6.4 Approximation in Sublinear Time
6.4.2 A Lower Bound
We now turn to provide a lower bound on the time it takes an algorithm to compute a β- approximation for the maximum expected influence problem. In particular, for any given budgetk, at leastΩ(n/β)queries are required to obtain approximation factor 1β with con- stant probability.
Theorem 6.12. Let0<e< 101e, β>1be given. Any randomized algorithm for the max-
imum influence problem that has runtime of m+n
24βmin{k,β} cannot return, with probability at
least1−1e −e, a set of nodes with approximation ratio better than 1β.
Proof. Note first that for a graph consisting of n singletons, an algorithm must return at leastk/βnodes to obtain an approximation ratio of 1β. Doing so in at mostn/2β2queries requires that2k/β≤n/β2, which implies2kβ≤n. We can therefore assume2kβ≤nfor the remainder of the proof.
The proof will invoke the application of Yao’s Minimax Principle for the performance of Las Vegas randomized algorithms on a family of inputs [119]. The lemma states that the least expected cost of deterministic Las Vegas algorithms on a distribution over a family of inputs is a lower bound on the expected cost of the optimal randomized Las Vegas algorithm over that family of inputs.
We define the cost of the algorithm as0if it returns a set nodes with approximation ratio better than 1
β and1otherwise. Note that the cost of an algorithm equals its probability of
failure and we can think of any Monte-Carlo randomized algorithm (with runtime at most m+n
24βmin{k,β}) as a Las Vegas one.
Assume for notational simplicity that βis an integer. We will build a family of lower bound graphs, one for each value of n(beginning from n=β+1); each graph will have
m≤n, so it will suffice to demonstrate a lower bound of 12βminn{k,β}.
We now consider the behavior of a deterministic algorithm Awith respect to the uni- form distribution on the constructed family of inputs. For a given valueβthe graph would
Figure 6.1: An illustration of the lower bound construction for approximate influence max- imization.
be made from k components of size 2β and n−2kβ singleton components (recall that 2kβ≤n). See Figure 6.1 for an illustration of the construction.
If algorithm Areturns nodes from`of thekcomponents of size2β, it achieves a total influence of 2`β+ (k−`). Thus, to attain approximation factor better than 1β, we must have2`β+ (k−`)≥ 1β2kβ, which implies`≥ 2βk−1 for anyβ>1.
Supposek>12β. The condition`≥ 2βk−1 implies that at least 2βk−1 of the large com- ponents must be queried by the algorithm, where each random query has probability 2knβ of hitting a large component. If the algorithm makes fewer than 12βn2 queries, then the ex-
pected number of components hit is 12βn2 · 2kβ
n =
k
6β. The Multiplicative chernoff bound
(Lemma A.1, part 3) then imply that the probability hitting more than 2βk components is no more thane−6kβ·2/3≤ 1
e4/3 <1−
1
e −e, a contradiction.
Ifk≤12βthen we need that`≥1, which occurs only if the algorithm queries at least one of thekβvertices in the large components. With 2nkβ queries, fornlarge enough, this happens with probability smaller than 1e −e, a contradiction.
We conclude that, in all cases, at least 12βminn{k,β} queries are necessary to obtain ap- proximation factor better than 1
β with probability at least1− 1
e −e, as required.
By Yao’s Minimax Principle this gives a lower bound of Ω(24βminnd{k,β}) on the ex- pected performance of any randomized algorithm, on at least one of the inputs.
Finally, we note that our construction can be modified to apply to non-sparse networks, as follows. For any d≤n, we can augment our graph by overlaying a d-regular graph with exponentially small weight on each edge. This does not significantly impact the in- fluence of any set of nodes, but it increases the time to determine whether a node is in a large component by a factor of O(d) (as edges must be traversed until one with non- exponentially-small weight is found). Thus, for each d≤n, we have a lower bound of
nd
24βmin{k,β} on th expected performance of A on a distribution of networks with m=nd
edges.
6.5
Discussion
We would like to end this chapter with a short discussion on the applicability of our results and techniques to other problems.
We would like to start by mentioning that although we have focused on the influence maximization problem under the independent cascades model, our results apply as they are to the probabilistick-coverage problem as well. In the probabilistic k-coverage problem, each edge comes with an exposure probability, and the goal is to choose a set ofk nodes such that its one-hop expected exposure (namely coverage) in maximized. One can clearly run our algorithms, growing diffusion trees only one hop away, namely stars. As now we grow stars instead of deep diffusion trees, the dependency of our results (for sublinear-time approximation) on the arboricity of the graph disappears.
We next turn to discuss how our results could potentially be used for addressing the influence maximization problem under other diffusion models. Our algorithms were based on designing a concise submodular function, the degree of a set in the hypergraphH, such that the influence of a set S can be computed much faster from the concise submodular
function representation. Looking back at the proof of theorem 6.1, only two ingredients need to hold in order to get a (1−1/e−e)-approximation in quasilinear time: the first, is that one can compute a distribution of concise submodular functions that approximate influence with weak probabilistic guarantee: only with probability at least 1−n3, the concise-submodular family approximates well the influence of every set of size at mostk. That is the guarantee of lemma 6.4. As we have shown, Lemma 6.5 can then be used for any such approximation to get good expected approximation! The second ingredient is to be able to compute, in at most quasilinear time, a maximizing set in the concise representation used (the hypergraph H in our results). As long as we use some type of graph as the concise-representation and are interested in node degrees in it, this could potentially be possible.
Unfortunately, it is not clear how to get the first ingredient (the equivalent of Lemma 6.4) from the methods we developed in this work. In particular, Lemma 6.4 heavily re- lies on key observation 6.2, which does not hold for non-independent cascading effects. Proving (or disproving) an equivalent to Lemma 6.4 for other diffusion models, as well as developing appropriate concise-representations of other diffusion processes is an interest- ing direction for future work.