Local algorithms in (weakly) coloured graphs

A local algorithm is a distributed algorithm that completes after a constant number of synchronous communication rounds. We present local approximation algorithms for the minimum dominating set problem and the maximum matching problem in 2-coloured and weakly 2-coloured graphs. In a weakly 2-coloured graph, both problems admit a local algorithm with the approximation factor $(\Delta+1)/2$, where $\Delta$ is the maximum degree of the graph. We also give a matching lower bound proving that there is no local algorithm with a better approximation factor for either of these problems. Furthermore, we show that the stronger assumption of a 2-colouring does not help in the case of the dominating set problem, but there is a local approximation scheme for the maximum matching problem in 2-coloured graphs.Comment: 14 pages, 3 figure

Fast Local Computation Algorithms

For input $x$, let $F(x)$ denote the set of outputs that are the "legal" answers for a computational problem $F$. Suppose $x$ and members of $F(x)$ are so large that there is not time to read them in their entirety. We propose a model of {\em local computation algorithms} which for a given input $x$, support queries by a user to values of specified locations $y_i$ in a legal output $y \in F(x)$. When more than one legal output $y$ exists for a given $x$, the local computation algorithm should output in a way that is consistent with at least one such $y$. Local computation algorithms are intended to distill the common features of several concepts that have appeared in various algorithmic subfields, including local distributed computation, local algorithms, locally decodable codes, and local reconstruction. We develop a technique, based on known constructions of small sample spaces of $k$-wise independent random variables and Beck's analysis in his algorithmic approach to the Lov{\'{a}}sz Local Lemma, which under certain conditions can be applied to construct local computation algorithms that run in {\em polylogarithmic} time and space. We apply this technique to maximal independent set computations, scheduling radio network broadcasts, hypergraph coloring and satisfying $k$-SAT formulas.Comment: A preliminary version of this paper appeared in ICS 2011, pp. 223-23

Almost Stable Matchings by Truncating the Gale–Shapley Algorithm

We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2 + ε)-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching.Peer reviewe

Hollow Heaps

We introduce the hollow heap, a very simple data structure with the same amortized efficiency as the classical Fibonacci heap. All heap operations except delete and delete-min take $O(1)$ time, worst case as well as amortized; delete and delete-min take $O(\log n)$ amortized time on a heap of $n$ items. Hollow heaps are by far the simplest structure to achieve this. Hollow heaps combine two novel ideas: the use of lazy deletion and re-insertion to do decrease-key operations, and the use of a dag (directed acyclic graph) instead of a tree or set of trees to represent a heap. Lazy deletion produces hollow nodes (nodes without items), giving the data structure its name.Comment: 27 pages, 7 figures, preliminary version appeared in ICALP 201

Local approximability of max-min and min-max linear programs

In a max-min LP, the objective is to maximise ω subject to Ax ≤ 1, Cx ≥ ω1, and x ≥ 0. In a min-max LP, the objective is to minimise ρ subject to Ax ≤ ρ1, Cx ≥ 1, and x ≥ 0. The matrices A and C are nonnegative and sparse: each row ai of A has at most ΔI positive elements, and each row ck of C has at most ΔK positive elements. We study the approximability of max-min LPs and min-max LPs in a distributed setting; in particular, we focus on local algorithms (constant-time distributed algorithms). We show that for any ΔI ≥ 2, ΔK ≥ 2, and ε > 0 there exists a local algorithm that achieves the approximation ratio ΔI (1 − 1/ΔK) + ε. We also show that this result is the best possible: no local algorithm can achieve the approximation ratio ΔI (1 − 1/ΔK) for any ΔI ≥ 2 and ΔK ≥ 2.Peer reviewe

Lower bounds for local approximation

In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique O(log n)-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient. Our result holds for so-called simple PO-checkable graph optimisation problems; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks. As a corollary of our result, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem. By prior work, there is a deterministic local algorithm that achieves the approximation factor of 4 − 1/⌊Δ/2⌋ in graphs of maximum degree Δ. This approximation ratio is known to be optimal in the port-numbering model—our main theorem implies that it is optimal also in the standard model in which each node has a unique identifier. Our main technical tool is an algebraic construction of homogeneously ordered graphs: We say that a graph is (α,r)-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius-r neighbourhoods. We show that there exists a finite (α,r)-homogeneous 2k-regular graph of girth at least g for any α < 1 and any r, k, and g.Peer reviewe