32 research outputs found
Consensus for nonlinear monotone networks with unilateral interactions
This paper deals with an extended framework of the distributed asymptotic agreement problem by allowing the presence of unilateral interactions (optimistic or pessimistic) in place of bilateral ones, for a large class of nonlinear monotone time-varying networks. In this original setup we firstly introduce notions of unilateral optimistic and/or pessimistic interaction, of associated bicolored edge in the interaction graph and a suitable graph-theoretical connectedness property. Secondly, we formulate a new assumption of integral connectivity and show that it is sufficient to guarantee exponential convergence towards the agreement subspace. Finally, we remark that the proposed conditions are also necessary for consensuability. Theoretical advances are emphasized through illustrative examples given both to support the discussion and to highlight how the proposed framework extends all existing conditions for consensus of monotone networks
Just Renormalizable TGFT's on U(1)^d with Gauge Invariance
We study the polynomial Abelian or U(1)^d Tensorial Group Field Theories
equipped with a gauge invariance condition in any dimension d. From our
analysis, we prove the just renormalizability at all orders of perturbation of
the phi^4_6 and phi^6_5 random tensor models. We also deduce that the phi^4_5
tensor model is super-renormalizable.Comment: 33 pages, 22 figures. One added paragraph on the different notions of
connectedness, preciser formulation of the proof of the power counting
theorem, more general statements about traciality of tensor graph
Assessing the Computational Complexity of Multi-Layer Subgraph Detection
Multi-layer graphs consist of several graphs (layers) over the same vertex
set. They are motivated by real-world problems where entities (vertices) are
associated via multiple types of relationships (edges in different layers). We
chart the border of computational (in)tractability for the class of subgraph
detection problems on multi-layer graphs, including fundamental problems such
as maximum matching, finding certain clique relaxations (motivated by community
detection), or path problems. Mostly encountering hardness results, sometimes
even for two or three layers, we can also spot some islands of tractability
Simultaneous Feedback Vertex Set: A Parameterized Perspective
Given a family of graphs , a graph , and a positive integer
, the -Deletion problem asks whether we can delete at most
vertices from to obtain a graph in . -Deletion
generalizes many classical graph problems such as Vertex Cover, Feedback Vertex
Set, and Odd Cycle Transversal. A graph ,
where the edge set of is partitioned into color classes, is called
an -edge-colored graph. A natural extension of the
-Deletion problem to edge-colored graphs is the
-Simultaneous -Deletion problem. In the latter problem, we
are given an -edge-colored graph and the goal is to find a set
of at most vertices such that each graph , where and , is in . In this work, we
study -Simultaneous -Deletion for being the
family of forests. In other words, we focus on the -Simultaneous
Feedback Vertex Set (-SimFVS) problem. Algorithmically, we show that,
like its classical counterpart, -SimFVS parameterized by is
fixed-parameter tractable (FPT) and admits a polynomial kernel, for any fixed
constant . In particular, we give an algorithm running in time and a kernel with vertices. The
running time of our algorithm implies that -SimFVS is FPT even when
. We complement this positive result by showing that for
, where is the number of vertices in the input graph,
-SimFVS becomes W[1]-hard. Our positive results answer one of the open
problems posed by Cai and Ye (MFCS 2014)