-monotone) submodular valuation functions and derive the first truthful, budget-feasible, and O(1)-approximation mechanisms that run in polynomial time in the value query model, for both offline and online auctions. Prior to our work, the only O(1)-approximation mechanism known for non-monotone submodular objectives required an exponential number of value queries. At the heart of our approach lies a novel greedy algorithm for non-monotone submodular maximization under a knapsack constraint. Our algorithm builds two candidate solutions simultaneously (to achieve a good approximation), yet ensures that agents cannot jump from one solution to the other (to implicitly enforce truthfulness). The fact that in our mechanism the agents are not ordered according to their marginal value per cost allows us to appropriately adapt these ideas to the online setting as well. To further illustrate the applicability of our approach, we also consider the case where additional feasibility constraints are present, for example, at most k agents can be selected. We obtain O(p)-approximation mechanisms for both monotone and non-monotone submodular objectives, when the feasible solutions are independent sets of a p-system. With the exception of additive valuation functions, no mechanisms were known for this setting prior to our work. Finally, we provide lower bounds suggesting that, when one cares about nontrivial approximation guarantees in polynomial time, our results are, asymptotically, the best possible.

<p>In combinatorial optimization, covering problems are those problems where given a ground set and a family of subsets of the ground set, the objective is to find a minimum cost set of subsets whose union contains the ground set. We consider covering problems in the context of Computational Geometry, which is a subfield of Computer Science that deals with problems associated with geometric objects such as points, lines, disks, polygons etc. In particular, geometric covering is an active research area, where the ground set and the family of subsets are induced by geometric objects. Covering problems in combinatorial optimizations often have a geometric analogue that arises naturally, and though such problems remain NP-hard, it is often possible to exploit the geometric properties of the set system to obtain better approximation algorithms.</p> <p>In this work, the fundamental problem that we consider is a generalization of geometric covering, where each element in the ground set may need to covered by more than one subset. To be precise, the problem is defined as follows: given two sets of points X, Y and a coverage function κ : X → Z+ ∪ {0}, construct balls centered on the points in Y such that each point in X is covered by at least κ(x) distinct balls. The objective in this problem is to minimize the total cost, which is a function of the radii of the balls. This problem is termed as the metric multi-cover (MMC) problem.</p> <p>We first consider version of the MMC problem where κ(x) = 1 for all clients, i.e. the 1-covering case. The known results that give a constant factor approximation for this problem are variations of LP-based primal-dual algorithm. We use a modified local search technique, motivated by geometric idea, to derive a simple, constant- factor approximation for this problem in Chapter 2.</p> <p>We then look at the MMC problem where the point sets X,Y are in the Euclidean plane, and each client x ∈ X needs to be covered by at least κ(x) distinct disks centered on the points in Y . In Chapter 4, we give the first polynomial time constant factor approximation for this problem, in which the constant is independent of the coverage function κ. Our solution also has an incremental property, which allows the algorithm to handle increases in the coverage requirement by increasing the radii of the current server disks, without affecting the approximation factor.</p> <p>In the next problem, we move from the Euclidean plane to arbitrary metric spaces where we consider the uniform MMC problem. In this problem, each client x has the demand κ(x) = k, where k > 0 is an integer. We give the first constant factor approximation (independent of k) for this problem. The key contribution that led to this result is the formulation of a partitioning scheme of the servers in the uniform MMC problem, that reduces the uniform MMC problem to k instances of 1-covering problem, while preserving the optimality of the solution to a constant multiplicative factor. We present the partitioning scheme as an independent result in Chapter 5, which we then use to solve the uniform MMC problem in Chapter 6.</p> <p>The MMC problem with arbitrary coverage function κ is then considered in Chapter 7. The key challenge that the non-uniform version presents is that the symmetry of the server partitioning scheme breaks down as the coverage demands of clients are independent of each other. We present a constant factor algorithm for this problem in Chapter 7.</p> <p>The last problem that we consider is the t-MMC problem, which is a restricted version of the uniform MMC problem. The objective is to compute a cover in which each client is covered by at least k distinct server disks, using atmost t server disks in total. This problem is a generalization of the clustering problem (where k = 1), and to our knowledge this is the first time this generalization has been considered. We give a constant factor approximation for this problem in Chapter 8.</p>

