The TCS Formalization Drive

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…

In Stackelberg network pricing a leader sets prices on some resources, and a follower then buys the cheapest structure available to them, paying the leader for the priced resources they use. The Stackelberg Minimum Spanning Tree problem, introduced by Cardinal, Demaine, Fiorini, Joret, Langerman, Newman and Weimann (Algorithmica 2011), is the version in…

A retailer or an airline chooses which products to offer, and customers choose among what is offered, or buy nothing. Choosing the offer set that maximizes expected revenue is the assortment problem, central to revenue management (Talluri and van Ryzin, 2004). It is NP-hard even for simple mixtures of logit models, so practice relies on heuristics. The most…

A retailer that can display only some of its products must choose an assortment: the set of products offered to an arriving customer. Customers substitute: whether a given product is bought depends on what else is on the shelf. The assortment problem asks for the offer set that maximises expected revenue under a model of this substitution behaviour. It is a…

Consensus asks $N$ processors, each holding an initial value, to agree on one value despite some of them failing. It underlies replicated databases, atomic commit and state-machine replication. Its solvability depends on the timing assumptions. With fully synchronous processors and communication, $t$ fail-stop faults can be tolerated for any $t < N$. With fully…

Consensus asks $N$ processors, each starting with a value, to agree on one value although some of them may be faulty. It underlies replicated databases, atomic commitment and state-machine replication. Two classical models bracket the problem. In a fully synchronous system, with known bounds on message delay and processor speed, consensus is solvable for many…

Graham (1966) showed that list scheduling finds a schedule within a factor $2$ of optimal for precedence-constrained jobs on identical parallel machines, the first performance guarantee for an approximation algorithm. When the machines run at different speeds (uniformly related machines), the same analysis breaks down, and for two decades the problem …

In online routing of virtual circuits, connection requests between pairs of nodes of a capacitated network arrive one at a time. Each request must be accepted and routed on a single path with bandwidth $1$, or rejected, immediately and irrevocably, and no edge may carry more than its capacity. The goal is to maximize the number of accepted requests (the…

In multiclass classification a learner sees examples $(x, y)$ with $x$ in a domain $\mathcal X$ and a label $y$ in a set $\mathcal Y$, and must predict labels of new points. When $\mathcal Y$ is finite, the Natarajan dimension characterizes PAC learnability, extending the role of the VC dimension in binary classification (Natarajan 1989; Ben-David, Cesa-Bianchi,…

The problem $1\,|\,\mathrm{prec}\,|\,\sum w_jC_j$ asks for a single-machine sequence of jobs, respecting precedence constraints, that minimizes the weighted sum of completion times. It has been known to be strongly NP-hard since Lawler (1978) and Lenstra and Rinnooy Kan (1978), several different 2-approximation algorithms are known, and closing the…

In the single-machine scheduling problem $1|\mathrm{prec}|\sum w_jC_j$, a set $N$ of $n$ jobs, each with a processing time $p_j\ge 0$ and a weight $w_j\ge 0$, is processed on one machine without interruption, subject to precedence constraints given by a partial order $P$ on $N$. The aim is to minimize the weighted sum of completion times $\sum_j w_jC_j$. The problem is…

The problem $1|\mathrm{prec}|\sum w_jC_j$ asks for an order in which to process jobs on one machine, respecting precedence constraints, so as to minimize the weighted sum of completion times. It is strongly NP-hard, and for decades the best approximation ratio known has been $2$, achieved by several unrelated algorithms (LP relaxations, Sidney decompositions,…

A retailer, an airline or an online platform decides which products to show a customer. Showing more is not always better: a customer who would have bought an expensive product may switch to a cheap one once it is offered. The assortment problem asks for the set of products that maximises expected revenue, given a model of how customers choose. It is a…

Sparse graphs with small nontrivial eigenvalues, expanders, are basic objects in combinatorics and theoretical computer science. They are used in error-correcting codes, derandomization, sorting networks, and the analysis of random walks. The Alon–Boppana bound says that a $d$-regular graph on $n$ vertices has a nontrivial eigenvalue of absolute value at least …

Linear programming, $\min\{c^\top x : Ax=b,\ x\ge0\}$ with $A\in\mathbb R^{d\times n}$, is the basic model of operations research, and the complexity of solving it is a central question of algorithm theory. Interior-point methods follow the central path: primal–dual pairs $(x,s)$ with $x,s>0$ and $x_is_i=t$ for every $i$, as the path parameter $t$ decreases to $0$. A…

We establish the polynomial simulation needed for WordRAM-based NP proofs. It reuses the same machines and Cook–Levin definitions, with uniform programs, logarithmic word widths, and standard bit input/output.

Single-machine sequencing is the base case of deterministic scheduling theory. Many multi-machine and shop problems are analysed by reduction to it, and many bounds and approximation algorithms for harder models use it as a subroutine. A central objective class is the bottleneck or minmax objective. Each job carries a nondecreasing cost of its completion…

The classical secretary problem asks how to select one valuable candidate when candidates arrive in random order and a decision must be made when each candidate appears. Many allocation settings have several goods of unequal quality instead of a single position. An employer may have roles of different desirability, or a seller may have placements with…

Set cover is one of the basic covering problems of combinatorial optimization: given a ground set and a family of subsets with costs, choose a cheapest subfamily whose union contains every element. In many applications the elements to be covered are not known in advance but appear over time: requests for a service that must be served by opening facilities,…

In the online set cover problem of Alon, Awerbuch, Azar, Buchbinder and Naor (SIAM J. Comput. 2009; preliminary version STOC 2003), a ground set and a family of subsets are known in advance, but the elements that actually need covering arrive one at a time, and each must be covered on arrival by sets chosen irrevocably. The paper's motivating example is a…

Set cover asks for the fewest sets from a family $\mathcal S$ of $m$ subsets of a ground set $X$ of $n$ elements whose union contains $X$. It is NP-hard, and the best ratio achievable in polynomial time is $\Theta(\log n)$ (Feige 1998, doi:10.1145/285055.285059).

A set function $f : 2^{\mathcal N} \to \mathbb R$ on a finite ground set $\mathcal N$ is submodular if it has diminishing returns, equivalently if $f(A) + f(B) \ge f(A \cup B) + f(A \cap B)$ for all $A, B \subseteq \mathcal N$. Cut functions of graphs and hypergraphs, coverage functions, entropy, and many facility-location and welfare objectives are submodular. Unco…

The simplex method solves linear programs by moving between vertices of a feasible polyhedron. Its worst-case number of moves can grow exponentially, yet it often performs well on ordinary inputs. Worst-case examples alone therefore give an incomplete account of the method’s behavior. Spielman and Teng introduced smoothed analysis to measure expected…