The Neighbor Collecting Problem

Inspired by prize-collecting network design problems, we next consider the objective of minimizing the total cost of the selected nodes plus the number of nodes left uncovered: choose a setSofGthat minimizes f(S) =c|S|+|V\D(S)|for a given parameterc>0. Note that whenc<1the problem reduces to the minimum dominating set problem: it is always worthwhile to cover all nodes. Assuming c≥1, the1+-local algorithm for the minimum dominating set problem achieves anO(cH(∆))approximation.

Theorem 4.4. For anyc≥1and setOP T minimizing f(OP T), algorithm AlternateRan- dom returns a setSfor whichE[f(S)]≤2c(2+H(∆))f(OP T).

Proof of Theorem 4.4. We have f(OP T) =|V−D(OP T )|+c|OP T |and f(OP T ∪ {V−D(OP T )}) =c|OP T ∪ {V−D(OP T)}|=c|OP T |+c|V −D(OP T)| ≥ c|T∗|whereT∗is a minimum dominating set of the graph. Next, we know from Theorem 4.1 that|T∗| ≥(2(2+H(∆)))−1E[|S|] = (2(2+H(∆)))−1c−1E[f(S)].

Finally, f(OP T) =|V−D(OP T )|+c|OP T |soc|OP T |+c|V−D(OP T)| ≤ c f(OP T). We conclude thatE[f(S)]≤2(2+H(∆))c f(OP T ).

Since the neighbor-collecting problem contains the minimum dominating set problem as a special case (i.e. when c=1), we cannot hope to avoid the dependency on H(∆) in the approximation factor in Theorem 4.4. As we next show, the dependence on c in the approximation factor we obtain in Theorem 4.4 is also unavoidable.

Theorem 4.5. Let c=o(n1/4). Then for any randomized 1+-local algorithm A for the neighbor-collecting problem, there exists an input instanceGfor which

E[f(S)] =Ω(max{c,H(∆)})· f(|T|),

where S denotes the output generated by A on input G, and T denotes the output of the best algorithm with full knowledge of the input graph.

Proof. As done previously, we shall invoke the application of Yao’s Minimax Principle for the performance of randomized algorithms [119].

Figure 4.2: An illustration of the lower bound construction for1+-local algorithms for the neighbor-collecting problem.

We will build the following family of inputs. Givennwe construct a connected graph on nnodes in the following way. Let k=o(√n/c) be a parameter to be set specifically later. Create two star subgraphs one on n−√n−2k nodes and one on √n nodes. We connect one arbitrary leaf of the big star subgraph to one arbitrary leaf of the smaller star subgraph. Next, choosekspoke nodes from the bigger star subgraph and connect each of them to one new node of degree one. To complete the construction, randomly assign labels to the nodes from[n]. This gives us a connected graph onnvertices. Note thatTis at most 6c+kas an algorithm can always cover the hubs of the two stars in four Crawl steps. See Figure 4.2 for an illustration of the construction.

We show that the expected cost of a deterministic algorithm on the distribution of inputs is at least(1+k/2+2)c. This happens when the1+ local algorithm starts from a spoke in the bigger star component (with occurs with probability1−o(1)) and needs to traverse half of thekspokes that were assigned one new neighbor. Only after traversing such nodes it moves into a spoke of the small star subgraph and then to the hub of the smaller star

subgraph. We note thatk=o(√n/c), so taking any Jump queries would result in a worse cost.

Thus the approximation ratio is at least (1+₆k_c/2₊+_k2)c. This expression is the biggest (as a function ofk=o(√n/c)) fork=Θ(c). In that case the expression isΘ(c). By Yao’s Minimax Principle this gives a lower bound of Ω(c)on the expected performance of any randomized1+-local algorithm, on at least one of the inputs.

Finally, one cannot move from1+-local algorithms to 1-local algorithms without sig- nificant loss: every1-local algorithm has a polynomial approximation factor.

Theorem 4.6. For any randomized1-local algorithm Afor the neighbor-collecting prob- lem, there exists an input instance G for which E[f(S)] =Ω(√n/c)· f(|T|), where S

denotes the output generated by Aon input G, andT denotes the output of the best algo- rithm with full knowledge of the input graph.

Proof. As done previously, we shall invoke the application of Yao’s Minimax Principle for the performance of randomized algorithms [119].

We will build the following family of inputs. Build a clique on n−√n vertices and remove one edge(u,v). Next build a star with√n−1 leaves, say with root r, and label one of the leaves v0. Finally, add edge (v,v0). To complete the construction, randomly assign labels to the nodes from[n]. See Figure 4.3 for an illustration of the construction.

For such graph, the set{r,v}has cost2cand can be found in five Crawls, so f(|T|)≤

5c. Consider the setSreturned by a 1-local deterministic algorithm; we will show that S

will have cost at least√nwith high probability. IfSdoes not includer then it must leave

√

nnodes uncovered (or else contain at least√nvertices), in which case it has cost at least

√

n. SoSmust contain some node in the star centered atr. A node in the star can be found either via a random query or by querying node v. Since the star contains √n nodes, it would takeΩ(√n)random queries to find a node in the star with high probability. On the other hand, nodevis indistinguishable from the other nodes butuin the(n−√n)-clique until after it has been queried; it would therefore take Ω(n) queries to the nodes in the clique to findv, again with high probability. We conclude that the cost ofSis at least √n

Figure 4.3: An illustration of the lower bound construction for1-local algorithms for the neighbor-collecting problem.

in expectation. By Yao’s Minimax Principle, the expected cost of a randomized 1-local algorithm is at leastΩ(√n)and its approximation ratio is thereforeΩ(√n/c).