The TCS Formalization Drive

A cubic graph has chromatic index 3 or 4 (Vizing). Cubic graphs of chromatic index 4 — the uncolourable ones — are where most of the open problems on cubic graphs live: a minimal counterexample to the cycle double cover conjecture or to Tutte's 5-flow conjecture would be among them. Uncolourable graphs with small edge cuts or short cycles reduce to smaller…

A constraint satisfaction problem (CSP) asks whether variables can be assigned values from a finite domain so that every constraint, drawn from a fixed list of allowed relations, is satisfied. A promise CSP (PCSP) relaxes the question: given two templates $\mathbf A$ and $\mathbf B$ with a homomorphism $\mathbf A\to\mathbf B$, distinguish instances satisfiable in …

A snark is a cubic graph that cannot be properly edge-coloured with three colours and that does not reduce to a smaller such graph. Snarks are the smallest possible counterexamples to several central conjectures about cubic graphs, among them the cycle double cover conjecture, Tutte's 5-flow conjecture and the Berge–Fulkerson conjecture, so structural…

A data curator holds a database of $n$ individual records and wants to publish something from which many statistics can be computed, while guaranteeing differential privacy: no single record should noticeably change what is published (Dwork, McSherry, Nissim, Smith 2006). Interactive mechanisms answer queries one at a time and add noise to each answer, so…

The prophet inequality of optimal stopping compares a gambler, who sees independent non-negative random values $X_1, \dots, X_n$ one at a time and must accept or reject each on arrival, with a prophet who sees them all in advance. Krengel, Sucheston and Garling showed that the gambler can secure $\mathbb E[X_\tau] \ge \tfrac12\,\mathbb E[\max_i X_i]$, and…

MAX-2SAT asks for an assignment of Boolean variables that maximizes the total weight of satisfied clauses, each clause being a disjunction of two literals. It is one of the basic constraint satisfaction problems of approximation theory, and its approximability has been studied alongside MAX-CUT since semidefinite programming entered the field.

Search engines sell advertising slots query by query. Each advertiser states a bid per keyword and a daily budget; queries arrive one at a time, and the engine must assign each to an advertiser immediately, without knowing which queries will come later. Mehta, Saberi, Vazirani and Vazirani (J. ACM 2007) called this the adwords problem and gave a…

Barak, Goldreich, Impagliazzo, Rudich, Sahai, Vadhan and Yang proved that general-purpose program obfuscation in the virtual black-box sense is impossible (J. ACM 59(2), 2012). A natural question is whether the impossibility proof relativizes, that is, whether it survives when every party gets access to the same oracle. Proposition 4.14 of the paper shows…

A linear projection can concentrate a measure even when the measure has no atoms; how much depends on how much mass sits near the affine subspaces that the projection collapses. Quantitative versions of this principle go back to the potential-theoretic projection method of Kaufman and Mattila, which relates ball-growth bounds $\nu(B(x,r)) \le K r^\beta$ to…

Given vectors $v_1,\dots,v_N$ in the Euclidean unit ball of $\mathbb R^d$ that sum to zero, can they always be reordered so that every partial sum stays in a ball whose radius depends only on $d$? Steinitz's lemma, from his work on rearrangements of conditionally convergent vector series, says yes: in any norm, radius $d$ suffices (Grinberg–Sevast'yanov 1980). For…

Randomized load balancing asks how much a dispatcher gains from a little information. In the supermarket model, jobs arrive at $n$ servers and each job inspects a few servers chosen at random before joining one. With one random choice every server is an independent M/M/1 queue, and the expected time a job spends in the system is $1/(1-\lambda)$ at load $\lambda$.…

The learning with errors (LWE) problem underlies most of post-quantum public-key cryptography. Its hardness rests on Regev's reduction from worst-case lattice problems to LWE (Regev, J. ACM 2009). The reduction is iterative: it repeatedly turns samples of a discrete Gaussian on a lattice into samples of a narrower one. Its classical step, Lemma 3.11, needs…

