On the approximability and hardness of minimum topic connected overlay and its special instances

Abstract

In the context of designing a scalable overlay network to support decentralized topic-based pub/sub communication, the Minimum Topic-Connected Overlay problem (Min-TCO in short) has been investigated: Given a set of t topics and a collection of n users together with the lists of topics they are interested in, the aim is to connect these users to a network by a minimum number of edges such that every graph induced by users interested in a common topic is connected. It is known that Min-TCO is N P-hard and approximable within O(log t) in polynomial time.In this paper, we further investigate the problem and some of its special instances. We give various hardness results for instances where the number of topics in which an user is interested in is bounded by a constant, and also for the instances where the number of users interested in a common topic is constant. For the latter case, we present a first constant approximation algorithm. We also present some polynomial-time algorithms for very restricted instances of Min-TCO.

Constraint Generation Algorithm for the Minimum Connectivity Inference Problem

“…Minimum Connectivity Inference (MCI) Input: a hypergraph H = (V, E) Output: a graph G = (V, E) such that G[S] is connected ∀S ∈ E Goal: minimize |E(G)| This optimization problem is NP-hard [11], and was first introduced for the design of vacuum systems [12]. It has then be studied independently in several different contexts, mainly dealing with network design: computer networks [13], social networks [3] (more precisely modeling the publish/subscribe communication paradigm [7,15,19]), but also other fields, such as auction systems [8] and structural biology [1,2]. Finally, we can mention the issue of hypergraph drawing, where, in addition to the connectivity constraints, one usually looks for graphs with additional properties (e.g.…”

Polynomial-Time Data Reduction for the Subset Interconnection Design Problem

“…Vertices are drawn as white circles, hyperedges by grouping their incident vertices together inside a closed curve filled semi-transparently, and solution edges are drawn as thick lines.Subset Interconnection Design is a fundamental problem concerning hypergraph and graph structures that has many applications. Indeed, Subset Interconnection Design has been studied in the context of designing vacuum systems [10,11], scalable overlay networks [6,18,26], reconfigurable interconnection networks [13,14], and, in variants, in the context of inferring a most likely social network [1], determining winners of combinatorial auctions [7] as well as drawing hypergraphs [3,19,21,22]. The respective research communities seem largely unaware of each other's work, for instance leading to multiple NP-hardness proofs.…”

“…To the best of our knowledge, the problem was first defined by Du [9] and the first NPhardness proof was presented by Du and Miller [11]. SID has been independently studied under the names Minimum Topic-Connected Overlay by the "scalable overlay networks community" [6,18,26], Subset Interconnection Design by the "vacuum systems community" [10,12], and Interconnection Graph Problem by the "reconfigurable interconnection systems community" [13,14]. The term "topic-connected" in Minimum Topic-Connected Overlay refers to the desired property of overlay networks that agents interested in some particular topic should be able to inform each other about updates concerning this topic without involving other agents [26].…”

“…polynomial-time solvability [18] because then the input hypergraph is itself an optimal solution. If the hypergraph is closed under intersections, that is, every intersection of hyperedges is also a hyperedge, then SID also becomes polynomial-time solvable [4, Lemma 1].…”

“…If the hypergraph is closed under intersections, that is, every intersection of hyperedges is also a hyperedge, then SID also becomes polynomial-time solvable [4, Lemma 1]. Polynomial-time approximability has also been intensely studied, providing various logarithmic-factor approximation algorithms [1,6,18] and inapproximability results (implying that logarithmic-factor approximation algorithms are likely to be optimal) [1,18]. The currently best exact algorithm for SID has a running time of O(n 2k /4 k + n 2 ) [18].…”