The TCS Formalization Drive

Many welfare-maximization problems in mechanism design are NP-hard: combinatorial auctions, packing integer programs, unsplittable flow, maximum traveling salesman. The standard algorithmic answer is relax-and-round: solve a fractional relaxation exactly, then round the fractional solution to a feasible one. Turning such an algorithm into a mechanism is the…

Data centers and communication networks spend a large share of their operating cost on energy. A powered-on device pays a fixed idle cost, and with speed scaling it pays a power cost that grows like $x^{\alpha}$ in its load $x$, with $\alpha$ typically between $1$ and $3$. Andrews, Fernández Anta, Zhang and Zhao (IEEE/ACM Trans. Networking 20(1), 2012) proposed the…

The statistical query (SQ) model describes algorithms that never see individual samples of a distribution $D$; they only ask for approximate expectations $\mathbf E_{w\sim D}[\phi(w)]$ of functions $\phi$ of their choice. It was introduced by Kearns for noise-tolerant learning (Kearns 1998), and it has since become the standard abstraction for algorithms that can…

Ride-sharing platforms match passengers who request rides at different times and places. Three features of that problem are hard to separate: two passengers far apart yield a poor match, two requests far apart in time cannot be matched at all, and the platform does not know future demand. Ashlagi, Burq, Dutta, Jaillet, Saberi and Sholley (arXiv:1808.03526v1,…

A Mastermind player sees guesses and feedback but cannot inspect the hidden sequence of colors. Before asking which guess to make next, one can ask whether the observations are consistent with any sequence at all. This is the Mastermind satisfiability problem, or MSP. It separates the logical content of the observations from the strategic problem of…

In Multiway Cut one is given an edge-weighted graph and $k$ terminals, and must delete a minimum-weight set of edges so that no two terminals remain connected. Equivalently, one labels every vertex by a terminal, each terminal by itself, and pays for the edges whose endpoints get different labels. The problem is APX-hard for every fixed $k\ge3$ (Dahlhaus et…

Training a classifier asks for parameters that fit observed examples. Even when the data can be classified perfectly, the task of finding suitable parameters can have substantial computational cost. The question here concerns exact training: deciding whether any parameter choice makes zero mistakes on the entire supplied data set. It concerns computation on…

A prophet inequality compares an online decision maker, who sees random values one at a time and must accept or reject each irrevocably, with a "prophet" who sees all values in advance. The classical single-item version (Krengel–Sucheston, Samuel-Cahn) guarantees half of the prophet's expected value. Since the work of Hajiaghayi, Kleinberg and Sandholm…

A network failure can remove several connections at once. When the connections share a physical resource, paying for or disabling that resource acts on an entire class of edges. The minimum label s-t cut problem captures this distinction: edges carry labels, and the cost of removing a label is paid once, regardless of how many edges have that label. Zhang…

A puzzle can have simple local rules while the choice of moves has global consequences. Clickomania, also known as Same Game, is an example: a player removes adjacent blocks of one color, and the remaining blocks move into the vacated space. The computational question asks whether a given starting position can be cleared completely. Biedl, Demaine, Demaine,…

Boolean satisfiability asks whether a collection of logical requirements can hold simultaneously. Maximum satisfiability instead asks how many requirements can hold, while sometimes retaining a separate collection that must hold. Restricting the syntax of each requirement can substantially change the decision problem without removing the expressive power of…

Quantum Max Cut asks for the highest energy of a two-qubit interaction Hamiltonian placed on the edges of a weighted graph. It is a quantum analogue of Max Cut: a classical cut assigns one of two labels to each vertex, while a quantum state can use correlations and entanglement across the qubits. Lee studies an approximation algorithm that first solves a…

Classical Max Cut asks for a partition of the vertices of a weighted graph that maximises the total weight of edges crossing it. Its best-known approximation algorithm, due to Goemans and Williamson, rounds a semidefinite relaxation and guarantees ratio $0.878$. Quantum Max Cut is the analogous problem for a family of local Hamiltonians: each edge contributes…

Approximate counting asks for the number $K$ of marked items in a list of $N$ items, to within a multiplicative factor $1\pm\varepsilon$, when the marked set can only be accessed through a membership oracle. Brassard, Høyer, Mosca and Tapp (arXiv:quant-ph/0005055) gave a quantum algorithm that uses $O(\sqrt{N/K}\,\varepsilon^{-1})$ queries, which is optimal. Their…

Many quantum algorithms act on a matrix $A$ that is not itself unitary: a Hamiltonian to be simulated, a linear system to be inverted, a projector to be amplified. The standard way to hand such a matrix to a quantum computer is to embed it as a block of a unitary $U$, so that $A=\widetilde\Pi U\Pi$ for two orthogonal projectors $\Pi,\widetilde\Pi$. The question is…

The Quantum Approximate Optimization Algorithm (QAOA) of Farhi, Goldstone and Gutmann (arXiv:1411.4028) is a family of shallow quantum circuits for combinatorial optimization. At its lowest level, $p = 1$, the circuit has two real parameters, and its output is a random bit string. The question is how many constraints that string satisfies, compared with the…

In a linear exchange market every agent owns one divisible good and has linear utilities over all goods; an equilibrium is a price vector at which every agent can spend its budget on goods that maximize utility per unit of money, and all goods are sold. Computing such an equilibrium in strongly polynomial time, with a number of arithmetic operations bounded…

Prove equivalence between two WordRAM models: one with unsigned pointers, bit operations, and shifts; the other with signed pointers and branching on sign. Both support bounded-word arithmetic and multiplication.

Many matrix algorithms spend most of their time reading their input. A matrix with few nonzero entries should admit routines whose cost scales with that number, rather than with the product of its dimensions. Clarkson and Woodruff study random maps that compress a matrix before a regression or low rank approximation calculation while retaining the geometric…

Many graph classes are defined by excluding a fixed graph as a minor. A minor can be formed by deleting vertices, deleting edges, and contracting edges. Planar graphs, for example, exclude $K_5$ as a minor. Édouard Bonnet, Eun Jung Kim, Stéphan Thomassé, and Rémi Watrigant proved that every class excluding a fixed clique minor has bounded twin-width, a graph…

The spectral norm of a matrix, its largest singular value, is computable in polynomial time to any accuracy. Its analogue for 3-tensors appears in the analysis of multilinear systems, in the best rank-1 approximation of a data array, in quantum information (the geometric measure of entanglement), and in the injective norm of tensor products. Hillar and Lim, …

In the edge-disjoint paths problem (EDP) one is given an undirected graph and a list of source–sink pairs, and asks how many pairs can be connected simultaneously by paths that share no edge. Its relaxation EDP with congestion (EDPwC) allows every edge to carry up to a fixed number of paths. Both are central problems of network routing and of approximation…

Many routing problems come with a hard limit on distance or time rather than a requirement to visit everything. A repair crew with a fixed shift wants to service as many stations as possible; a vehicle with a fixed range wants to cover as many customers as possible. The budgeted prize-collecting traveling salesman problem captures this: given edge costs and…

The maximum weight matching (MWM) problem asks, for a graph with edge weights, for a set of vertex-disjoint edges of largest total weight. It is a central problem of combinatorial optimization, with applications to transportation, assignment and scheduling, and as a subroutine for shortest paths, planar max cut, Chinese postman tours and metric TSP.…

Quantum Max-Cut asks for a high-energy state of an antiferromagnetic spin system specified by a weighted graph. Vertices carry qubits, and an edge rewards the two qubits for occupying their antisymmetric singlet state. A single qubit participates in several interactions, so maximizing each edge separately generally gives incompatible demands. The problem…