On minimizing budget and time in influence propagation over social networks

Abstract

In recent years, study of influence propagation in social networks has gained tremendous attention. In this context, we can identify three orthogonal dimensions—the number of seed nodes activated at the beginning (known as budget), the expected number of activated nodes at the end of the propagation (known as expected spread or coverage), and the time taken for the propagation. We can constrain one or two of these and try to optimize the third. In their seminal paper, Kempe et al. constrained the budget, left time unconstrained, and maximized the coverage: this problem is known as Influence Maximization (or MAXINF for short). In this paper, we study alternative optimization problems which are naturally motivated by resource and time constraints on viral marketing campaigns. In the first problem, termed minimum target set selection (or MINTSS for short), a coverage threshold η is given and the task is to find the minimum size seed set such that by activating it, at least η nodes are eventually activated in the expected sense. This naturally captures the problem of deploying a viral campaign on a budget. In the second problem, termed MINTIME, the goal is to minimize the time in which a predefined coverage is achieved. More precisely, in MINTIME, a coverage threshold η and a budget threshold k are given, and the task is to find a seed set of size at most k such that by activating it, at least η nodes are activated in the expected sense, in the minimum possible time. This problem addresses the issue of timing when deploying viral campaigns. Both these problems are NP-hard, which motivates our interest in their approximation. For MINTSS, we develop a simple greedy algorithm and show that it provides a bicriteria approximation. We also establish a generic hardness result suggesting that improving this bicriteria approximation is likely to be hard. For MINTIME, we show that even bicriteria and tricriteria approximations are hard under several conditions. We show, however, that if we allow the budget for number of seeds k to be boosted by a logarithmic factor and allow the coverage to fall short, then the problem can be solved exactly in PTIME, i.e., we can achieve the required coverage within the time achieved by the optimal solution to MINTIME with budget k and coverage threshold η. Finally, we establish the value of the approximation algorithms, by conducting an experimental evaluation, comparing their quality against that achieved by various heuristics.

Appendix

1.1 A Proof of Lemma 2

Suppose there exists an algorithm $\mathcal{A}$ that selects β k sets which covers γ η elements. Apply $\mathcal{A}$ to an arbitrary instance $\langle \mathcal{U}, \mathcal{S}, \eta \rangle$ of PSC. The output is a collection of sets $\mathcal{C}_1$ such that $|\mathcal{C}_1| \le \beta k$ and $\left| { \cup _{{s \in c_{1} }} S} \right|{ \ge } \gamma \eta$ Next, discard the sets that have been selected and the elements they cover, and apply again the algorithm $\mathcal{A}$ on the remaining universe. Repeat this process until 1 or fewer elements are left uncovered.⁷

Let η _i denote the number of elements uncovered after iteration i. In iteration i, the algorithm picks β k sets and covers at least γ η _i−1 elements. Hence, $\eta_i \le \eta_{i-1} \cdot (1 - \gamma).$ Expanding, $\eta_i \le \eta \cdot (1 - \gamma)^i.$ Suppose after l iterations, η _l  = 1. The total number of sets picked is $l\beta k. \eta \cdot (1 - \gamma)^l = 1$ implies $l = \frac{\ln \eta}{\ln \frac{1}{1-\gamma}}.$

We now prove the first claim. Let γ > 1 − 1/e ^β, then $\ln \left( \frac{1}{1-\gamma} \right) > \beta.$ This yields a PTIME algorithm for PSC which outputs a solution of size $l \beta k = \beta k \cdot \ln \eta / \ln \frac{1}{1-\gamma} \le c \cdot k \ln \eta$ (for some c < 1) This yields an $c \cdot \ln \eta$-approximation for PSC for some c < 1, which is not possible unless ${\rm NP} \subseteq \text{DTIME}(n^{O(\log \log n)})$ (Feige 1998).

To prove the second claim, assume $\beta \le (1 - \delta) \ln \left( \frac{1}{1 -\gamma} \right).$ This gives a PTIME algorithm for PSC which outputs a solution of size $l \beta k = \beta k \cdot \ln \eta / \ln \frac{1}{1-\gamma} \le (1 - \delta) k \cdot \ln \eta$ which is not possible unless ${\rm NP} \subseteq \text{DTIME}(n^{O(\log \log n)}).$ $\quad\square$

1.2 B Example illustrating performance of Wolsey’s solution

Wolsey (1982) studied the RSSC problem and showed, among many things, that the greedy algorithm provides a solution that is within a factor of $1 + \ln (\eta/(\eta-f(S_{t-1}))$ of the optimal solution. Unfortunately, this does not yield an approximation algorithm with any guaranteed bounds. The following example shows the greedy solution with threshold η can be arbitrarily worse than the optimum.

Example

(Illustrated also in Fig. 4). Consider a ground set $\mathcal{X} = \{w_1, w_2, v_1, v_2, \ldots, v_l\}$ with elements having unit costs. Figure 4 geometrically depicts the definition of a function ${f: 2^{\mathcal{X}} {\rightarrow}\;\mathbb{R}, }$ where for any set $S \subset \mathcal{X},\;f(S)$ is defined to be the area (shown shaded) covered by the elements of S. Specifically, f(w ₁) = f(w ₂) = 1 − 1/2^l+1 and f(v _i ) = 1/2ⁱ⁻¹, 1 ≤ i ≤ l. Notice, $f(\{v_1, \ldots, v_l\}) = \Upsigma_{i=1}^l 1/2^{i-1} = 2 - 1/2^{l-1} < 2 - 1/2^l = f(\{w_1, w_2\}).$ The greedy algorithm will first pick v ₁. Suppose it picks $S = \{v_1,\ldots, v_i\}$ in i rounds. Then f(S∪{v _i+1}) − f(S) = 1/2ⁱ > 1 − 1/2^l+1 − 1 + 1/2ⁱ = 1 − 1/2^l+1 − 1/2(2 − 1/2ⁱ⁻¹) = f(S ∪ {w ₁}) − f(S). Thus, greedy will never pick w ₁ or w ₂ before it picks $v_1,\ldots, v_l.$ Suppose η = 2 − 1/2^l. Clearly, the greedy solution is $\mathcal{X}$ whereas the optimal solution is {w ₁, w ₂}. Here l can be arbitrarily large.

Fig. 4

Example. Rectangles represent the elements in the universe. The shaded area within a rectangle represents the coverage function f for the element. e.g., f(v ₁) = 1/2 + 1/2 = 1