The TCS Formalization Drive

A retailer chooses which products to carry and how many units of each to stock, under a shelf or warehouse capacity, before a selling season in which inventory is not replenished. Customers who find their favourite product sold out may buy another one instead: this dynamic substitution couples the demand of all products through the stock-out history.…

A matching in a graph is a set of edges no two of which share a vertex. Maximum-cardinality matching is one of the basic problems of combinatorial optimization, and the fastest known algorithm for it in general graphs is the Micali–Vazirani (MV) algorithm (Micali–Vazirani, FOCS 1980), which runs in $O(m\sqrt n)$ time. It follows the phase structure of…

Min-sum set cover asks for an order of the elements of a ground set that minimizes the average time at which a hyperedge is first covered. Feige, Lovász and Tetali showed that the greedy algorithm is a 4-approximation and that no better constant is possible unless P = NP (Feige, Lovász, Tetali 2004). Min-sum vertex cover, pipelined set cover,…

The Nash social welfare of an allocation of indivisible items is the geometric mean of the agents' valuations. Nash proposed it for the bargaining problem as the objective characterized by a set of natural axioms, and it has since become a standard objective in fair division because it balances efficiency against fairness. Maximizing it exactly is NP-hard,…

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…