Half-Duplex Routing is NP-hard

Routing is a widespread approach to transfer information from a source node to a destination node in many deployed wireless ad-hoc networks. Today's implemented routing algorithms seek to efficiently find the path/route with the largest Full-Duplex (FD) capacity, which is given by the minimum among the point-to-point link capacities in the path. Such an approach may be suboptimal if then the nodes in the selected path are operated in Half-Duplex (HD) mode. Recently, the capacity (up to a constant gap that only depends on the number of nodes in the path) of an HD line network i.e., a path) has been shown to be equal to half of the minimum of the harmonic means of the capacities of two consecutive links in the path. This paper asks the questions of whether it is possible to design a polynomial-time algorithm that efficiently finds the path with the largest HD capacity in a relay network. This problem of finding that path is shown to be NP-hard in general. However, if the number of cycles in the network is polynomial in the number of nodes, then a polynomial-time algorithm can indeed be designed

Blocking optimal arborescences

The problem of covering minimum cost common bases of two matroids is NP-complete, even if the two matroids coincide, and the costs are all equal to 1. In this paper we show that the following special case is solvable in polynomial time: given a digraph $D=(V,A)$ with a designated root node $r\in V$ and arc-costs $c:A\to \mathbb{R}$, find a minimum cardinality subset $H$ of the arc set $A$ such that $H$ intersects every minimum $c$-cost $r$-arborescence. By an $r$-arborescence we mean a spanning arborescence of root $r$. The algorithm we give solves a weighted version as well, in which a nonnegative weight function $w:A\to \mathbb{R}_+$ (unrelated to $c$) is also given, and we want to find a subset $H$ of the arc set such that $H$ intersects every minimum $c$-cost $r$-arborescence, and $w(H)=\sum_{a\in H}w(a)$ is minimum. The running time of the algorithm is $O(n^3T(n,m))$, where $n$ and $m$ denote the number of nodes and arcs of the input digraph, and $T(n,m)$ is the time needed for a minimum $s-t$ cut computation in this digraph. A polyhedral description is not given, and seems rather challenging

Blocking unions of arborescences

Given a digraph $D=(V,A)$ and a positive integer $k$, a subset $B\subseteq A$ is called a \textbf{$k$-union-arborescence}, if it is the disjoint union of $k$ spanning arborescences. When also arc-costs $c:A\to \mathbb{R}$ are given, minimizing the cost of a $k$-union-arborescence is well-known to be tractable. In this paper we take on the following problem: what is the minimum cardinality of a set of arcs the removal of which destroys every minimum $c$-cost $k$-union-arborescence. Actually, the more general weighted problem is also considered, that is, arc weights $w:A\to \mathbb{R}_+$ (unrelated to $c$) are also given, and the goal is to find a minimum weight set of arcs the removal of which destroys every minimum $c$-cost $k$-union-arborescence. An equivalent version of this problem is where the roots of the arborescences are fixed in advance. In an earlier paper [A. Bern\'ath and Gy. Pap, \emph{Blocking optimal arborescences}, Integer Programming and Combinatorial Optimization, Springer, 2013] we solved this problem for $k=1$. This work reports on other partial results on the problem. We solve the case when both $c$ and $w$ are uniform -- that is, find a minimum size set of arcs that covers all $k$-union-arbosercences. Our algorithm runs in polynomial time for this problem. The solution uses a result of [M. B\'ar\'asz, J. Becker, and A. Frank, \emph{An algorithm for source location in directed graphs}, Oper. Res. Lett. \textbf{33} (2005)] saying that the family of so-called insolid sets (sets with the property that every proper subset has a larger in-degree) satisfies the Helly-property, and thus can be (efficiently) represented as a subtree hypergraph. We also give an algorithm for the case when only $c$ is uniform but $w$ is not. This algorithm is only polynomial if $k$ is not part of the input

Parameterized Algorithms for Directed Maximum Leaf Problems

We prove that finding a rooted subtree with at least $k$ leaves in a digraph is a fixed parameter tractable problem. A similar result holds for finding rooted spanning trees with many leaves in digraphs from a wide family $\cal L$ that includes all strong and acyclic digraphs. This settles completely an open question of Fellows and solves another one for digraphs in $\cal L$. Our algorithms are based on the following combinatorial result which can be viewed as a generalization of many results for a `spanning tree with many leaves' in the undirected case, and which is interesting on its own: If a digraph $D\in \cal L$ of order $n$ with minimum in-degree at least 3 contains a rooted spanning tree, then $D$ contains one with at least $(n/2)^{1/5}-1$ leaves

Min (A)cyclic Feedback Vertex Sets and Min Ones Monotone 3-SAT

In directed graphs, we investigate the problems of finding: 1) a minimum feedback vertex set (also called the Feedback Vertex Set problem, or MFVS), 2) a feedback vertex set inducing an acyclic graph (also called the Vertex 2-Coloring without Monochromatic Cycles problem, or Acyclic FVS) and 3) a minimum feedback vertex set inducing an acyclic graph (Acyclic MFVS). We show that these problems are strongly related to (variants of) Monotone 3-SAT and Monotone NAE 3-SAT, where monotone means that all literals are in positive form. As a consequence, we deduce several NP-completeness results on restricted versions of these problems. In particular, we define the 2-Choice version of an optimization problem to be its restriction where the optimum value is known to be either D or D+1 for some integer D, and the problem is reduced to decide which of D or D+1 is the optimum value. We show that the 2-Choice versions of MFVS, Acyclic MFVS, Min Ones Monotone 3-SAT and Min Ones Monotone NAE 3-SAT are NP-complete. The two latter problems are the variants of Monotone 3-SAT and respectively Monotone NAE 3-SAT requiring that the truth assignment minimize the number of variables set to true. Finally, we propose two classes of directed graphs for which Acyclic FVS is polynomially solvable, namely flow reducible graphs (for which MFVS is already known to be polynomially solvable) and C1P-digraphs (defined by an adjacency matrix with the Consecutive Ones Property)

