The discrete polymatroid is a multiset analogue of the matroid. This paper studies the combinatorics and the algebra on discrete polymatroids. An algebraic aspect of this paper is to study Ehrhart rings and base rings together with toric ideals of discrete polymatroids. One result on toric ideals of discrete polymatroids given in this paper is Theorem. (a) Suppose that each matroid has the property that the toric ideal of its base ring is generated by symmetric exchange relations, then this is also true for each discrete polymatroid. (b) If \(P \subset \mathbb{Z}^n_+\) is a discrete polymatroid whose set of base \(B\) satisfies the strong exchange property (i.e., if \(u =(u_1,\ldots, u_n),\;v=(v_1,\ldots, v_n)\in B\), then for all \(i\) and \(j\) with \(u_i >v_i\) and \(u_j