Edoardo Amaldi, Viggo Kann

We investigate the computational complexity of two classes of

combinatorial optimization problems related to linear systems

and study the relationship between their approximability properties.

In the first class (MIN ULR) one wishes, given a possibly infeasible

system of linear relations, to find ...
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Sanjeev Arora, Madhu Sudan

NP = PCP(log n, 1) and related results crucially depend upon

the close connection between the probability with which a

function passes a ``low degree test'' and the distance of

this function to the nearest degree d polynomial. In this

paper we study a test ...
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Peter Jonsson, Paolo Liberatore

We study the computational complexity of an optimization

version of the constraint satisfaction problem: given a set $F$ of

constraint functions, an instance consists of a set of variables $V$

related by constraints chosen from $F$ and a natural number $k$. The

problem is to decide whether there exists a ...
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Andrea E. F. Clementi, Paolo Penna, Riccardo Silvestri

Given a finite set $S$ of points (i.e. the stations of a radio

network) on a $d$-dimensional Euclidean space and a positive integer

$1\le h \le |S|-1$, the \minrangeh{d} problem

consists of assigning transmission ranges to the stations so as

to minimize the total power consumption, provided ...
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Christian Glaßer, Christian Reitwießner, Heinz Schmitz, Maximilian Witek

We systematically study the hardness and the approximability of combinatorial multi-objective NP optimization problems (multi-objective problems, for short).

We define solution notions that precisely capture the typical algorithmic tasks in multi-objective optimization. These notions inherit polynomial-time Turing reducibility from multivalued functions, which allows us to compare the solution notions and ... more >>>

Libor Barto, Marcin Kozik

An algorithm for a constraint satisfaction problem is called robust if it outputs an assignment satisfying at least $(1-g(\varepsilon))$-fraction of the constraints given a $(1-\varepsilon)$-satisfiable instance, where $g(\varepsilon) \rightarrow 0$ as $\varepsilon \rightarrow 0$, $g(0)=0$.

Guruswami and Zhou conjectured a characterization of constraint languages for which the corresponding constraint satisfaction ...
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Vikraman Arvind, Sebastian Kuhnert, Johannes Köbler, Yadu Vasudev

We study optimization versions of Graph Isomorphism. Given two graphs $G_1,G_2$, we are interested in finding a bijection $\pi$ from $V(G_1)$ to $V(G_2)$ that maximizes the number of matches (edges mapped to edges or non-edges mapped to non-edges). We give an $n^{O(\log n)}$ time approximation scheme that for any constant ... more >>>

Venkatesan Guruswami, Ali Kemal Sinop

Convex relaxations based on different hierarchies of

linear/semi-definite programs have been used recently to devise

approximation algorithms for various optimization problems. The

approximation guarantee of these algorithms improves with the number

of {\em rounds} $r$ in the hierarchy, though the complexity of solving

(or even writing down the solution for) ...
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Vijay Bhattiprolu, Mrinalkanti Ghosh, Venkatesan Guruswami, Euiwoong Lee, Madhur Tulsiani

We consider the following basic problem: given an $n$-variate degree-$d$ homogeneous polynomial $f$ with real coefficients, compute a unit vector $x \in \mathbb{R}^n$ that maximizes $|f(x)|$. Besides its fundamental nature, this problem arises in many diverse contexts ranging from tensor and operator norms to graph expansion to quantum information ... more >>>

Vijay Bhattiprolu, Mrinalkanti Ghosh, Venkatesan Guruswami, Euiwoong Lee, Madhur Tulsiani

We consider the $(\ell_p,\ell_r)$-Grothendieck problem, which seeks to maximize the bilinear form $y^T A x$ for an input matrix $A \in {\mathbb R}^{m \times n}$ over vectors $x,y$ with $\|x\|_p=\|y\|_r=1$. The problem is equivalent to computing the $p \to r^\ast$ operator norm of $A$, where $\ell_{r^*}$ is the dual norm ... more >>>

Hermann Gruber , Markus Holzer, Simon Wolfsteiner

Finite languages lie at the heart of literally every regular expression. Therefore, we investigate the approximation complexity of minimizing regular expressions without Kleene star, or, equivalently, regular expressions describing finite languages. On the side of approximation hardness, given such an expression of size~$s$, we prove that it is impossible to ... more >>>

Venkatesan Guruswami, Sai Sandeep

A famous conjecture of Tuza states that the minimum number of edges needed to cover all the triangles in a graph is at most twice the maximum number of edge-disjoint triangles. This conjecture was couched in a broader setting by Aharoni and Zerbib who proposed a hypergraph version of this ... more >>>

Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Santhoshini Velusamy

A Boolean constraint satisfaction problem (CSP), Max-CSP$(f)$, is a maximization problem specified by a constraint $f:\{-1,1\}^k\to\{0,1\}$. An instance of the problem consists of $m$ constraint applications on $n$ Boolean variables, where each constraint application applies the constraint to $k$ literals chosen from the $n$ variables and their negations. The goal ... more >>>

Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Santhoshini Velusamy

A constraint satisfaction problem (CSP), Max-CSP$({\cal F})$, is specified by a finite set of constraints ${\cal F} \subseteq \{[q]^k \to \{0,1\}\}$ for positive integers $q$ and $k$. An instance of the problem on $n$ variables is given by $m$ applications of constraints from ${\cal F}$ to subsequences of the $n$ ... more >>>

Chi-Ning Chou, Alexander Golovnev, Madhu Sudan, Ameya Velingker, Santhoshini Velusamy

We consider the approximability of constraint satisfaction problems in the streaming setting. For every constraint satisfaction problem (CSP) on $n$ variables taking values in $\{0,\ldots,q-1\}$, we prove that improving over the trivial approximability by a factor of $q$ requires $\Omega(n)$ space even on instances with $O(n)$ constraints. We also identify ... more >>>

Joanna Boyland, Michael Hwang, Tarun Prasad, Noah Singer, Santhoshini Velusamy

A Boolean maximum constraint satisfaction problem, Max-CSP\((f)\), is specified by a predicate \(f:\{-1,1\}^k\to\{0,1\}\). An \(n\)-variable instance of Max-CSP\((f)\) consists of a list of constraints, each of which applies \(f\) to \(k\) distinct literals drawn from the \(n\) variables. For \(k=2\), Chou, Golovnev, and Velusamy [CGV20, FOCS 2020] obtained explicit ratios ... more >>>

Chi-Ning Chou, Alexander Golovnev, Amirbehshad Shahrasbi, Madhu Sudan, Santhoshini Velusamy

We analyze the sketching approximability of constraint satisfaction problems on Boolean domains, where the constraints are balanced linear threshold functions applied to literals. In particular, we explore the approximability of monarchy-like functions where the value of the function is determined by a weighted combination of the vote of the first ... more >>>