The TCS Formalization Drive

Search engines sell advertising space query by query. Each advertiser states a bid for each keyword and a daily budget; queries arrive one at a time, and each must be assigned to an advertiser immediately, without knowledge of the queries still to come. The revenue-maximizing assignment of the day can only be computed offline. The adwords problem asks how…

In shop scheduling, jobs consist of chains of operations, each to be processed on a prescribed machine, and the goal is to minimize the makespan, the time at which the last operation finishes. Almost every approximation algorithm for job shops, acyclic job shops and flow shops is analysed against one quantity: the trivial lower bound $\mathrm{lb}=\max(C,D)$,…

A sparse graph that approximates a dense one, in the sense that both have nearly the same Laplacian quadratic form, can stand in for it in every algorithm that only reads cuts or solves Laplacian linear systems. Spielman and Teng introduced such spectral sparsifiers and built them in nearly linear time; the construction is the sparsification step of their…

Many graph algorithms run in time proportional to the number of edges. A sparsifier of a weighted graph $G$ is a graph $H$ on the same vertices with far fewer edges that approximates $G$ for the purpose at hand, so that the algorithm can be run on $H$ instead. Benczúr and Karger (1996) showed that every graph has a sparsifier with $O(n\log n/\varepsilon^2)$ edges…

In a Maker–Breaker achievement game, two players alternately claim cells of a board; Maker wins by owning every cell of some copy of a fixed shape, and Breaker only tries to prevent this. Polyomino achievement games on the infinite square grid, introduced by Frank Harary, ask which small polyominoes are winners (Maker can force a copy) and which are losers (…

A matroid abstracts linear independence: a finite ground set with a family of "independent" subsets that is closed under subsets and satisfies the augmentation axiom. Its maximal independent sets, the bases, all have the same size, the rank. Matroid intersection, finding a common basis of two matroids, is one of the central polynomially solvable problems of…

In the local model of differential privacy, each individual randomizes their own data before handing it to an untrusted learner. Local protocols are the deployed form of private data collection, and in practice every round of interaction with the population is expensive: the learner must broadcast new instructions and wait for new reports. Kasiviswanathan,…

Automatic verification of programs that manipulate pointers needs procedure specifications: preconditions saying which part of the heap a procedure touches and postconditions saying what it leaves behind. Writing them by hand does not scale to millions of lines of code. Calcagno, Distefano, O'Hearn and Yang (J. ACM 58(6), 2011) infer them compositionally…

The scheduling problem $R\,|\,r_j\,|\sum C_j$ asks for a nonpreemptive schedule of jobs on unrelated parallel machines, each job with a machine-dependent processing time and a release date, so that the total completion time is minimal. It is NP-hard, and the question that remains is how well it can be approximated in polynomial time. Positive results include…

The Learning with Errors (LWE) problem, introduced by Regev in On Lattices, Learning with Errors, Random Linear Codes, and Cryptography (J. ACM 56(6), 2009, Article 34; doi:10.1145/1568318.1568324), asks to recover a secret vector $s \in \mathbb{Z}_p^n$ from noisy inner products $\langle a, s\rangle/p + e \bmod 1$. The noise $e$ is drawn from a periodic normal…

In sequential prediction, several forecasters can issue different answers for the same instance. A master algorithm must combine their advice before the true label is known. Littlestone and Warmuth's Weighted Majority framework gives worst-case guarantees for this setting: the instances and labels need not come from a probability distribution. The paper…

An oracle lets an algorithm ask for the value of a function without giving it a description of that function. An oracle lower bound isolates how many questions are needed, even when computation between questions is unrestricted. Bennett, Bernstein, Brassard and Vazirani used this viewpoint to compare classical and quantum access to randomly chosen functions…

Scheduling independent jobs on unrelated parallel machines to minimize the makespan is one of the basic NP-hard problems of scheduling theory: each job has its own processing time on each machine, with no relation between machines, and the goal is to finish all jobs as early as possible. When the number of machines is part of the input, Lenstra, Shmoys and…

Many graph problems in routing, VLSI layout and scheduling ask for a cut that separates a lot of demand while cutting little capacity. For a single commodity, the max-flow min-cut theorem of Ford and Fulkerson says that the largest flow equals the smallest cut. For several commodities sharing one network this equality fails, and the question becomes how far…

For a single commodity, the Ford–Fulkerson theorem says that the maximum flow between two nodes equals the minimum capacity of a cut separating them. Many routing and partitioning problems involve many source–sink pairs at once, and for those the equality fails: the largest fraction of every demand that can be routed simultaneously can be strictly smaller…

Programs often reveal information through the memory locations they touch, even when the contents of those locations are hidden. An oblivious RAM simulation aims to conceal this access pattern. Goldreich and Ostrovsky studied the cost of doing so for arbitrary RAM programs and showed that the access sequence itself imposes a lower bound on simulation…

A seller with $n$ distinct items faces $m$ consumers, each of whom wants at most one item. Choosing item prices to maximise revenue in this setting is the unit-demand envy-free pricing problem, introduced by Rusmevichientong and studied in algorithmic game theory and revenue management since the mid-2000s. Optimal prices are hard to compute in general, so the…

Simulation, Monte Carlo integration and randomized algorithms consume long streams of numbers that should behave like independent uniform draws. A deterministic generator with finite state is eventually periodic, so its period is the first figure of merit: a stream cannot be longer than its cycle. G. Marsaglia and A. Zaman, A new class of random number…

Prediction with expert advice asks a master algorithm to combine the binary predictions of a pool of $n$ prediction algorithms, trial after trial, so that it makes few more mistakes than the best member of the pool, without any probabilistic assumption on the data. Littlestone and Warmuth's Weighted Majority Algorithm WM (Littlestone and Warmuth 1994) keeps a…

Factoring a polynomial with rational coefficients into irreducible factors is one of the basic tasks of computer algebra. Before 1982 the standard method (Berlekamp's algorithm modulo a small prime, Hensel lifting, then recombination of the $p$-adic factors, as in Zassenhaus's approach) could take exponential time, because the number of ways to recombine…

In a store-and-forward network, messages are cut into packets that travel from node to node along wires, one wire per time step, and wait in buffers between moves. Routing such traffic splits into two problems: choosing a path for each packet, and scheduling the packets along their paths, deciding at every step which packets move and which wait. Leighton,…

A flow shop is the simplest multi-stage production model: every job passes through the same machines in the same order, and the question is how to sequence the work so that everything is finished as early as possible. Gonzalez and Sahni's 1978 paper in Operations Research settled the computational complexity of the preemptive version of this problem, in…

An online matching algorithm must commit to each assignment when a request arrives, before it sees later requests. This limitation arises whenever waiting for the full instance is impossible: an available resource can be assigned to a current request or saved for an unknown future request. The quality of the assignment is measured by how many requests can…