arXiv subjects

Magnús M. Halldórsson

Publications and source records attributed to Magnús M. Halldórsson.

32 records · Page 2Linked to original sources

arXiv2021

Query Minimization under Stochastic Uncertainty

We study problems with stochastic uncertainty information on intervals for which the precise value can be queried by paying a cost. The goal is to devise an adaptive decision tree to find a correct solution to the problem in consideration while minimizing the expected total query cost. We show that, for the sorting problem, such a decision tree can be found in polynomial time. For the problem of finding the data item with minimum value, we have some evidence for hardness. This contradicts intuition, since the minimum problem is easier both in the online setting with adversarial inputs and in the offline verification setting. However, the stochastic assumption can be leveraged to beat both deterministic and randomized approximation lower bounds for the online setting.

cs.DS↗

arXiv2021

Ultrafast Distributed Coloring of High Degree Graphs

We give a new randomized distributed algorithm for the $Δ+1$-list coloring problem. The algorithm and its analysis dramatically simplify the previous best result known of Chang, Li, and Pettie [SICOMP 2020]. This allows for numerous refinements, and in particular, we can color all $n$-node graphs of maximum degree $Δ\ge \log^{2+Ω(1)} n$ in $O(\log^* n)$ rounds. The algorithm works in the CONGEST model, i.e., it uses only $O(\log n)$ bits per message for communication. On low-degree graphs, the algorithm shatters the graph into components of size $\operatorname{poly}(\log n)$ in $O(\log^* Δ)$ rounds, showing that the randomized complexity of $Δ+1$-list coloring in CONGEST depends inherently on the deterministic complexity of related coloring problems.

cs.DC↗

arXiv2021

Query-Competitive Sorting with Uncertainty

We study the problem of sorting under incomplete information, when queries are used to resolve uncertainties. Each of $n$ data items has an unknown value, which is known to lie in a given interval. We can pay a query cost to learn the actual value, and we may allow an error threshold in the sorting. The goal is to find a nearly-sorted permutation by performing a minimum-cost set of queries. We show that an offline optimum query set can be found in poly time, and that both oblivious and adaptive problems have simple competitive algorithms. The competitive ratio for the oblivious problem is $n$ for uniform query costs, and unbounded for arbitrary costs; for the adaptive problem, the ratio is 2. We present a unified adaptive strategy for uniform costs that yields the following improved results: (1) a 3/2-competitive randomized algorithm; (2) a 5/3-competitive deterministic algorithm if the dependency graph has no 2-components after some preprocessing, which has competitive ratio $3/2+\mathrm{O}(1/k)$ if the components obtained have size at least $k$; and (3) an exact algorithm for laminar families of intervals. The first two results have matching lower bounds, and we have a lower bound of 7/5 for large components. We also give a randomized adaptive algorithm with competitive ratio $1+\frac{4}{3\sqrt{3}}\approx 1.7698$ for arbitrary query costs, and we show that the 2-competitive deterministic adaptive algorithm can be generalized for queries returning intervals and for a more general vertex cover problem, by using the local ratio technique. Moreover, we prove that the advice complexity of the adaptive problem is $\lfloor n/2\rfloor$ if no error threshold is allowed, and $\lceil n/3\cdot\lg 3\rceil$ for the general case. Finally, we present some graph-theoretical results on co-threshold tolerance graphs, and we discuss uncertainty variants of some classical interval problems.

cs.DS↗

arXiv2021

Efficient Randomized Distributed Coloring in CONGEST

Distributed vertex coloring is one of the classic problems and probably also the most widely studied problems in the area of distributed graph algorithms. We present a new randomized distributed vertex coloring algorithm for the standard CONGEST model, where the network is modeled as an $n$-node graph $G$, and where the nodes of $G$ operate in synchronous communication rounds in which they can exchange $O(\log n)$-bit messages over all the edges of $G$. For graphs with maximum degree $Δ$, we show that the $(Δ+1)$-list coloring problem (and therefore also the standard $(Δ+1)$-coloring problem) can be solved in $O(\log^5\log n)$ rounds. Previously such a result was only known for the significantly more powerful LOCAL model, where in each round, neighboring nodes can exchange messages of arbitrary size. The best previous $(Δ+1)$-coloring algorithm in the CONGEST model had a running time of $O(\logΔ+ \log^6\log n)$ rounds. As a function of $n$ alone, the best previous algorithm therefore had a round complexity of $O(\log n)$, which is a bound that can also be achieved by a naïve folklore algorithm. For large maximum degree $Δ$, our algorithm hence is an exponential improvement over the previous state of the art.

