The TCS Formalization Drive

A processor can save energy by running slowly when work is light, but a deadline may force it to run faster. Yao, Demers, and Shenker formulated this tradeoff as an offline and online scheduling problem for a variable-speed processor in their 1995 FOCS paper. Their Average Rate Heuristic (AVR) uses only each job's arrival, deadline, and required work to set…

A network designer may need a cheap connection that keeps the number of links at each site within a local capacity. The minimum bounded-degree spanning tree problem asks for the cheapest spanning tree of a weighted, undirected graph whose vertex degrees respect prescribed upper bounds. The capacity constraints matter in settings where a low-cost tree alone…

In online prediction, a learner makes a forecast before seeing the next label. A collection of other forecasters may offer useful advice, but the learner need not know beforehand which forecaster will perform well. Littlestone and Warmuth's Weighted Majority Algorithm (WM) gives a deterministic way to combine such advice. Its guarantees hold for every…

A job shop is the basic model of a factory floor in which every job follows its own route through the machines. Minimizing the time at which the last job is finished (the finish time, or makespan) is one of the central problems of deterministic scheduling theory, and the question of which special cases admit efficient algorithms has organized much of the…

A multicommodity flow problem asks to route many source–sink demands through one capacitated network at the same time. Its min-cut (sparsest cut) is the smallest ratio, over all cuts, of the capacity crossing the cut to the demand separated by it. Routing is always limited by every cut, so the max-flow never exceeds the min-cut. For a single commodity the…

A finite directed graph on $d$ vertices is recorded by its adjacency matrix, a $d \times d$ array of zeros and ones. Many questions about the graph are questions about reachability: can vertex $j$ be reached from vertex $i$ by following one or more edges? In the late 1950s this question arose in the analysis of programs. Prosser (Applications of Boolean matrices to…

Processors in a distributed system may hold separate readings of a sensor, clock, or diagnostic result while some processors give misleading reports. The aim of interactive consistency is for every functioning processor to compute the same vector of reported values and to put each functioning processor's actual private value in its corresponding position. A…

A fault-tolerant system built from several processors must let its working processors act on the same data, even though some processors may have failed in ways nobody can predict: a failed processor may stop relaying information, relay something it invented, or tell different processors different things. Pease, Shostak and Lamport (J. ACM 27 (1980) 228–234)…

A fault-tolerant system often replicates a computation on several processors and needs the nonfaulty ones to act on the same data, even though some processors may fail in arbitrary ways: send wrong values, send different values to different recipients, or lie about what they were told. Pease, Shostak and Lamport posed the question in 1980, motivated by the…

Prediction with expert advice asks a learner to predict a binary label in each of a sequence of trials, having seen the predictions of a pool of $n$ prediction algorithms (the experts), and to make not many more mistakes than the best of them, with no probabilistic assumption on how the instances and labels are generated. Littlestone and Warmuth's Weighted…

In prediction with expert advice, a master algorithm predicts a binary label in each of a sequence of trials, after seeing the predictions of a pool of other algorithms (the "experts"), and is charged one mistake whenever its prediction differs from the label. No probabilistic assumption is made on the sequence: the goal is a bound on the master's mistakes,…

All-pairs shortest paths (APSP) asks for the exact distance between every ordered pair of vertices in an $n$-vertex weighted directed graph. Floyd–Warshall solves it in $O(n^3)$ time, and for decades no algorithm beat $n^{3-\delta}$ for any constant $\delta>0$ on graphs with polynomially bounded integer weights. The APSP conjecture of fine-grained complexity…

Scheduling unrelated parallel machines to minimize the makespan, written $R\,\|\,C_{\max}$, is one of the basic NP-hard problems of scheduling theory. A set of jobs must each be run on one of several machines, and the time a job takes depends arbitrarily on the machine; the goal is to finish all jobs as early as possible. Before the work formalized here, the…

A flow shop is the simplest model of a production line: every job passes through the same sequence of machines in the same order. Minimizing the time at which the last job leaves the last machine (the finish time, or makespan) is easy for two machines, where Johnson's rule gives an optimal schedule in time $O(n\log n)$ (Johnson 1954), and NP-hard from three…

Flow shops and job shops are the basic models of multi-stage production: every job passes through several machines, and each machine handles one task at a time. Gonzalez and Sahni (Operations Research 26(1), 1978) showed that minimizing the finish time of such shops is NP-complete even with preemption, and then asked how well simple heuristics do. One of…

In a job shop, each job has an ordered list of tasks, and each task must use a specified machine. A production planner may be able to interrupt a task and resume it later, yet still be unable to find a short schedule efficiently. Gonzalez and Sahni studied how the ability to preempt changes the difficulty of minimizing the time at which all jobs are…

Shop scheduling asks how to process a set of jobs, each a sequence of tasks on prescribed machines, so that all work is completed as early as possible. The flow shop, in which every job visits the machines in the same order, is the most studied of these models in operations research. For two machines, Johnson's rule (1954) computes a schedule of minimum…

Scheduling work on several identical processors creates a plausible expectation: completing tasks sooner, removing constraints, or adding a processor should not delay the overall finish. A fixed priority-list rule defeats that expectation. When a task becomes ready earlier, it can occupy a processor that would otherwise have run another task; that decision…

Parallel processing poses a basic allocation question: given tasks of known lengths and several identical processors, how much time is lost when tasks are assigned by a simple list rule instead of an optimal allocation? The finishing time is the moment the last processor completes its work. For independent tasks, a list rule starts each next task on a…

3SUM asks whether three distinct positions in a list of integers contain values summing to zero. It is a central problem in fine-grained complexity.

Many network optimization algorithms spend most of their time in a heap (priority queue): a set of items with real keys that supports inserting an item, finding and deleting an item of minimum key, melding two heaps, decreasing the key of an item, and deleting an arbitrary item. Dijkstra's shortest-path algorithm performs at most one delete min per vertex…

All-pairs shortest paths (APSP) computes the shortest distance between every pair of vertices in a weighted graph. It is a fundamental problem in graph algorithms and a fundamental problem in fine-grained complexity.

Minimum makespan scheduling on unrelated parallel machines, written $R\,||\,C_{\max}$, is one of the basic models of scheduling theory: $n$ jobs must each be run on one of $m$ machines, and the time a job takes depends arbitrarily on the machine. Lenstra, Shmoys and Tardos (CWI Report OS-R8714, 1987; journal version Math. Programming 46, 1990) gave a polynomial…

Minimum makespan scheduling on unrelated parallel machines asks for an assignment of $n$ jobs to $m$ machines, where job $j$ takes $p_{ij}$ time units on machine $i$, so that the largest machine load is as small as possible. In the three-field notation of Graham, Lawler, Lenstra and Rinnooy Kan (1979) it is $R\,\|\,C_{\max}$. It models load balancing across…