The TCS Formalization Drive

The simplex method, introduced by G. B. Dantzig in 1947, is the standard algorithm for linear programming and remains the core of commercial solvers. It moves from one basic feasible solution to another by pivot steps, and at each step a pivot rule chooses which variable enters and which leaves the basis. For several natural rules, including Dantzig's…

We formalize the equivalence between Turing machines and the WordRAM model of computation. Turing machines are the standard model for studying computability, but algorithms are rarely described in terms of tape operations. WordRAM is much closer to assembly, with memory accesses, arithmetic instructions, branches, and loops. The goal is to connect this…

Scheduling indivisible jobs on parallel machines to finish all of them as early as possible is a basic problem in operations research and in the theory of algorithms. In the unrelated machines model each job may take a different time on each machine, with no relation between the rows of the time table, as when machines of different types (black-and-white,…

The discrete logarithm problem modulo a prime asks, given a prime $p$, a generator $g$ of the multiplicative group modulo $p$, and a nonzero residue $x$, for the exponent $r$ with $g^r\equiv x \pmod p$. Its presumed classical hardness underlies Diffie–Hellman key exchange, ElGamal encryption and the Digital Signature Algorithm. The best classical algorithm known when…

Shor's 1997 paper (SIAM J. Comput. 26(5), arXiv:quant-ph/9508027) gives a polynomial-time quantum algorithm for factoring integers. The quantum computer does not factor directly: it finds the multiplicative order of an element modulo $n$. The step from order finding to factoring is classical and randomized, and goes back to Miller's 1976 work on primality…

Shor's factoring and discrete logarithm algorithms (Shor 1997) reduce both problems to sampling from the output of a quantum Fourier transform: a register holding a superposition with a hidden period is transformed, then measured, and the measured value carries information about the period. The whole speed-up rests on one engineering fact: for $q = 2^l$ the …

A process on a shared-memory multiprocessor that finds a lock held must decide what to do while it waits. It can spin, repeatedly testing the lock and occupying its processor, or it can block, giving the processor to another process and paying a fixed context-switch cost $C$ to be descheduled and later restored. Spinning is cheap when the lock is released…

Minimizing the total weighted completion time $\sum_j w_j C_j$ of jobs with release dates on a single machine, written $1\,|\,r_j\,|\,\sum w_j C_j$ in scheduling notation, is strongly NP-hard. Most constant-factor approximation algorithms for it, and for many related scheduling problems, follow one pattern: solve a linear programming relaxation, which gives a…

Two optimization problems on weighted networks sit at the base of operations research and algorithm design. The first asks for the cheapest way to connect every node of a network, such as a cable, pipeline or communication network. The answer is a minimum weight spanning tree. The second asks for the shortest route from a depot to every other node of a road…

Insertion heuristics build a traveling salesman tour one city at a time: start from a single city, and at each step choose a city not yet on the subtour and splice it into the subtour where it lengthens the subtour least. Local search heuristics start from a tour and repeatedly replace a few of its edges by others while this shortens the tour. Both families…

MAXIMUM SATISFIABILITY asks for a truth assignment satisfying as many clauses of a given set as possible. It was among the first NP-hard optimization problems for which an approximation algorithm came with a proven worst-case guarantee. In Approximation Algorithms for Combinatorial Problems (J. Comput. System Sci. 9, 1974, doi:10.1016/S0022-0000(74)80044-9),…

The traveling salesman problem (TSP) asks for a shortest closed route through a finite set of points. It is NP-hard, and in practice tours are built by fast constructive heuristics whose output is then improved or used as is. A central question in the analysis of algorithms, raised in this form by Rosenkrantz, Stearns and Lewis in 1977, is how far such a…

Supervisory control theory, introduced by Ramadge and Wonham (SIAM J. Control Optim. 25(1), 1987), models a manufacturing cell, a communication protocol or a resource-sharing system as an automaton whose transitions are events, some of which an external controller may disable. A controller, the supervisor, is itself an automaton that watches the event…

The edit distance between two strings is the least total cost of a sequence of single-character insertions, deletions and replacements that turns one string into the other. It underlies spelling correction, sequence alignment in computational biology, and file comparison. Wagner and Fischer (J. ACM 21, 1974) computed it for strings of length $n$ in time $O(n^2)$…

Many optimization problems ask for a subset of a finite ground set that maximizes a function with diminishing returns: the cut of a graph or a directed graph, the value of a facility-location configuration, the entropy of a set of random variables, or a welfare function in combinatorial auctions. These functions are submodular but typically non-monotone:…

The k-median problem asks to open $k$ facilities among a set of candidate sites so that the total distance from clients to their nearest open facility is as small as possible. It is a basic model of facility location and of clustering, and it is NP-hard, so the question studied in approximation algorithms is how close a polynomial-time method can get to the…

Deciding whether a graph has a perfect matching, and finding one, are basic problems of combinatorial optimization; Edmonds' blossom algorithm solves them sequentially in polynomial time. The question behind this paper is whether they can also be solved in parallel, in polylogarithmic time on polynomially many processors (the class NC, or RNC when random…

Set covering asks for the fewest members of a family of sets whose union is everything the family covers. It models crew scheduling, facility siting, test-suite reduction, logic minimization and fault testing; Johnson names the last two as its practical applications. Karp showed in 1972 that the decision version is NP-complete (Karp 1972), so in practice…

Many problems in scientific computing and statistics ask for a solution of a linear system $Ax\approx b$ that uses as few unknowns as possible. In statistics this is subset selection (Golub and Van Loan, Matrix Computations, 1983). In coding theory over binary matrices it is the minimum weight solution problem (Gallager, 1968). Natarajan's own motivation was r…

Maximum satisfiability asks for a truth assignment satisfying as many clauses of a propositional formula as possible. The paper notes that the restriction $MS(k)$, in which every clause has at least $k$ literals, is polynomial complete for every $k \ge 1$, so exact optimization is out of reach in general and one asks instead how close a fast algorithm is…

David S. Johnson's 1974 paper Approximation Algorithms for Combinatorial Problems (J. Comput. System Sci. 9 (1974) 256–278) is one of the founding papers of the theory of approximation algorithms. It asks, for optimization problems whose decision versions Karp had just shown to be polynomial complete, how close a fast heuristic can be guaranteed to come to…

A single machine must process a set of jobs, each with a processing time and a due-date, and a job that finishes after its due-date is late. Counting late jobs is the natural objective when a late order is simply lost, whatever its lateness. In the three-field notation of scheduling theory this is the problem $1\,\|\,\sum U_j$, and it is one of the few…

The nearest common ancestor problem asks, for a rooted tree and two of its vertices $v$ and $w$, for the deepest vertex that is an ancestor of both, written $\operatorname{nca}(v,w)$. It is a subroutine in string and graph algorithms.