Using structural equation modeling (SEM), this study investigates the factors influencing consumer purchase intention toward smart home products, with a particular focus on the role of high-performance electronic component optimization in product strategy. Anchored in the stimuli-organism-response theoretical framework, the study constructs a multi-dimensional analytical model incorporating six key variables: Consumer demographics, product quality, economic status, social environment, emotional value and social value. Data were collected through structured questionnaires and analyzed using AMOS 24.0 and SPSS 27.0. The results indicate that all six factors significantly affect purchase intention, with product quality, economic status and social environment exerting the strongest influence. Further analysis reveals that social environment, emotional value and social value function as important mediating variables. Notably, high pricing and privacy concerns are identified as major barriers to adoption, while affordability and the integration of intelligent, responsive functionalities—enabled by optimized electronic components—are seen as critical areas for improvement. This study offers practical insights for smart home manufacturers aiming to align technical innovation with consumer expectations through targeted product enhancement.

The paper studies properties of best-response (BR) dynamics in potential games with continuous action sets. It is known that BR dynamics converge to the set of Nash equilibria (NE) in potential games. The set of NE in potential games is composed of local maximizers and saddle points of the potential function. The paper studies non-convergence of BR dynamics to saddle points of the potential function. Under relatively mild assumptions it is shown that BR dynamics may only converge to an interior saddle-point from a measure-zero set of initial conditions. This provides a weak stable manifold theorem in this context.

We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds Bv on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a (2, 2Bv+3)-approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most 2Bv +3. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly k-Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain’s iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost λ-edge-connected subgraph with at least k vertices, which we call the (k, λ)-subgraph problem. This generalizes some well-studied classical problems such as the k-MST and the minimum cost λ-edgeconnected subgraph problems. We give a polylogarithmic approximation for the (k, 2)-subgraph problem. However, by relating it to the Densest k-Subgraph problem, we provide evidence that the (k, λ)-subgraph problem might be hard to approximate for arbitrary λ.

This research examines Sasando as a symbol of cultural identity and pride of the people of Rote Island by exploring its philosophical values, social functions, and the challenges of its preservation in the midst of modernization. The present study employed a qualitative case study approach, with data collected through participatory observation and in-depth interviews with Sasando players, craftsmen, and cultural figures. The research findings reveal three key aspects: the value of cultural identity reflected in the use of natural materials (bamboo and palm) and its role in traditional rituals as a marker of local wisdom, social function as a medium of community unification in collaborative events and a means of intergenerational value transmission, and preservation challenges in the form of declining interest from the younger generation, lack of integration in formal education, and limited promotion. The research also identified opportunities through cultural festivals and tourism-based economic potential. The research conclusions emphasize that Sasando is not merely a musical instrument, but rather a symbol of collective identity that represents the history, social values, and cultural resilience of the Rote people. The following policy recommendations are put forth: first, the integration of a Sasando-based curriculum; second, the strengthening of youth training programs; and third, multi-stakeholder collaboration for sustainable promotion strategies. This research makes a theoretical contribution to the discussion of the relationship between traditional musical instruments, identity construction, and cultural preservation strategies in the midst of globalization.

We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem. Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. Our techniques build upon the constant-factor approximation algorithm for the special case of node-weighted metrics. Specifically, we give a generic reduction to structured instances that resemble but are more general than those arising from node-weighted metrics. For those instances, we then solve Local-Connectivity ATSP, a problem known to be equivalent (in terms of constant-factor approximation) to the asymmetric traveling salesman problem.

