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Samuel Fiorini

Publications and source records attributed to Samuel Fiorini.

At least 37 records · Page 2Linked to original sources
arXiv2018

Strengthening Convex Relaxations of 0/1-Sets Using Boolean Formulas

In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set $ S $, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set $ Q \subseteq \mathbb{R}^n $ containing a set $ S \subseteq \{0,1\}^n $ by exploiting certain additional information about $ S $. Namely, the required extra information will be in the form of a Boolean formula $ ϕ$ defining the target set $ S $. The aim of this work is to analyze various aspects regarding the strength of our procedure. As one result, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems.

math.CO↗
arXiv2018

Bounds on the number of 2-level polytopes, cones and configurations

We prove an upper bound of the form $2^{O(d^2 \mathrm{polylog}\,d)}$ on the number of affine (resp. linear) equivalence classes of, by increasing order of generality, 2-level d-polytopes, d-cones and d-configurations. This in particular answers positively a conjecture of Bohn et al. on 2-level polytopes. We obtain our upper bound by relating affine (resp. linear) equivalence classes of 2-level d-polytopes, d-cones and d-configurations to faces of the correlation cone. We complement this with a $2^{Ω(d^2)}$ lower bound, by estimating the number of nonequivalent stable set polytopes of bipartite graphs.

math.CO↗
arXiv2017

Characterizing Polytopes Contained in the $0/1$-Cube with Bounded Chvátal-Gomory Rank

Let $S \subseteq \{0,1\}^n$ and $R$ be any polytope contained in $[0,1]^n$ with $R \cap \{0,1\}^n = S$. We prove that $R$ has bounded Chvátal-Gomory rank (CG-rank) provided that $S$ has bounded notch and bounded gap, where the notch is the minimum integer $p$ such that all $p$-dimensional faces of the $0/1$-cube have a nonempty intersection with $S$, and the gap is a measure of the size of the facet coefficients of $\mathsf{conv}(S)$. Let $H[\bar{S}]$ denote the subgraph of the $n$-cube induced by the vertices not in $S$. We prove that if $H[\bar{S}]$ does not contain a subdivision of a large complete graph, then both the notch and the gap are bounded. By our main result, this implies that the CG-rank of $R$ is bounded as a function of the treewidth of $H[\bar{S}]$. We also prove that if $S$ has notch $3$, then the CG-rank of $R$ is always bounded. Both results generalize a recent theorem of Cornuéjols and Lee, who proved that the CG-rank is bounded by a constant if the treewidth of $H[\bar{S}]$ is at most $2$.

math.OC↗
arXiv2017

Extended Formulations for Order Polytopes through Network Flows

Mathematical psychology has a long tradition of modeling probabilistic choice via distribution-free random utility models and associated random preference models. For such models, the predicted choice probabilities often form a bounded and convex polyhedral set, or polytope. Polyhedral combinatorics have thus played a key role in studying the mathematical structure of these models. However, standard methods for characterizing the polytopes of such models are subject to a combinatorial explosion in complexity as the number of choice alternatives increases. Specifically, this is the case for random preference models based on linear, weak, semi- and interval orders. For these, a complete, linear description of the polytope is currently known only for, at most, 5--8 choice alternatives. We leverage the method of extended formulations to break through those boundaries. For each of the four types of preferences, we build an appropriate network, and show that the associated network flow polytope provides an extended formulation of the polytope of the choice model. This extended formulation has a simple linear description that is more parsimonious than descriptions obtained by standard methods for large numbers of choice alternatives. The result is a computationally less demanding way of testing the probabilistic choice model on data. We sketch how the latter interfaces with recent developments in contemporary statistics.

math.OC↗
arXiv2017

Extension complexity of stable set polytopes of bipartite graphs

The extension complexity $\mathsf{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that affinely projects to $P$. Let $G$ be a bipartite graph with $n$ vertices, $m$ edges, and no isolated vertices. Let $\mathsf{STAB}(G)$ be the convex hull of the stable sets of $G$. It is easy to see that $n \leqslant \mathsf{xc} (\mathsf{STAB}(G)) \leqslant n+m$. We improve both of these bounds. For the upper bound, we show that $\mathsf{xc} (\mathsf{STAB}(G))$ is $O(\frac{n^2}{\log n})$, which is an improvement when $G$ has quadratically many edges. For the lower bound, we prove that $\mathsf{xc} (\mathsf{STAB}(G))$ is $Ω(n \log n)$ when $G$ is the incidence graph of a finite projective plane. We also provide examples of $3$-regular bipartite graphs $G$ such that the edge vs stable set matrix of $G$ has a fooling set of size $|E(G)|$.

