In this thesis we explore two methods of computing lower bounds. We first discuss the Lagrangian Relaxation as it applies to the Golomb ruler problem, and then we explore adding multi-valued decision diagrams to an additive bounding scheme. The Golomb Ruler Problem asks to position n integer marks on a ruler such that all pairwise distances between the marks are distinct and the ruler has minimum total length. It is a notoriously challenging combinatorial problem, and provably optimal rulers

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... only known for n up to 27. Lower bounds can be obtained using linear programming (LP) formulations, but these are computationally expensive for large n. In Chapter 2 of this thesis, we propose a new method for finding lower bounds based on a Lagrangian relaxation. We apply a subgradient optimization scheme to find good bounds quickly, and we show experimentally that our method can find bounds that are very close to the optimal LP bound in a fraction of the time that is needed to compute the LP bound. We furthermore embed our Lagrangian bounds into a constraint programming search procedure, and show that these can help reduce the constraint programming search tree considerably. Additive bounding is a method of taking several algorithms for computing lower bounds, each of which typically exploits a different substructure of the problem, and combining them to produce a single lower bound which is larger than the lower bound that any of the individual algorithms can produce alone. Approximate multi-valued decision diagrams (MDDs) have recently been used to compute upper and lower bounds on several optimization problems. In Chapter 3 of this thesis, we show how we can integrate MDDs into an addivite bounding scheme.