In this paper we consider the capacitated vertex cover problem, which is the variant of vertex cover where each node is allowed to cover a limited number of edges. We present an efficient, deterministic, distributed approximation algorithm for the problem. Our algorithm computes a $(2+\epsilon)$-approximate solution which violates the capacity constraints by a factor of $(4+\epsilon)$ in a polylogarithmic number of communication rounds. On the other hand, we also show that every efficient distributed approximation algorithm for this problem must violate the capacity constraints. Our result is achieved in two steps. We first develop a 2-approximate, sequential primal-dual algorithm that violates the capacity constraints by a factor of 2. Subsequently, we present a distributed version of this algorithm. We demonstrate that the sequential algorithm has an inherent need for synchronization which forces any naive distributed implementation to use a linear number of communication rounds. The challenge in this step is therefore to achieve a reduction of the communication complexity to a polylogarithmic number of rounds without worsening the approximation guarantee.

How to efficiently and reliably spread information in a system is one of the most fundamental problems in distributed computing. Recently, inspired by biological scenarios, several works focused on identifying the minimal communication resources necessary to spread information under faulty conditions. Here we study the self-stabilizing bit-dissemination problem, introduced by Boczkowski, Korman, and Natale in [SODA 2017]. The problem considers a fully-connected network of nagents, with a binary world of opinions, one of which is called correct. At any given time, each agent holds an opinion bit as its public output. The population contains a source agent which knows which opinion is correct. This agent adopts the correct opinion and remains with it throughout the execution. We consider the basic PULL\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {PULL}$$\\end{document} model of communication, in which each agent observes relatively few randomly chosen agents in each round. The goal of the non-source agents is to quickly converge on the correct opinion, despite having an arbitrary initial configuration, i.e., in a self-stabilizing manner. Once the population converges on the correct opinion, it should remain with it forever. Motivated by biological scenarios in which animals observe and react to the behavior of others, we focus on the extremely constrained model of passive communication, which assumes that when observing another agent the only information that can be extracted is the opinion bit of that agent. We prove that this problem can be solved in a poly-logarithmic in n number of rounds with high probability, while sampling a logarithmic number of agents at each round. Previous works solved this problem faster and using fewer samples, but they did that by decoupling the messages sent by agents from their output opinion, and hence do not fit the framework of passive communication. Moreover, these works use complex recursive algorithms with refined clocks that are unlikely to be used by biological entities. In contrast, our proposed algorithm has a natural appeal as it is based on letting agents estimate the current tendency direction of the dynamics, and then adapt to the emerging trend.

45 A THEORETICAL MODEL FOR DIAGNOSTIC TEST ASSESSMENT AND ITS APPLICATION TO PROSTATE-SPECIFIC ANTIGEN TEST USE AMONG MEDICARE BENEFICIARIES

Submodular optimization generalizes many classic problems in combinatorial\noptimization and has recently found a wide range of applications in machine\nlearning (e.g., feature engineering and active learning). For many large-scale\noptimization problems, we are often concerned with the adaptivity complexity of\nan algorithm, which quantifies the number of sequential rounds where\npolynomially-many independent function evaluations can be executed in parallel.\nWhile low adaptivity is ideal, it is not sufficient for a distributed algorithm\nto be efficient, since in many practical applications of submodular\noptimization the number of function evaluations becomes prohibitively\nexpensive. Motivated by these applications, we study the adaptivity and query\ncomplexity of adaptive submodular optimization.\n Our main result is a distributed algorithm for maximizing a monotone\nsubmodular function with cardinality constraint $k$ that achieves a\n$(1-1/e-\\varepsilon)$-approximation in expectation. This algorithm runs in\n$O(\\log(n))$ adaptive rounds and makes $O(n)$ calls to the function evaluation\noracle in expectation. The approximation guarantee and query complexity are\noptimal, and the adaptivity is nearly optimal. Moreover, the number of queries\nis substantially less than in previous works. Last, we extend our results to\nthe submodular cover problem to demonstrate the generality of our algorithm and\ntechniques.\n