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We obtain new transference bounds that connect two active areas of research: proximity and sparsity of solutions to integer programs. Specifically, we study the additive integrality gap of the integer linear programs $min\{\mathit{c}·\mathit{x}:\mathit{x}\in P\cap {\mathbb{Z}}^n\}$, where $P=\{\mathit{x}\in {\mathbb{R}}^n:...\mathit{}\mathit{}\mathit{}\mathit{}\mathbf{}$$$$\mathit{}$$\mathit{}$${}_{\mathrm{}}$

Given a metric space (V, d) along with an integer k, the $k\text{-}\mathrm{M}\mathrm{E}\mathrm{D}\mathrm{I}\mathrm{A}\mathrm{N}$ problem asks to open k centers $C\subseteq V$ to minimize ${\sum }_{v\in V}d(v,C)$, where $d(v,C):={min}_{c\in C}d(v,c)$. While the best-known approximation ratio 2.613 holds for the more general supplier version ...$$$$$\mathrm{}\mathrm{}\mathrm{}$$\mathrm{}\mathrm{}\mathrm{}$$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}$

Prize-Collecting TSP is a variant of the traveling salesperson problem where one may drop vertices from the tour at the cost of vertex-dependent penalties. The quality of a solution is then measured by adding the length of the tour and the sum of ...$\mathrm{}\sqrt{\mathrm{}}\mathrm{}\mathrm{}$

This paper introduces a framework to study discrete optimization problems which are parametric in the following sense: their constraint matrices correspond to matrices over the ring $\mathbb{Z}[x]$ of polynomials in one variable. We investigate in particular ...$\mathbb{}$$\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}$$\mathbb{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}$$\mathrm{}\mathrm{}\mathrm{}\mathrm{}$$\mathbb{}\mathrm{}\mathrm{}$

Various first order approaches have been proposed in the literature to solve Linear Programming (LP) problems, recently leading to practically efficient solvers for large-scale LPs. From a theoretical perspective, linear convergence rates have ...

In a digraph, a dicut is a cut where all the arcs cross in one direction. A dijoin is a subset of arcs that intersects each dicut. Woodall conjectured in 1976 that in every digraph, the minimum size of a dicut equals to the maximum number of ...$$$\frac{}{}$$\frac{}{\mathrm{}}$$$$\frac{\mathrm{}\mathrm{}}{}$$\frac{}{\mathrm{}\mathrm{}}$$$

We consider the capacitated facility location problem with outliers when facility costs are uniform. Our main result is the first constant factor approximation for this problem. We give a local search algorithm that requires only 2 operations and ...$$$$

We investigate the semigroup of integer points inside a convex cone. We extend classical results in integer linear programming to integer conic programming. We show that the semigroup associated with nonpolyhedral cones can sometimes have a notion ...

We show that any bounded integral function $f:A\times B\mapsto \{0,1,\cdots ,\Delta \}$ with rank r has deterministic communication complexity ${\Delta }^{O(\Delta )}·\sqrt{r}·logr$, where the rank of f is defined to be the rank of the $A\times B$ matrix whose entries are the function values. As a corollary, ...$\mathrm{}\mathrm{}$${}^{}\sqrt{}$

Motivated by applications in e-retail and online advertising, we study the problem of assortment optimization under visibility constraints, referred to as APV. We are given a universe of substitutable products and a stream of T customers. The ...

We study a two-sided assortment optimization framework to address the challenge of choice congestion faced by matching platforms. The goal is to decide the assortments to offer to agents in order to maximize the expected number of matches. We ...$\mathrm{}\mathrm{}$

We introduce and study a two-stage stochastic stable matching problem between students and schools. A decision maker chooses a stable matching in a marriage instance; then, after some agents enter or leave the market following a probability ...$\mathcal{}$$\mathcal{}$

Gale and Shapley’s stability criterion enjoys a rich mathematical structure, which propelled its application in various settings. Although immensely popular, the approach by Gale and Shapley cannot encompass all the different features that arise ...