cs.DS↗

arXiv2021

Superfast Coloring in CONGEST via Efficient Color Sampling

We present a procedure for efficiently sampling colors in the {\congest} model. It allows nodes whose number of colors exceeds their number of neighbors by a constant fraction to sample up to $Θ(\log n)$ semi-random colors unused by their neighbors in $O(1)$ rounds, even in the distance-2 setting. This yields algorithms with $O(\log^* Δ)$ complexity for different edge-coloring, vertex coloring, and distance-2 coloring problems, matching the best possible. In particular, we obtain an $O(\log^* Δ)$-round CONGEST algorithm for $(1+ε)Δ$-edge coloring when $Δ\ge \log^{1+1/\log^*n} n$, and a poly($\log\log n$)-round algorithm for $(2Δ-1)$-edge coloring in general. The sampling procedure is inspired by a seminal result of Newman in communication complexity.

cs.DC↗

arXiv2020

Tight Bounds on Subexponential Time Approximation of Set Cover and Related Problems

We show that Set Cover on instances with $N$ elements cannot be approximated within $(1-γ)\ln N$-factor in time exp($N^{γ-δ})$, for any $0 < γ< 1$ and any $δ> 0$, assuming the Exponential Time Hypothesis. This essentially matches the best upper bound known by Cygan et al.\ (IPL, 2009) of $(1-γ)\ln N$-factor in time $exp(O(N^γ))$. The lower bound is obtained by extracting a standalone reduction from Label Cover to Set Cover from the work of Moshkovitz (Theory of Computing, 2015), and applying it to a different PCP theorem than done there. We also obtain a tighter lower bound when conditioning on the Projection Games Conjecture. We also treat three problems (Directed Steiner Tree, Submodular Cover, and Connected Polymatroid) that strictly generalize Set Cover. We give a $(1-γ)\ln N$-approximation algorithm for these problems that runs in $exp(\tilde{O}(N^γ))$ time, for any $1/2 \le γ< 1$.

cs.DS↗

arXiv2019

The Capacity of Smartphone Peer-to-Peer Networks

We study three capacity problems in the mobile telephone model, a network abstraction that models the peer-to-peer communication capabilities implemented in most commodity smartphone operating systems. The capacity of a network expresses how much sustained throughput can be maintained for a set of communication demands, and is therefore a fundamental bound on the usefulness of a network. Because of this importance, wireless network capacity has been active area of research for the last two decades. The three capacity problems that we study differ in the structure of the communication demands. The first problem is pairwise capacity, where the demands are (source, destination) pairs. Pairwise capacity is one of the most classical definitions, as it was analyzed in the seminal paper of Gupta and Kumar on wireless network capacity. The second problem we study is broadcast capacity, in which a single source must deliver packets to all other nodes in the network. Finally, we turn our attention to all-to-all capacity, in which all nodes must deliver packets to all other nodes. In all three of these problems we characterize the optimal achievable throughput for any given network, and design algorithms which asymptotically match this performance. We also study these problems in networks generated randomly by a process introduced by Gupta and Kumar, and fully characterize their achievable throughput. Interestingly, the techniques that we develop for all-to-all capacity also allow us to design a one-shot gossip algorithm that runs within a polylogarithmic factor of optimal in every graph. This largely resolves an open question from previous work on the one-shot gossip problem in this model.

cs.DC↗

arXiv2018

Distributed Algorithms for Minimum Degree Spanning Trees

The minimum degree spanning tree (MDST) problem requires the construction of a spanning tree $T$ for graph $G=(V,E)$ with $n$ vertices, such that the maximum degree $d$ of $T$ is the smallest among all spanning trees of $G$. In this paper, we present two new distributed approximation algorithms for the MDST problem. Our first result is a randomized distributed algorithm that constructs a spanning tree of maximum degree $\hat d = O(d\log{n})$. It requires $O((D + \sqrt{n}) \log^2 n)$ rounds (w.h.p.), where $D$ is the graph diameter, which matches (within log factors) the optimal round complexity for the related minimum spanning tree problem. Our second result refines this approximation factor by constructing a tree with maximum degree $\hat d = O(d + \log{n})$, though at the cost of additional polylogarithmic factors in the round complexity. Although efficient approximation algorithms for the MDST problem have been known in the sequential setting since the 1990's, our results are first efficient distributed solutions for this problem.

