We focus on algorithms whose runtime and performance guarantee are independent of the number of nodes in the network. For a quick summary of negative results for deterministic local algorithms, see Tables I and II. A problem is said to be of distributed constant size if G is a graph of bounded degree and the size of the local input is bounded by a constant.
When we study local algorithms, we assume that the problem instance is given in a distributed way: each node in the communication graph G knows part of the input. For unweighted graph problems, we do not need any task-specific information in the local input; the structure of the communication graph is enough.
Partitions
Covering Problems
Packing Problems
Mixed Packing and Covering
The maximum degree of an agent v ∈ V is V, the maximum degree of a constraint i ∈ I is I, and the maximum degree of a customer k∈KisK.
AUXILIARY INFORMATION AND LOCAL VIEWS
- Symmetry Breaking
- Covering Graphs and Unfoldings
- Local View
- Graphs with Orientation
- Graphs with Unique Identifiers
After roundi, each node merges the local views received from its neighbors; this leads to the radius ilocal view. If the local views of nodesuandv differ, then the local outputs of nodesuandv may also differ. The bad news is that choosing the local output based on the local view is in a sense the only thing one can do in a local algorithm [Angluin 1980; Boldi and Vigna 2001;.
In general, we can construct ad-regular graphs for any constant such that the local appearance of each node is identical [Naor and Stockmeyer 1995], regardless of a port count and an orientation; see Figure 5 for an example. Letv be an arbitrary node; we show that the local view of v is different from the local view of at least one neighbor of v. In Figure 7(a) , the node and its neighbor have different degrees outside; so the local view of vv differs from the local view of u.
In this case, the local view of sis is necessarily different from the local view of t: both have exactly one predecessor, and the port numbers assigned to these unique incoming flanks are different because p(v,s) = p(v, h). Therefore, the local view of v is different from the local view of black t (or both). Of course, we require the local algorithm to solve the problem for any choice of unique identifiers.
If the local output cannot be determined based on G(v,r) for a constant, then there is no local algorithm for the problem; if it can be determined based on G(v,r) for some constants, then there is a local algorithm.
NEGATIVE RESULTS
- Preliminary Observations
- Comparable Identifiers
- Numerical Identifiers
- Approximations for Combinatorial Problems
- Approximations for LPs
As a simple generalization, there is no localo()-approximation algorithm for the minimal dominating set problem in anonymous graphs with bounded degree. We immediately get the same negative results as we had for anonymous networks in ann-cycle: for example, there is no local algorithm for finding the largest matching, largest independent set, node coloring, edge coloring, or weak coloring. These results show that there is no local constant-factor approximation algorithm for the ann-cycle maximum cut problem.
As another corollary, there is no local (2−ε) approximation algorithm for the edge covering problem in a cycle. An analogous argument shows that there is no local (2−ε) approximation algorithm for the top coverage problem in an n-cycle. By exchanging the roles of edges and nodes, the same argument shows that there is no local (3−ε) approximation algorithm for the edge-dominating set problem in a cycle.
There is no local (2k+1−ε)-approximation algorithm for the dominating set problem on regular 2k (case=2) graphs. Moreover, because it is a dominating set, there is no path with more than 2 nodes in the subgraph of Ginduced byX (Figure 10(e)). There is no local (k+1−ε)-approximation algorithm for the dominating set problem on regular (2k+1) graphs (case=2).
2004, 2006b, 2010] and Moscibroda [2006] show that it is not possible to find a constant factor approximation to a minimum vertex covering, a minimum dominating set or a maximum matching in general graphs with a local algorithm, if there no degree bound.
POSITIVE RESULTS
- Bicolored Matchings and Vertex Covers
- Linear Programs and Vertex Covers
- Weak Coloring
- Color Reduction
- Matchings
- Domination
- Trivial Algorithms
- Local Verification and Locally Checkable Proofs
- Other Problems
A communication graph G is said to be bicolored if the 2-coloring of a node of G is given as part of the local input; each node knows whether it is black or white. One of the following applies: (i) never sent a proposal or (ii) rejected a proposal from you. Therefore, we can find a factor 1+ε approximation of the LP relaxation of the minimal node covering problem in bounded-degree graphs.
Note that the vertex coverage problem is a special case of the set coverage problem with K=2; The same technique of deterministic LP rounding can be applied to design a local (K+ε) approximation algorithm for the set coverage problem in bounded degree graphs, for an arbitrary K. Note that the elements are gate numbers of the form(· , v), but they are ordered according to the port numbers p(v,·). To see that the vectors x(v) are a weak coloring of the graph, we generalize the argument we saw in section 5.4.
