On polyhedral approximations of the positive semidefinite cone

Let $D$ be the set of $n\times n$ positive semidefinite matrices of trace equal to one, also known as the set of density matrices. We prove two results on the hardness of approximating $D$ with polytopes. First, we show that if $0 < \epsilon < 1$ and $A$ is an arbitrary matrix of trace equal to one, any polytope $P$ such that $(1-\epsilon)(D-A) \subset P \subset D-A$ must have linear programming extension complexity at least $\exp(c\sqrt{n})$ where $c > 0$ is a constant that depends on $\epsilon$. Second, we show that any polytope $P$ such that $D \subset P$ and such that the Gaussian width of $P$ is at most twice the Gaussian width of $D$ must have extension complexity at least $\exp(cn^{1/3})$. The main ingredient of our proofs is hypercontractivity of the noise operator on the hypercube.Comment: 12 page

Query-to-Communication Lifting for BPP

For any $n$-bit boolean function $f$, we show that the randomized communication complexity of the composed function $f\circ g^n$, where $g$ is an index gadget, is characterized by the randomized decision tree complexity of $f$. In particular, this means that many query complexity separations involving randomized models (e.g., classical vs. quantum) automatically imply analogous separations in communication complexity.Comment: 21 page

Subexponential LPs Approximate Max-Cut

We show that for every $\varepsilon > 0$, the degree-$n^\varepsilon$ Sherali-Adams linear program (with $\exp(\tilde{O}(n^\varepsilon))$ variables and constraints) approximates the maximum cut problem within a factor of $(\frac{1}{2}+\varepsilon')$, for some $\varepsilon'(\varepsilon) > 0$. Our result provides a surprising converse to known lower bounds against all linear programming relaxations of Max-Cut, and hence resolves the extension complexity of approximate Max-Cut for approximation factors close to $\frac{1}{2}$ (up to the function $\varepsilon'(\varepsilon)$). Previously, only semidefinite programs and spectral methods were known to yield approximation factors better than $\frac 12$ for Max-Cut in time $2^{o(n)}$. We also show that constant-degree Sherali-Adams linear programs (with $\text{poly}(n)$ variables and constraints) can solve Max-Cut with approximation factor close to $1$ on graphs of small threshold rank: this is the first connection of which we are aware between threshold rank and linear programming-based algorithms. Our results separate the power of Sherali-Adams versus Lov\'asz-Schrijver hierarchies for approximating Max-Cut, since it is known that $(\frac{1}{2}+\varepsilon)$ approximation of Max Cut requires $\Omega_\varepsilon (n)$ rounds in the Lov\'asz-Schrijver hierarchy. We also provide a subexponential time approximation for Khot's Unique Games problem: we show that for every $\varepsilon > 0$ the degree-$(n^\varepsilon \log q)$ Sherali-Adams linear program distinguishes instances of Unique Games of value $\geq 1-\varepsilon'$ from instances of value $\leq \varepsilon'$, for some $\varepsilon'( \varepsilon) >0$, where $q$ is the alphabet size. Such guarantees are qualitatively similar to those of previous subexponential-time algorithms for Unique Games but our algorithm does not rely on semidefinite programming or subspace enumeration techniques

The combined basic LP and affine IP relaxation for promise VCSPs on infinite domains

Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami [SODA'20] for promise (non-valued) CSPs (on finite domains).Comment: Full version of an MFCS'20 pape

Strengths and Limitations of Linear Programming Relaxations

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than $2$ and $\omega(1)$, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than $1.5$ for the Vertex Cover Problem, and better than $2$ for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of $2-\varepsilon$ for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture

Lifting Theorems Meet Information Complexity: Known and New Lower Bounds of Set-disjointness

Set-disjointness problems are one of the most fundamental problems in communication complexity and have been extensively studied in past decades. Given its importance, many lower bound techniques were introduced to prove communication lower bounds of set-disjointness. Combining ideas from information complexity and query-to-communication lifting theorems, we introduce a density increment argument to prove communication lower bounds for set-disjointness: We give a simple proof showing that a large rectangle cannot be $0$-monochromatic for multi-party unique-disjointness. We interpret the direct-sum argument as a density increment process and give an alternative proof of randomized communication lower bounds for multi-party unique-disjointness. Avoiding full simulations in lifting theorems, we simplify and improve communication lower bounds for sparse unique-disjointness. Potential applications to be unified and improved by our density increment argument are also discussed.Comment: Working Pape

A Tight Approximation Algorithm for the Cluster Vertex Deletion Problem

We give the first $2$-approximation algorithm for the cluster vertex deletion problem. This is tight, since approximating the problem within any constant factor smaller than $2$ is UGC-hard. Our algorithm combines the previous approaches, based on the local ratio technique and the management of true twins, with a novel construction of a 'good' cost function on the vertices at distance at most $2$ from any vertex of the input graph. As an additional contribution, we also study cluster vertex deletion from the polyhedral perspective, where we prove almost matching upper and lower bounds on how well linear programming relaxations can approximate the problem.Comment: 23 pages, 3 figure