Correlation Clustering Based Coalition Formation For Multi-Robot Task Allocation

In this paper, we study the multi-robot task allocation problem where a group of robots needs to be allocated to a set of tasks so that the tasks can be finished optimally. One task may need more than one robot to finish it. Therefore the robots need to form coalitions to complete these tasks. Multi-robot coalition formation for task allocation is a well-known NP-hard problem. To solve this problem, we use a linear-programming based graph partitioning approach along with a region growing strategy which allocates (near) optimal robot coalitions to tasks in a negligible amount of time. Our proposed algorithm is fast (only taking 230 secs. for 100 robots and 10 tasks) and it also finds a near-optimal solution (up to 97.66% of the optimal). We have empirically demonstrated that the proposed approach in this paper always finds a solution which is closer (up to 9.1 times) to the optimal solution than a theoretical worst-case bound proved in an earlier work

Clustering Partially Observed Graphs via Convex Optimization

This paper considers the problem of clustering a partially observed unweighted graph---i.e., one where for some node pairs we know there is an edge between them, for some others we know there is no edge, and for the remaining we do not know whether or not there is an edge. We want to organize the nodes into disjoint clusters so that there is relatively dense (observed) connectivity within clusters, and sparse across clusters. We take a novel yet natural approach to this problem, by focusing on finding the clustering that minimizes the number of "disagreements"---i.e., the sum of the number of (observed) missing edges within clusters, and (observed) present edges across clusters. Our algorithm uses convex optimization; its basis is a reduction of disagreement minimization to the problem of recovering an (unknown) low-rank matrix and an (unknown) sparse matrix from their partially observed sum. We evaluate the performance of our algorithm on the classical Planted Partition/Stochastic Block Model. Our main theorem provides sufficient conditions for the success of our algorithm as a function of the minimum cluster size, edge density and observation probability; in particular, the results characterize the tradeoff between the observation probability and the edge density gap. When there are a constant number of clusters of equal size, our results are optimal up to logarithmic factors.Comment: This is the final version published in Journal of Machine Learning Research (JMLR). Partial results appeared in International Conference on Machine Learning (ICML) 201

Unifying Sparsest Cut, Cluster Deletion, and Modularity Clustering Objectives with Correlation Clustering

Graph clustering, or community detection, is the task of identifying groups of closely related objects in a large network. In this paper we introduce a new community-detection framework called LambdaCC that is based on a specially weighted version of correlation clustering. A key component in our methodology is a clustering resolution parameter, $\lambda$, which implicitly controls the size and structure of clusters formed by our framework. We show that, by increasing this parameter, our objective effectively interpolates between two different strategies in graph clustering: finding a sparse cut and forming dense subgraphs. Our methodology unifies and generalizes a number of other important clustering quality functions including modularity, sparsest cut, and cluster deletion, and places them all within the context of an optimization problem that has been well studied from the perspective of approximation algorithms. Our approach is particularly relevant in the regime of finding dense clusters, as it leads to a 2-approximation for the cluster deletion problem. We use our approach to cluster several graphs, including large collaboration networks and social networks

Image Segmentation by Edge Partitioning over a Nonsubmodular Markov Random Field

Edge weight-based segmentation methods, such as normalized cut or minimum cut, require a partition number specification for their energy formulation. The number of partitions plays an important role in the segmentation overall quality. However, finding a suitable partition number is a nontrivial problem, and the numbers are ordinarily manually assigned. This is an aspect of the general partition problem, where finding the partition number is an important and difficult issue. In this paper, the edge weights instead of the pixels are partitioned to segment the images. By partitioning the edge weights into two disjoints sets, that is, cut and connect, an image can be partitioned into all possible disjointed segments. The proposed energy function is independent of the number of segments. The energy is minimized by iterating the QPBO-α-expansion algorithm over the pairwise Markov random field and the mean estimation of the cut and connected edges. Experiments using the Berkeley database show that the proposed segmentation method can obtain equivalently accurate segmentation results without designating the segmentation numbers