According to the pigeon principle, there are at least two successors of v, call them sandt, and indices with the following property: the element i of the vector x(s) is p(v,s) and the element of the vectorjax(t) is p(v,t). A much more efficient algorithm can be designed by exploiting the numerical values of the original colors. Using the two-color double cover of Section 7.1, this gives a (2+ε)-approximation of the simple maximum size 2-match in general bounded-degree graphs.
For each node, the proof consists of:. i) the identity of the root node, (ii) the distance to the root node in the spanning tree, and (iii) the edge pointing towards the root node.
RANDOMIZED LOCAL ALGORITHMS
- Nonconstant Guarantees
- Negative Results
- Matchings and Independent Sets
- Maximum Cut and Maximum Satisfiability
- LP Rounding
This provides a partial answer to Elkin's [2004] question about the distributed approximation of the maximum cut problem. Kuhn's [2005, Section 4.5] algorithm for one-round graph coloring is a good example of the latter case. As explained in Section 1.1, our focus is on algorithms for which both runtime and performance guarantees are independent of the number of nodes; that is, we would get a limited performance guarantee even on an infinitely large network.
First, the probability of an “error”—the event that an infeasible output occurs—can be made negligible by adapting the algorithm to the global properties of the input; for example, the number of iterations may depend on the size of the input. Indeed, if a random local algorithm has a non-zero probability of failure given the input, then we can simply take multiple copies of the input to increase the probability that the algorithm fails in at least one copy; see for example Kuhn [2005, section 4.5]. If we are satisfied with the guarantee of the probabilistic approximation, some of the negative results in Section 6 can be overcome.
First, for each clause, remove all but one literal value; essentially we end up with a MAX-1-SAT instance. Second, at least half of the clauses must be satisfied using a local version of Johnson's 2-approximation algorithm from [1974]. The solution is obtained in three stages: (1) Approximately solve the LP relaxation of the problem with a local algorithm.
As discussed earlier in Chapter 7, the LP relaxations of the set-cover and set-packing problems have local approximation schemes in bounded-degree graphs.
GEOMETRIC PROBLEMS
- Models
- Partial Geometric Information
- Algorithms from Simple Tilings
- Other Algorithms
- Planar Subgraphs and Geographic Routing
- Spanners
- Colored Subgraphs
2007, 2008d] study scheduling problems in a semi-geometric environment in which the coordinates of nodes are unknown but a small amount of symmetry-breaking information is available. As a concrete example, we can divide the plane into rectangles of size 2×1 and paint them with 3 colors, so that the distance between a pair of nodes in two different rectangles of the same color is greater than 1. By construction, we know that if nodes and v are in two different rectangles of the same color , then there is no edge in the (quasi) graph of the unit disk, v}.
The edges that cross the borders of the tiles can be safely ignored in the algorithm. Put together, we get a feasible vertex coloring, and the number of colors we use is within a factor of 3 of the optimum; see Figure 15. At each conflict resolution, we lose at most one half of the nodes; thus the remaining nodes provide a factor 4 approximation.
Kuhn and Moscibroda [2007] present a local approximation algorithm for the capacitated dominating set problem in unit disk graphs; it is a variant of the dominating set problem in which each node has a limited capacity that determines how many neighbors it can dominate. Once we have constructed a planar subgraph of a unit disk graph, it is possible to send messages with local geometric rules, assuming we know the coordinates of the target node [Bose et al. In a t-key, for any pair u, v of nodes inG, the shortest path between uandv in His is at most as long as the shortest path between uandfinG; here the length of a path is the sum of the Euclidean lengths of the edges.
2008] further guarantees that the total edge length of the key is at most a constant factor larger than the total edge length of a minimal spanning tree.
OPEN PROBLEMS
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InProceedings of the 4thInternational Workshop on Algorithmic Aspects of Wireless Sensor Networks.Lecture Notes in Computer Science, vol. InProceedings of the 8th International Conference on Distributed Computing and Networking (ICDCN).Lecture Notes in Computer Science, vol. InProceedings of the 22nd International Symposium on Distributed Computing (DISC). Lecture Notes in Computer Science, vol.
In Proceedings of the 24th International Symposium on Distributed Computing (DISC). Lecture Notes in Computer Science, vol.