cs.DM↗
arXiv2017

Enumeration of $2$-level polytopes

A (convex) polytope $P$ is said to be $2$-level if for every direction of hyperplanes which is facet-defining for $P$, the vertices of $P$ can be covered with two hyperplanes of that direction. The study of these polytopes is motivated by questions in combinatorial optimization and communication complexity, among others. In this paper, we present the first algorithm for enumerating all combinatorial types of $2$-level polytopes of a given dimension $d$, and provide complete experimental results for $d \leqslant 7$. Our approach is inductive: for each fixed $(d-1)$-dimensional $2$-level polytope $P_0$, we enumerate all $d$-dimensional $2$-level polytopes $P$ that have $P_0$ as a facet. This relies on the enumeration of the closed sets of a closure operator over a finite ground set. By varying the prescribed facet $P_0$, we obtain all $2$-level polytopes in dimension $d$.

math.CO↗
arXiv2017

A $\frac{3}{2}$-Approximation Algorithm for Tree Augmentation via Chvátal-Gomory Cuts

The weighted tree augmentation problem (WTAP) is a fundamental network design problem. We are given an undirected tree $G = (V,E)$, an additional set of edges $L$ called links and a cost vector $c \in \mathbb{R}^L_{\geq 1}$. The goal is to choose a minimum cost subset $S \subseteq L$ such that $G = (V, E \cup S)$ is $2$-edge-connected. In the unweighted case, that is, when we have $c_\ell = 1$ for all $\ell \in L$, the problem is called the tree augmentation problem (TAP). Both problems are known to be APX-hard, and the best known approximation factors are $2$ for WTAP by (Frederickson and JáJá, '81) and $\tfrac{3}{2}$ for TAP due to (Kortsarz and Nutov, TALG '16). In the case where all link costs are bounded by a constant $M$, (Adjiashvili, SODA '17) recently gave a $\approx 1.96418+\varepsilon$-approximation algorithm for WTAP under this assumption. This is the first approximation with a better guarantee than $2$ that does not require restrictions on the structure of the tree or the links. In this paper, we improve Adjiashvili's approximation to a $\frac{3}{2}+\varepsilon$-approximation for WTAP under the bounded cost assumption. We achieve this by introducing a strong LP that combines $\{0,\frac{1}{2}\}$-Chvátal-Gomory cuts for the standard LP for the problem with bundle constraints from Adjiashvili. We show that our LP can be solved efficiently and that it is exact for some instances that arise at the core of Adjiashvili's approach. This results in the improved guarantee of $\frac{3}{2}+\varepsilon$. For TAP, this is the best known LP-based result, and matches the bound of $\frac{3}{2}+\varepsilon$ achieved by the best SDP-based algorithm due to (Cheriyan and Gao, arXiv '15).

cs.DM↗
arXiv2016

Small Extended Formulation for Knapsack Cover Inequalities from Monotone Circuits

Initially developed for the min-knapsack problem, the knapsack cover inequalities are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield linear programming (LP) relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. In this paper 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. For the min-knapsack cover problem, our main result can be stated formally as follows: for any $\varepsilon >0$, there is a $(1/\varepsilon)^{O(1)}n^{O(\log n)}$-size LP relaxation with an integrality gap of at most $2+\varepsilon$, where $n$ is the number of items. Prior to this work, there was no known relaxation of subexponential size with a constant upper bound on the integrality gap. Our construction is inspired by a connection between extended formulations and monotone circuit complexity via Karchmer-Wigderson games. In particular, our LP is based on $O(\log^2 n)$-depth monotone circuits with fan-in~$2$ for evaluating weighted threshold functions with $n$ inputs, as constructed by Beimel and Weinreb. We believe that a further understanding of this connection may lead to more positive results complementing the numerous lower bounds recently proved for extended formulations.

cs.DM↗
arXiv2016

The excluded minors for isometric realizability in the plane

Let $G$ be a graph and $p \in [1, \infty]$. The parameter $f_p(G)$ is the least integer $k$ such that for all $m$ and all vectors $(r_v)_{v \in V(G)} \subseteq \mathbb{R}^m$, there exist vectors $(q_v)_{v \in V(G)} \subseteq \mathbb{R}^k$ satisfying $$\|r_v-r_w\|_p=\|q_v-q_w\|_p, \ \text{ for all }\ vw\in E(G).$$ It is easy to check that $f_p(G)$ is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour, there are a finite number of excluded minors for the property $f_p(G) \leq k$. In this paper, we determine the complete set of excluded minors for $f_\infty(G) \leq 2$. The two excluded minors are the wheel on $5$ vertices and the graph obtained by gluing two copies of $K_4$ along an edge and then deleting that edge. We also show that the same two graphs are the complete set of excluded minors for $f_1(G) \leq 2$. In addition, we give a family of examples that show that $f_\infty$ is unbounded on the class of planar graphs and $f_\infty$ is not bounded as a function of tree-width.