cs.DS↗

arXiv2018

Simple and Local Independent Set Approximation

We bound the performance guarantees that follow from Turán-like bounds for unweighted and weighted independent sets in bounded-degree graphs. In particular, a randomized approach of Boppana forms a simple 1-round distributed algorithm, as well as a streaming and preemptive online algorithm. We show it gives a tight $(Δ+1)/2$-approximation in unweighted graphs of maximum degree $Δ$, which is best possible for 1-round distributed algorithms. For weighted graphs, it gives only a $Δ$-approximation, but a simple modification results in an asymptotic expected $0.529 Δ$-approximation. This compares with a recent, more complex $Δ$-approximation~\cite{BCGS17}, which holds deterministically.

cs.DS↗

arXiv2015

Max Point-Tolerance Graphs

A graph $G$ is a \emph{max point-tolerance (MPT)} graph if each vertex $v$ of $G$ can be mapped to a \emph{pointed-interval} $(I_v, p_v)$ where $I_v$ is an interval of $\mathbb{R}$ and $p_v \in I_v$ such that $uv$ is an edge of $G$ iff $I_u \cap I_v \supseteq \{p_u, p_v\}$. MPT graphs model relationships among DNA fragments in genome-wide association studies as well as basic transmission problems in telecommunications. We formally introduce this graph class, characterize it, study combinatorial optimization problems on it, and relate it to several well known graph classes. We characterize MPT graphs as a special case of several 2D geometric intersection graphs; namely, triangle, rectangle, L-shape, and line segment intersection graphs. We further characterize MPT as having certain linear orders on their vertex set. Our last characterization is that MPT graphs are precisely obtained by intersecting special pairs of interval graphs. We also show that, on MPT graphs, the maximum weight independent set problem can be solved in polynomial time, the coloring problem is NP-complete, and the clique cover problem has a 2-approximation. Finally, we demonstrate several connections to known graph classes; e.g., MPT graphs strictly contain interval graphs and outerplanar graphs, but are incomparable to permutation, chordal, and planar graphs.

cs.DM↗

arXiv2015

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 $n/2$.

math.CO↗

arXiv2014

Wireless Scheduling Algorithms in Complex Environments

Efficient spectrum use in wireless sensor networks through spatial reuse requires effective models of packet reception at the physical layer in the presence of interference. Despite recent progress in analytic and simulations research into worst-case behavior from interference effects, these efforts generally assume geometric path loss and isotropic transmission, assumptions which have not been borne out in experiments. Our paper aims to provide a methodology for grounding theoretical results into wireless interference in experimental reality. We develop a new framework for wireless algorithms in which distance-based path loss is replaced by an arbitrary gain matrix, typically obtained by measurements of received signal strength (RSS). Gain matrices allow for the modeling of complex environments, e.g., with obstacles and walls. We experimentally evaluate the framework in two indoors testbeds with 20 and 60 motes, and confirm superior predictive performance in packet reception rate for a gain matrix model over a geometric distance-based model. At the heart of our approach is a new parameter $ζ$ called metricity which indicates how close the gain matrix is to a distance metric, effectively measuring the complexity of the environment. A powerful theoretical feature of this parameter is that all known SINR scheduling algorithms that work in general metric spaces carry over to arbitrary gain matrices and achieve equivalent performance guarantees in terms of $ζ$ as previously obtained in terms of the path loss constant. Our experiments confirm the sensitivity of $ζ$ to the nature of the environment. Finally, we show analytically and empirically how multiple channels can be leveraged to improve metricity and thereby performance. We believe our contributions will facilitate experimental validation for recent advances in algorithms for physical wireless interference models.

cs.NI↗

arXiv2014

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.

cs.NI↗

arXiv2012

Algorithms for Wireless Capacity

In this paper we address two basic questions in wireless communication: First, how long does it take to schedule an arbitrary set of communication requests? Second, given a set of communication requests, how many of them can be scheduled concurrently? Our results are derived in an interference model with geometric path loss and consist of efficient algorithms that find a constant approximation for the second problem and a logarithmic approximation for the first problem. In addition, we analyze some important properties of the interference model and show that it is robust to various factors that can influence the signal attenuation. More specifically, we prove that as long as such influences on the signal attenuation are constant, they affect the capacity only by a constant factor.

cs.NI↗