Some intractable problems in digital signal processing

Over the past decade a large class of problems, called NP-complete[1], have been shown to be equivalent in the sense that if a fast algorithm can be found for one, fast algorithms can be found for all. At the same time, despite much effort, no fast algorithms have been found for any, and these problems are widely regarded as intractable. This class includes such notoriously difficult problems as the traveling salesman problem, graph coloring, and satisfiability of Boolean expressions. This paper describes some problems in digital signal processing which are NP-complete. These include: (1) Minimize the number of registers required to implement a signal flow graph; (2) Minimize the time to perform the additions (multiplications) of a signal flow graph using P adders (multipliers); (3) Minimize the computational cost for multiplication by a fixed matrix. Large-scale instances of such problems may become important with the use of VLSI technology to implement signal processing.