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Submodular functions are an important class of functions in combinatorialoptimization which satisfy the natural properties of decreasing marginal costs.The study of these functions has led to strong structural properties withapplications in many areas. Recently, there has been significant interest inextending the theory of algorithms for optimizing combinatorial problems (suchas network design problem of spanning tree) over submodular functions.Unfortunately, the lower bounds under the general class of submodular functionsare known to be very high for many of the classical problems.  In this paper, we introduce and study an important subclass of submodularfunctions, which we call discounted price functions. These functions aresuccinctly representable and generalize linear cost functions. In this paper westudy the following fundamental combinatorial optimization problems: EdgeCover, Spanning Tree, Perfect Matching and Shortest Path, and obtain tightupper and lower bounds for these problems.  The main technical contribution of this paper is designing novel adaptivegreedy algorithms for the above problems. These algorithms greedily build thesolution whist rectifying mistakes made in the previous steps.

Optimal Approximation Algorithms for Multi-agent Combinatorial Problems with Discounted Price Functions