Effective bounds on the union probability are well known to be beneficial in the analysis of stochastic problems in many areas, including probability theory, information theory, statistical communications, computing and operations research. In this work we present new results on bounding the probability of a finite union of events, ${P\left(\bigcup_{i=1}^N A_i\right)}$, for a fixed positive integer ${N}$, using partial information on the events in terms of ${\{P(A_i)\}}$ and ${\{\sum_j c_j P(A_i\cap A_j)\}}$ where ${c_1}$, ${\dots}$, ${c_N}$ are given weights. We derive two new classes of lower bounds of at most pseudo-polynomial computational complexity. These classes of lower bounds generalize the existing bound in \cite{Kuai2000} and recent bounds in \cite{Yang2014,Yang2014ISIT} and are numerically shown to be tighter in some cases than the Gallot-Kounias bound \cite{Gallot1966,Kounias1968} and the Pr{é}kopa-Gao bound \cite{Prekopa2005} which require more information on the events probabilities.