The coalition structure formation problem represents an active research area in multi-agent systems. A coalition structure is defined as a partition of the agents involved in a system into disjoint coalitions. The problem of finding the optimal coalition structure is NP-complete. In order to find the optimal solution in a combinatorial optimization problem it is theoretically possible to enumerate the solutions and evaluate each. But this approach is infeasible since the number of solutions often grows exponentially with the size of the problem. In this paper we present a greedy adaptive search procedure (GRASP) to efficiently search the space of coalition structures in order to find an optimal one. Experiments and comparisons to other algorithms prove the validity of the proposed method in solving this hard combinatorial problem.
A deterministic algorithm must systematically explore the search space of candidate solutions. One of the first algorithm returning an optimal solution is the dynamic programming algorithm (DP) proposed in @cite_4 for the set partitioning problem.This algorithm is polynomial in the size of the input ( @math ) and it runs in @math time, which is significantly less than an exhaustive enumeration ( @math ). However, DP is not an anytime algorithm, and has a large memory requirement. Indeed, for each coalition @math it computes @math and @math . It computes all the possible splits of the coalition @math and assign to @math the best split and to @math its value. @cite_2 the authors proposed an improved version of the DP algorithm (IDP) performing fewer operations and requiring less memory than DP. IDP, as shown by the authors, is considered one of the fastest available exact algorithm in the literature computing an optimal solution.
@cite_4 @cite_2
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We present a deterministic (1+o(1))-approximation O(n1 2+o(1)+D1+o(1))-time algorithm for solving the single-source shortest paths problem on distributed weighted networks (the CONGEST model); here n is the number of nodes in the network and D is its (hop) diameter. This is the first non-trivial deterministic algorithm for this problem. It also improves (i) the running time of the randomized (1+o(1))-approximation O(n1 2D1 4+D)-time algorithm of Nanongkai [STOC 2014] by a factor of as large as n1 8, and (ii) the O(є−1logє−1)-approximation factor of Lenzen and Patt-Shamir’s O(n1 2+є+D)-time algorithm [STOC 2013] within the same running time. Our running time matches the known time lower bound of Ω(n1 2 logn + D) [Das STOC 2011] modulo some lower-order terms, thus essentially settling the status of this problem which was raised at least a decade ago [Elkin SIGACT News 2004]. It also implies a (2+o(1))-approximation O(n1 2+o(1)+D1+o(1))-time algorithm for approximating a network’s weighted diameter which almost matches the lower bound by [PODC 2012]. In achieving this result, we develop two techniques which might be of independent interest and useful in other settings: (i) a deterministic process that replaces the “hitting set argument” commonly used for shortest paths computation in various settings, and (ii) a simple, deterministic, construction of an (no(1), o(1))-hop set of size O(n1+o(1)). We combine these techniques with many distributed algorithmic techniques, some of which from problems that are not directly related to shortest paths, e.g. ruling sets [ STOC 1987], source detection [Lenzen, Peleg PODC 2013], and partial distance estimation [Lenzen, Patt-Shamir PODC 2015]. Our hop set construction also leads to single-source shortest paths algorithms in two other settings: (i) a (1+o(1))-approximation O(no(1))-time algorithm on congested cliques, and (ii) a (1+o(1))-approximation O(no(1)logW)-pass O(n1+o(1)logW)-space streaming algorithm, when edge weights are in 1, 2, …, W . The first result answers an open problem in [Nanongkai, STOC 2014]. The second result partially answers an open problem raised by McGregor in 2006 [ sublinear.info , Problem 14]. Forming effective coalitions is a major research challenge in the field of multi-agent systems. Central to this endeavour is the problem of partitioning the set of agents into exhaustive and disjoint coalitions such that the social welfare is maximized. This coalition structure generation problem is extremely challenging due to the exponential number of partitions that need to be examined. Specifically, given n agents, there are O(nn) possible partitions. To date, the only algorithm that can find an optimal solution in O(3n) is the Dynamic Programming (DP) algorithm, due to However, one of the main limitations of DP is that it requires a significant amount of memory. In this paper, we devise an Improved Dynamic Programming algorithm (IDP) that is proved to perform fewer operations than DP (e.g. 38.7 of the operations given 25 agents), and is shown to use only 33.3 of the memory in the best case, and 66.6 in the worst. We present a new Dynamic Programming (DP) formulation of the Coalition Structure Generation (CSG) problem based on imposing a hierarchical organizational structure over the agents. We show the efficiency of this formulation by deriving DyPE, a new optimal DP algorithm which significantly outperforms current DP approaches in speed and memory usage. In the classic case, in which all coalitions are feasible, DyPE has half the memory requirements of other DP approaches. On graph-restricted CSG, in which feasibility is restricted by a (synergy) graph, DyPE has either the same or lower computational complexity depending on the underlying graph structure of the problem. Our empirical evaluation shows that DyPE outperforms the state-of-the-art DP approaches by several orders of magnitude in a large range of graph structures (e.g. for certain scalefree graphs DyPE reduces the memory requirements by @math and solves problems that previously needed hours in minutes). We consider dynamic evaluation of algebraic functions such as computing determinant, matrix adjoint, matrix inverse and solving linear system of equations. We show that in the dynamic setup the above problems can be solved faster than evaluating everything from scratch. In the case when rows and columns of the matrix can change we show an algorithm that achieves O(n sup 2 ) arithmetic operations per update and O(1) arithmetic operations per query. Next, we describe two algorithms, with different tradeoffs, for updating the inverse and determinant when single entries of the matrix are changed. The fastest update for the first tradeoff is O(n sup 1.575 ) arithmetic operations per update and O(n sup 0.575 ) arithmetic operations per query. The second tradeoff gives O(n sup 1.495 ) arithmetic operations per update and O(n sup 1.495 ) arithmetic operations per query. We also consider the case when some number of columns or rows can change. We use dynamic determinant computations to solve the following problems in the dynamic setup: computing the number of spanning trees in a graph and testing if an edge in a graph is contained in some perfect matching. These are the first dynamic algorithms for these problems. Next, with the use of dynamic matrix inverse, we solve fully dynamic transitive closure in general directed graphs. The bounds on arithmetic operations for dynamic matrix inverse translate directly to time bounds for dynamic transitive closure. Thus we obtain the first known algorithm with O(n sup 2 ) worst-case update time and constant query time and two algorithms for transitive closure in general digraphs with subquadratic update and query times. Our algorithms for transitive closure are randomized with one-sided error. We also consider for the first time the case when the edges incident with a part of vertices of the graph can be changed.