Learning with errors (LWE) asks to recover a secret vector $s \in \mathbb Z_p^n$ from noisy inner products $(a_i, \langle a_i, s\rangle + e_i)$, where the $a_i$ are uniform and the errors $e_i$ are drawn from a narrow distribution $\chi$ on $\mathbb Z_p$. Regev introduced the problem and showed that solving it is at least as hard as solving worst-case lattice problems…

Prediction with expert advice asks a learner to predict a binary label on each of a sequence of trials, after seeing the predictions of a pool of candidate functions, and to make few more mistakes than the best member of the pool. Littlestone and Warmuth's The Weighted Majority Algorithm (Inform. and Comput. 108, 1994) gave the deterministic algorithm WM…

Biometric readings, physical unclonable functions and typed passphrases share two defects as cryptographic keys: they are not uniformly distributed, and they are never reproduced exactly. Dodis, Ostrovsky, Reyzin and Smith (SIAM J. Comput. 38(1), 2008; preprint arXiv:cs/0602007v4) introduced the secure sketch to handle the second defect without destroying…

Many linear programs that arise in combinatorial optimization ask only whether a convex set contains a point satisfying a family of nonnegative linear constraints. In the fractional covering problem the constraints have the form $Ax\ge b$ with $A\ge 0$: the fractional relaxation of set cover, the Held–Karp lower bound for the travelling salesman problem, and…

A pseudorandom number generator replaces repeated random sampling with a deterministic update of a small state. Its useful lifetime depends on the length and structure of the resulting orbit. In Marsaglia and Zaman’s 1991 paper, the subtract-with-borrow generators use only digit subtraction and a one-bit borrow while attaining cycles that can be almost as…

A lattice is the set of integer combinations of $n$ linearly independent vectors in $\mathbb R^n$. Many computational problems — factoring polynomials over $\mathbb Q$, simultaneous diophantine approximation, integer programming in fixed dimension, breaking knapsack cryptosystems, and the analysis of lattice-based cryptography — reduce to finding a good basis of…

A multicommodity flow problem asks to route several commodities simultaneously through a capacitated network, each from its own source to its own sink. For a single commodity, the max-flow min-cut theorem of Ford and Fulkerson says that the largest routable flow equals the capacity of the smallest separating cut. For many commodities the natural analogue…

Communication networks must route many source–destination demands through shared edges. A multicommodity flow describes how much of each demand can be carried simultaneously without exceeding an edge's capacity. In an undirected network, a cut bounds the traffic that can pass between two groups of vertices, but many commodities can interfere even when no…

Many dense combinatorial optimization problems (Max-Cut, graph bisection, quadratic assignment on dense instances) have objective functions that are sums of matrix entries over rectangles $S\times T$. If a matrix can be replaced by a simple surrogate that agrees with it on every rectangle up to a small error, such problems can be solved approximately on the…

Nearest neighbor searching asks for a data structure that stores $n$ points $S\subset\mathbb R^d$ and, given a query point $q$, returns a point of $S$ closest to $q$. It underlies classification, vector quantization, data compression, pattern recognition and similarity search in databases. In dimension $d\ge 3$, exact methods either use space growing like …

An online learner predicts a binary label for each instance before seeing that instance's correct label. Its guarantee is usually measured by its worst possible number of mistakes against a target function. Such guarantees can differ greatly across target functions: an algorithm might learn one target quickly while taking much longer on another. Littlestone…

Routing several commodities through one capacitated network at the same time is a basic problem of network optimization, with applications to communication networks, VLSI layout and divide-and-conquer graph algorithms. Its optimum is the value of a linear program, but the natural certificates of an upper bound are combinatorial: every cut of the network…

Online matching describes allocation when requests must be answered as they arrive. A matching decision uses only the edges revealed so far and cannot be revised when later requests appear. In the bipartite setting studied by Karp, U. Vazirani, and V. Vazirani, the arriving vertices are girls and the possible partners are boys. Even when the full graph has…