Disimplicial arcs, transitive vertices, and disimplicial eliminations

In this article we deal with the problems of finding the disimplicial arcs of a digraph and recognizing some interesting graph classes defined by their existence. A diclique of a digraph is a pair $V \to W$ of sets of vertices such that $v \to w$ is an arc for every $v \in V$ and $w \in W$. An arc $v \to w$ is disimplicial when $N^-(w) \to N^+(v)$ is a diclique. We show that the problem of finding the disimplicial arcs is equivalent, in terms of time and space complexity, to that of locating the transitive vertices. As a result, an efficient algorithm to find the bisimplicial edges of bipartite graphs is obtained. Then, we develop simple algorithms to build disimplicial elimination schemes, which can be used to generate bisimplicial elimination schemes for bipartite graphs. Finally, we study two classes related to perfect disimplicial elimination digraphs, namely weakly diclique irreducible digraphs and diclique irreducible digraphs. The former class is associated to finite posets, while the latter corresponds to dedekind complete finite posets.Comment: 17 pags., 3 fig

Resilience of Complex Networks

This article determines and characterizes the minimal number of actuators needed to ensure structural controllability of a linear system under structural alterations that can severe the connection between any two states. We assume that initially the system is structurally controllable with respect to a given set of controls, and propose an efficient system-synthesis mechanism to find the minimal number of additional actuators required for resilience of the system w.r.t such structural changes. The effectiveness of this approach is demonstrated by using standard IEEE power networks

A Maximum Linear Arrangement Problem on Directed Graphs

We propose a new arrangement problem on directed graphs, Maximum Directed Linear Arrangement (MaxDLA). This is a directed variant of a similar problem for undirected graphs, in which however one seeks maximum and not minimum; this problem known as the Minimum Linear Arrangement Problem (MinLA) has been much studied in the literature. We establish a number of theorems illustrating the behavior and complexity of MaxDLA. First, we relate MaxDLA to Maximum Directed Cut (MaxDiCut) by proving that every simple digraph $D$ on $n$ vertices satisfies $\frac{n}{2}$$maxDiCut(D) \leq MaxDLA(D) \leq (n-1)MaxDiCut(D)$. Next, we prove that MaxDiCut is NP-Hard for planar digraphs (even with the added restriction of maximum degree 15); it follows from the above bounds that MaxDLA is also NP-Hard for planar digraphs. In contrast, Hadlock (1975) and Dorfman and Orlova (1972) showed that the undirected Maximum Cut problem is solvable in polynomial time on planar graphs. On the positive side, we present a polynomial-time algorithm for solving MaxDLA on orientations of trees with degree bounded by a constant, which translates to a polynomial-time algorithm for solving MinLA on the complements of those trees. This pairs with results by Goldberg and Klipker (1976), Shiloach (1979) and Chung (1984) solving MinLA in polynomial time on trees. Finally, analogues of Harper's famous isoperimetric inequality for the hypercube, in the setting of MaxDLA, are shown for tournaments, orientations of graphs with degree at most two, and transitive acyclic digraphs.Comment: 12 page

Packing and domination parameters in digraphs

Given a digraph $D=(V,A)$, a set $B\subset V$ is a packing set in $D$ if there are no arcs joining vertices of $B$ and for any two vertices $x,y\in B$ the sets of in-neighbors of $x$ and $y$ are disjoint. The set $S$ is a dominating set (an open dominating set) in $D$ if every vertex not in $S$ (in $V$) has an in-neighbor in $S$. Moreover, a dominating set $S$ is called a total dominating set if the subgraph induced by $S$ has no isolated vertices. The packing sets of maximum cardinality and the (total, open) dominating sets of minimum cardinality in digraphs are studied in this article. We prove that the two optimal sets concerning packing and domination achieve the same value for directed trees, and give some applications of it. We also show analogous equalities for all connected contrafunctional digraphs, and characterize all such digraphs $D$ for which such equalities are satisfied. Moreover, sharp bounds on the maximum and the minimum cardinalities of packing and dominating sets, respectively, are given for digraphs. Finally, we present solutions for two open problems, concerning total and open dominating sets of minimum cardinality, pointed out in [Australas. J. Combin. 39 (2007), 283--292]

Problem collection from the IML programme: Graphs, Hypergraphs, and Computing

This collection of problems and conjectures is based on a subset of the open problems from the seminar series and the problem sessions of the Institut Mitag-Leffler programme Graphs, Hypergraphs, and Computing. Each problem contributor has provided a write up of their proposed problem and the collection has been edited by Klas Markstr\"om.Comment: This problem collection is published as part of the IML preprint series for the research programme and also available there http://www.mittag-leffler.se/research-programs/preprint-series?course_id=4401. arXiv admin note: text overlap with arXiv:1403.5975, arXiv:0706.4101 by other author