One of the most fundamental problems in Computer Science is the Knapsack
problem. Given a set of n items with different weights and values, it asks to
pick the most valuable subset whose total weight is below a capacity threshold
T. Despite its wide applicability in various areas in Computer Science,
Operations Research, and Finance, the best known running time for the problem
is O(Tn). The main result of our work is an improved algorithm running in time
O(TD), where D is the number of distinct weights. Previously, faster runtimes
for Knapsack were only possible when both weights and values are bounded by M
and V respectively, running in time O(nMV) [Pisinger'99]. In comparison, our
algorithm implies a bound of O(nM^2) without any dependence on V, or O(nV^2)
without any dependence on M. Additionally, for the unbounded Knapsack problem,
we provide an algorithm running in time O(M^2) or O(V^2). Both our algorithms
match recent conditional lower bounds shown for the Knapsack problem [Cygan et
al'17, K\"unnemann et al'17].
We also initiate a systematic study of general capacitated dynamic
programming, of which Knapsack is a core problem. This problem asks to compute
the maximum weight path of length k in an edge- or node-weighted directed
acyclic graph. In a graph with m edges, these problems are solvable by dynamic
programming in time O(km), and we explore under which conditions the dependence
on k can be eliminated. We identify large classes of graphs where this is
possible and apply our results to obtain linear time algorithms for the problem
of k-sparse Delta-separated sequences. The main technical innovation behind our
results is identifying and exploiting concavity that appears in relaxations and
subproblems of the tasks we consider