We study the classical load balancing problem in a fully dynamic setting where jobs both arrive and depart. Each job can only be assigned to a subset of machines and can be reassigned at any time step. The goal is to maintain a near-optimal ...

We study online contention resolution schemes (OCRSs) and prophet inequalities for non-product distributions. Specifically, when the active set is sampled according to a pairwise-independent (PI) distribution, we show a $(1-o_k(1))$-selectable OCRS ...$\mathrm{}$$\mathrm{}$

In the connectivity interdiction problem, we are asked to find a global graph cut and remove a subset of edges under a budget constraint, so that the total weight of the remaining edges in this cut is minimized. This problem easily includes the ...

We study fundamental block-structured integer programs called tree-fold and multi-stage IPs. Tree-fold IPs admit a constraint matrix with independent blocks linked together by few constraints in a recursive pattern; and transposing their ...

One of the most famous conjectures in combinatorial optimization is the four-thirds conjecture, which states that the integrality gap of the subtour LP relaxation of the TSP is equal to $\frac{4}{3}$. For 40 years, the best known upper bound was 1.5, due to ...$\mathrm{}{\mathrm{}}^{\mathrm{}}$

The simplex algorithm is one of the most popular algorithms to solve linear programs (LPs). Starting at an extreme point solution of an LP, it performs a sequence of basis exchanges (called pivots) that allows one to move to a better extreme point ...

A Low-rank Spectral Optimization Problem (LSOP) minimizes a linear objective function subject to multiple two-sided linear inequalities intersected with a low-rank and spectral constrained domain. Although solving LSOP is, in general, NP-hard, its ...

Consider a matroid where all elements are labeled with an element in $\mathbb{Z}$. We are interested in finding a base where the sum of the labels is congruent to $g(modm)$. We show that this problem can be solved in $\tilde{O}(2^{4m}n{r}^{5/6})$ time for a matroid with n ...

We consider an online multi-weighted generalization of several classic online optimization problems called the online combinatorial assignment problem. We are given an independence system over a ground set of elements and agents that arrive online ...$$$$$$$\sqrt{}$$\mathrm{}{}^{}$$\mathrm{}\mathrm{}$${}^{\mathrm{}}$

Consider the triplet $(E,\mathcal{P},\pi )$, where E is a finite ground set, $\mathcal{P}\subseteq 2^E$ is a collection of subsets of E and $\pi :\mathcal{P}\to [0,1]$ is a requirement function. Given a vector of marginals$\rho \in {[0,1]}^E$, our goal is to find a distribution for a random subset $S\subseteq E$ such that $\mathbf{...}\left[\right]{}_{}$$$$\mathbf{}\left[\mathrm{}\right]{}_{}$$\mathcal{}$

It is a fundamental result in combinatorial optimization that submodular functions can be minimized in polynomial-time. This paper considers the minimization problem for a more general class of set functions that contains all submodular functions. ...

The Maximum Leaf Spanning Arborescence problem (MLSA) is defined as follows: Given a directed graph G and a vertex $r\in V(G)$ from which every other vertex is reachable, find a spanning arborescence rooted at r maximizing the number of leaves (...

A set family $\mathcal{F}$ is uncrossable if $A\cap B,A\cup B\in \mathcal{F}$ or $A\backslash B,B\backslash A\in \mathcal{F}$ for any $A,B\in \mathcal{F}$. A classic result of Williamson, Goemans, Mihail, and Vazirani [STOC 1993:708-717] states that the problem of covering an uncrossable set family by a min-cost edge set admits ...$\mathcal{}$$\mathcal{}$$$$\mathcal{}$$\mathcal{}$

Electric vehicle (EV) adoption in long-distance logistics faces challenges such as range anxiety and uneven distribution of charging stations. Two pivotal questions emerge: How can EVs be efficiently routed in a charging network considering range ...

Standard mixed-integer programming formulations for the stable set problem on n-node graphs require n integer variables. We prove that this is almost optimal: We give a family of n-node graphs for which every polynomial-size MIP formulation ...${}^{\mathrm{}}$$\sqrt{}$