On the Approximability of Digraph Ordering

Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $\ell : V \to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $\ell$(u) < $\ell$(v). For different values of k, this reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k={2,..., n}, improving on the known 2k/(k-1)-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,..., n} and constant $\varepsilon > 0$, Max-k-Ordering has an LP integrality gap of 2 - $\varepsilon$ for $n^{\Omega\left(1/\log\log k\right)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v \subseteq \mathbb{Z}^+$. We prove an LP rounding based $4\sqrt{2}/(\sqrt{2}+1) \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $k\in [n]$.Comment: 21 pages, Conference version to appear in ESA 201

Approximation Limits of Linear Programs (Beyond Hierarchies)

We develop a framework for approximation limits of polynomial-size linear programs from lower bounds on the nonnegative ranks of suitably defined matrices. This framework yields unconditional impossibility results that are applicable to any linear program as opposed to only programs generated by hierarchies. Using our framework, we prove that O(n^{1/2-eps})-approximations for CLIQUE require linear programs of size 2^{n^\Omega(eps)}. (This lower bound applies to linear programs using a certain encoding of CLIQUE as a linear optimization problem.) Moreover, we establish a similar result for approximations of semidefinite programs by linear programs. Our main ingredient is a quantitative improvement of Razborov's rectangle corruption lemma for the high error regime, which gives strong lower bounds on the nonnegative rank of certain perturbations of the unique disjointness matrix.Comment: 23 pages, 2 figure

Hardness of Graph Pricing through Generalized Max-Dicut

The Graph Pricing problem is among the fundamental problems whose approximability is not well-understood. While there is a simple combinatorial 1/4-approximation algorithm, the best hardness result remains at 1/2 assuming the Unique Games Conjecture (UGC). We show that it is NP-hard to approximate within a factor better than 1/4 under the UGC, so that the simple combinatorial algorithm might be the best possible. We also prove that for any $\epsilon > 0$, there exists $\delta > 0$ such that the integrality gap of $n^{\delta}$-rounds of the Sherali-Adams hierarchy of linear programming for Graph Pricing is at most 1/2 + $\epsilon$. This work is based on the effort to view the Graph Pricing problem as a Constraint Satisfaction Problem (CSP) simpler than the standard and complicated formulation. We propose the problem called Generalized Max-Dicut($T$), which has a domain size $T + 1$ for every $T \geq 1$. Generalized Max-Dicut(1) is well-known Max-Dicut. There is an approximation-preserving reduction from Generalized Max-Dicut on directed acyclic graphs (DAGs) to Graph Pricing, and both our results are achieved through this reduction. Besides its connection to Graph Pricing, the hardness of Generalized Max-Dicut is interesting in its own right since in most arity two CSPs studied in the literature, SDP-based algorithms perform better than LP-based or combinatorial algorithms --- for this arity two CSP, a simple combinatorial algorithm does the best.Comment: 28 page

Polynomial integrality gaps for strong SDP relaxations of Densest k

The Densest k-subgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest k-subgraph: the current best algorithm gives an ≈ O(n 1/4) approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than P ̸ = NP. In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of Densest k-subgraph and its variants. Thus, understanding the approximability of Densest k-subgraph is an important challenge. In this work, we give evidence for the hardness of approximating Densest k-subgraph within polynomial factors. Specifically, we expose the limitations of strong semidefinite programs from SDP hierarchies in solving Densest k-subgraph. Our results include: • A lower bound of Ω ( n 1/4 / log 3 n) on the integrality gap for Ω(log n / log log n) rounds of the Sherali-Adams relaxation for Densest k-subgraph. This also holds for the relaxation obtained from Sherali-Adams with an added SDP constraint. Our gap instances are i

A Union of Euclidean Metric Spaces is Euclidean

A Union of Euclidean Metric Spaces is Euclidean, Discrete Analysis 2016:14, 15pp. A major theme in metric geometry concerns conditions under which it is possible to embed one metric space into another with small distortion. More precisely, if $M_1$ and $M_2$ are metric spaces, one would like to find a function $f:M_1\to M_2$ and constants $c$ and $C$ such that for every $x,y\in M_1$ the inequalities $cd(x,y)\leq d(f(x),f(y))\leq Cd(x,y)$ hold, with the ratio $C/c$ being as small as possible. For example, the famous [Johnson-Lindenstrauss lemma](https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma) states that if $a_1,\dots,a_n$ are points in a Euclidean space and $\epsilon>0$, then there are points $b_1,\dots,b_n$ that live in a Euclidean space of dimension at most $8\log n/\epsilon^2$ such that $$(1-\epsilon)\|a_i-a_j\|^2\leq\|b_i-b_j\|^2\leq(1+\epsilon)^2\|a_i-a_j\|^2$$ for every $i,j$. That is, a finite subset $A$ of a Euclidean space can be embedded with small distortion into a Euclidean space with dimension logarithmic in the size of $A$. A rather basic question one can ask about embeddings of small distortion is the following. Let $A$ and $B$ be two subspaces of a metric space $X$ and suppose that each can be embedded into Euclidean space with bounded distortion. Can their union $A\cup B$ also be embedded with bounded distortion? One might think that this question would have either a simple proof or a simple counterexample, but surprisingly it was a question asked by Assaf Naor that had been open for some time. It is solved in this paper. If $Z$ is a metric space, write $D_Z$ for the smallest distortion with which one can embed $Z$ into Euclidean space -- that is, the smallest possible ratio $C/c$ in the discussion above. The main result of the paper is that $D_{A\cup B}\leq 7D_AD_B+2(D_A+D_B)$. In particular, if $D_A$ and $D_B$ are finite, then so is $D_{A\cup B}$. This result leaves open several further interesting questions, mentioned at the end of the paper. In particular, the ingenious proof uses a specifically Hilbertian tool, the [Kirszbraun extension theorem](https://en.wikipedia.org/wiki/Kirszbraun_theorem), so it does not solve the problem for embeddings into other spaces, such as $\ell_1$. It would also be interesting to know what the best possible dependence is of $D_{A\cup B}$ on $D_A$ and $D_B$. It is likely that this paper will stimulate many further interesting developments in metric geometry

Local global tradeoffs in metric embeddings

Suppose that every k points in a n point metric space X are D-distortion embeddable into ℓ1. We give upper and lower bounds on the distortion required to embed the entire space X into ℓ1. This is a natural mathematical question and is also motivated by the study of relaxations obtained by lift-and-project methods for graph partitioning problems. In this setting, we show that X can be embedded into ℓ1 with distortion O(D×log(n/k)). Moreover, we give a lower bound showing that this result is tight if D is bounded away from 1. For D = 1+δ we give a lower bound of Ω(log(n/k) / log(1/δ)); and for D = 1, we give a lower bound of Ω(log n/(log k + log log n)). Our bounds significantly improve on the results of Arora, Lovász, Newman, Rabani, Rabinovich and Vempala, who initiated a study of these questions