Beyond Geometry : Towards Fully Realistic Wireless Models

Signal-strength models of wireless communications capture the gradual fading of signals and the additivity of interference. As such, they are closer to reality than other models. However, nearly all theoretic work in the SINR model depends on the assumption of smooth geometric decay, one that is true in free space but is far off in actual environments. The challenge is to model realistic environments, including walls, obstacles, reflections and anisotropic antennas, without making the models algorithmically impractical or analytically intractable. We present a simple solution that allows the modeling of arbitrary static situations by moving from geometry to arbitrary decay spaces. The complexity of a setting is captured by a metricity parameter Z that indicates how far the decay space is from satisfying the triangular inequality. All results that hold in the SINR model in general metrics carry over to decay spaces, with the resulting time complexity and approximation depending on Z in the same way that the original results depends on the path loss term alpha. For distributed algorithms, that to date have appeared to necessarily depend on the planarity, we indicate how they can be adapted to arbitrary decay spaces. Finally, we explore the dependence on Z in the approximability of core problems. In particular, we observe that the capacity maximization problem has exponential upper and lower bounds in terms of Z in general decay spaces. In Euclidean metrics and related growth-bounded decay spaces, the performance depends on the exact metricity definition, with a polynomial upper bound in terms of Z, but an exponential lower bound in terms of a variant parameter phi. On the plane, the upper bound result actually yields the first approximation of a capacity-type SINR problem that is subexponential in alpha

Semi-Transitive Orientations and Word-Representable Graphs

A graph $G=(V,E)$ is a \emph{word-representable graph} if there exists a word $W$ over the alphabet $V$ such that letters $x$ and $y$ alternate in $W$ if and only if $(x,y)\in E$ for each $x\neq y$. In this paper we give an effective characterization of word-representable graphs in terms of orientations. Namely, we show that a graph is word-representable if and only if it admits a \emph{semi-transitive orientation} defined in the paper. This allows us to prove a number of results about word-representable graphs, in particular showing that the recognition problem is in NP, and that word-representable graphs include all 3-colorable graphs. We also explore bounds on the size of the word representing the graph. The representation number of $G$ is the minimum $k$ such that $G$ is a representable by a word, where each letter occurs $k$ times; such a $k$ exists for any word-representable graph. We show that the representation number of a word-representable graph on $n$ vertices is at most $2n$, while there exist graphs for which it is $n/2$.Comment: arXiv admin note: text overlap with arXiv:0810.031

Strongly simplicial vertices of powers of trees

AbstractFor a tree T and an integer k⩾1, it is well known that the kth power Tk of T is strongly chordal and hence has a strong elimination ordering of its vertices. In this note we obtain a complete characterization of strongly simplicial vertices of Tk, thereby characterizing all strong elimination orderings of the vertices of Tk

Independent sets in bounded-degree hypergraphs

AbstractIn this paper we analyze several approaches to the Maximum Independent Set (MIS) problem in hypergraphs with degree bounded by a parameter Δ. Since independent sets in hypergraphs can be strong and weak, we denote by MIS (MSIS) the problem of finding a maximum weak (strong) independent set in hypergraphs, respectively. We propose a general technique that reduces the worst case analysis of certain algorithms on hypergraphs to their analysis on ordinary graphs. This technique allows us to show that the greedy algorithm for MIS that corresponds to the classical greedy set cover algorithm has a performance ratio of (Δ+1)/2. It also allows us to apply results on local search algorithms on graphs to obtain a (Δ+1)/2 approximation for the weighted MIS and (Δ+3)/5−ϵ approximation for the unweighted case. We improve the bound in the weighted case to ⌈(Δ+1)/3⌉ using a simple partitioning algorithm. We also consider another natural greedy algorithm for MIS that adds vertices of minimum degree and achieves only a ratio of Δ−1, significantly worse than on ordinary graphs. For MSIS, we give two variations of the basic greedy algorithm and describe a family of hypergraphs where both algorithms approach the bound of Δ

Deterministic Digital Clustering of Wireless Ad Hoc Networks

We consider deterministic distributed communication in wireless ad hoc networks of identical weak devices under the SINR model without predefined infrastructure. Most algorithmic results in this model rely on various additional features or capabilities, e.g., randomization, access to geographic coordinates, power control, carrier sensing with various precision of measurements, and/or interference cancellation. We study a pure scenario, when no such properties are available. As a general tool, we develop a deterministic distributed clustering algorithm. Our solution relies on a new type of combinatorial structures (selectors), which might be of independent interest. Using the clustering, we develop a deterministic distributed local broadcast algorithm accomplishing this task in $O(\Delta \log^*N \log N)$ rounds, where $\Delta$ is the density of the network. To the best of our knowledge, this is the first solution in pure scenario which is only polylog$(n)$ away from the universal lower bound $\Omega(\Delta)$, valid also for scenarios with randomization and other features. Therefore, none of these features substantially helps in performing the local broadcast task. Using clustering, we also build a deterministic global broadcast algorithm that terminates within $O(D(\Delta + \log^* N) \log N)$ rounds, where $D$ is the diameter of the network. This result is complemented by a lower bound $\Omega(D \Delta^{1-1/\alpha})$, where $\alpha > 2$ is the path-loss parameter of the environment. This lower bound shows that randomization or knowledge of own location substantially help (by a factor polynomial in $\Delta$) in the global broadcast. Therefore, unlike in the case of local broadcast, some additional model features may help in global broadcast

Fast Coloring Despite Congested Relays

We provide a $O(\log^6 \log n)$-round randomized algorithm for distance-2 coloring in CONGEST with $\Delta^2+1$ colors. For $\Delta\gg\operatorname{poly}\log n$, this improves exponentially on the $O(\log\Delta+\operatorname{poly}\log\log n)$ algorithm of [Halld\'orsson, Kuhn, Maus, Nolin, DISC'20]. Our study is motivated by the ubiquity and hardness of local reductions in CONGEST. For instance, algorithms for the Local Lov\'asz Lemma [Moser, Tardos, JACM'10; Fischer, Ghaffari, DISC'17; Davies, SODA'23] usually assume communication on the conflict graph, which can be simulated in LOCAL with only constant overhead, while this may be prohibitively expensive in CONGEST. We hope our techniques help tackle in CONGEST other coloring problems defined by local relations.Comment: 37 pages. To appear in proceedings of DISC 202

Approximation algorithms for hard variants of the stable marriage and hospitals/residents problems

When ties and incomplete preference lists are permitted in the Stable Marriage and Hospitals/Residents problems, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is known to be NP-hard, even under very severe restrictions on the number, size and position of ties. In this paper, we describe polynomial-time 5/3-approximation algorithms for variants of these problems in which ties are on one side only and at the end of the preference lists. The particular variant is motivated by important applications in large scale centralised matching schemes