Towards an Entropy-based Analysis of Log Variability

Rules, decisions, and workflows are intertwined components depicting the overall process. So far imperative workflow modelling languages have played the major role for the description and analysis of business processes. Despite their undoubted efficacy in representing sequential executions, they hide circumstantial information leading to the enactment of activities, and obscure the rationale behind the verification of requirements, dependencies, and goals. This workshop aimed at providing a platform for the discussion and introduction of new ideas related to the development of a holistic approach that encompasses all those aspects. The objective was to extend the reach of the business process management audience towards the decisions and rules community and increase the integration between different imperative, declarative and hybrid modelling perspectives. Out of the high-quality submitted manuscripts, three papers were accepted for publication, with an acceptance rate of 50%. They contributed to foster a fruitful discussion among the participants about the respective impact and the interplay of decision perspective and the process perspective

Overview of the Relational Analysis approach in Data-Mining and Multi-criteria Decision Making

International audienceIn this chapter we introduce a general framework called the Relational Analysis approach and its related contributions and applications in the fields of data analysis, data mining and multi-criteria decision making. This approach was initiated by J.F. Marcotorchino and P. Michaud at the end of the 70's and has generated many research activities. However, the aspects of this framework that we would like to focus on are of a theoretical kind. Indeed, we are aimed at recalling the background and the basics of this framework, the unifying results and the modeling contributions that it has allowed to achieve. Besides, the main tasks that we are interested in are the ranking aggregation problem, the clustering problem and the block seriation problem. Those problems are combinatorial ones and the computational considerations of such tasks in the context of the RA methodology will not be covered. However, among the list of references that we give thoughout this chapter, there are numerous articles that the interested reader could consult to this end

Simple Games on Networks

In the same way that traditional game theory captured the minds of economists and allowed complex problems to be studied using simple models, so have games on networks come to be used in computer science. In particular, previous work has focused on extending a two-player base game M over a network G by having each vertex in G chose a strategy from the base game and play it simultaneously against all adjacent nodes. Similarly, the utility for each vertex becomes the sum of the utility of that node in each of the games it plays against its neighbors. The resulting networked game is called M + G. This paper seeks to augment the body of existing work by studying a few similar networked games and finding key characteristics like the existence of Nash equilibria, the price of anarchy, the price of stability, the convergence speed of best response dynamic, and the difficulty of finding the optimal solution. We also reproduce the result that the class of exact potential games is isomorphic to the class of congestion games with a proof that is drastically more readable than the original. Co-authored by Tom Wexler

Valued Constraint Satisfaction Problems over Infinite Domains

The object of the thesis is the computational complexity of certain combinatorial optimisation problems called \emph{valued constraint satisfaction problems}, or \emph{VCSPs} for short. The requirements and optimisation criteria of these problems are expressed by sums of \emph{(valued) constraints} (also called \emph{cost functions}). More precisely, the input of a VCSP consists of a finite set of variables, a finite set of cost functions that depend on these variables, and a cost $u$; the task is to find values for the variables such that the sum of the cost functions is at most $u$. By restricting the set of possible cost functions in the input, a great variety of computational optimisation problems can be modelled as VCSPs. Recently, the computational complexity of all VCSPs for finite sets of cost functions over a finite domain has been classified. Many natural optimisation problems, however, cannot be formulated as VCSPs over a finite domain. We initiate the systematic investigation of infinite-domain VCSPs by studying the complexity of VCSPs for piecewise linear (PL) and piecewise linear homogeneous (PLH) cost functions. The VCSP for a finite set of PLH cost functions can be solved in polynomial time if the cost functions are improved by fully symmetric fractional operations of all arities. We show this by (polynomial-time many-one) reducing the problem to a finite-domain VCSP which can be solved using a linear programming relaxation. We apply this result to show the polynomial-time tractability of VCSPs for {\it submodular} PLH cost functions, for {\it convex} PLH cost functions, and for {\it componentwise increasing} PLH cost functions; in fact, we show that submodular PLH functions and componentwise increasing PLH functions form maximally tractable classes of PLH cost functions. We define the notion of {\it expressive power} for sets of cost functions over arbitrary domains, and discuss the relation between the expressive power and the set of fractional operations improving the same set of cost functions over an arbitrary countable domain. Finally, we provide a polynomial-time algorithm solving the restriction of the VCSP for {\it all} PL cost functions to a fixed number of variables