math.MG↗
arXiv2016

Average case polyhedral complexity of the maximum stable set problem

We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via linear programs (LPs), in two distinct models. In the uniform model, the constraints of the LP are not allowed to depend on the input graph, which should be encoded solely in the objective function. There we prove a $2^{Ω(n/ \log n)}$ lower bound with probability at least $1 - 2^{-2^n}$ for every LP that is exact for a randomly selected set of instances; each graph on at most n vertices being selected independently with probability $p \geq 2^{-\binom{n/4}{2}+n}$. In the non-uniform model, the constraints of the LP may depend on the input graph, but we allow weights on the vertices. The input graph is sampled according to the G(n, p) model. There we obtain upper and lower bounds holding with high probability for various ranges of p. We obtain a super-polynomial lower bound all the way from $p = Ω(\log^{6+\varepsilon} / n)$ to $p = o (1 / \log n)$. Our upper bound is close to this as there is only an essentially quadratic gap in the exponent, which currently also exists in the worst-case model. Finally, we state a conjecture that would close this gap, both in the average-case and worst-case models.

cs.CC↗
arXiv2015

No Small Linear Program Approximates Vertex Cover within a Factor $2 - ε$

The vertex cover problem is one of the most important and intensively studied combinatorial optimization problems. Khot and Regev (2003) proved that the problem is NP-hard to approximate within a factor $2 - ε$, assuming the Unique Games Conjecture (UGC). This is tight because the problem has an easy 2-approximation algorithm. Without resorting to the UGC, the best inapproximability result for the problem is due to Dinur and Safra (2002): vertex cover is NP-hard to approximate within a factor 1.3606. We prove the following unconditional result about linear programming (LP) relaxations of the problem: every LP relaxation that approximates vertex cover within a factor $2-ε$ has super-polynomially many inequalities. As a direct consequence of our methods, we also establish that LP relaxations (as well as SDP relaxations) that approximate the independent set problem within any constant factor have super-polynomial size.

cs.CC↗
arXiv2015

Exponential Lower Bounds for Polytopes in Combinatorial Optimization

We solve a 20-year old problem posed by Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs.

math.CO↗
arXiv2015

Cut dominants and forbidden minors

The cut dominant of a graph is the unbounded polyhedron whose points are all those that dominate some convex combination of proper cuts. Minimizing a nonnegative linear function over the cut dominant is equivalent to finding a minimum weight cut in the graph. We give a forbidden-minor characterization of the graphs whose cut dominant can be defined by inequalities with integer coefficients and right-hand side at most 2. Our result is related to the forbidden-minor characterization of TSP-perfect graphs by Fonlupt and Naddef (Math. Prog., 1992). We prove that our result implies theirs, with a shorter proof. Furthermore, we establish general properties of forbidden minors for right-hand sides larger than 2.

math.CO↗
arXiv2015

Small Extended Formulations for Cyclic Polytopes

We provide an extended formulation of size O(log n)^{\lfloor d/2 \rfloor} for the cyclic polytope with dimension d and n vertices (i,i^2,\ldots,i^d), i in [n]. First, we find an extended formulation of size log(n) for d= 2. Then, we use this as base case to construct small-rank nonnegative factorizations of the slack matrices of higher-dimensional cyclic polytopes, by iterated tensor products. Through Yannakakis's factorization theorem, these factorizations yield small-size extended formulations for cyclic polytopes of dimension d>2.

math.OC↗
arXiv2014

Poset Entropy versus Number of Linear Extensions: the Width-$2$ Case

Kahn and Kim (J. Comput. Sci., 1995) have shown that for a finite poset $P$, the entropy of the incomparability graph of $P$ (normalized by multiplying by the order of $P$) and the base-$2$ logarithm of the number of linear extensions of $P$ are within constant factors from each other. The tight constant for the upper bound was recently shown to be $2$ by Cardinal, Fiorini, Joret, Jungers and Munro (STOC 2010, Combinatorica). Here, we refine this last result in case $P$ has width $2$: we show that the constant can be replaced by $2-\varepsilon$ if one also takes into account the number of connected components of size $2$ in the incomparability graph of $P$. Our result leads to a better upper bound for the number of comparisons in algorithms for the problem of sorting under partial information.

math.CO↗
arXiv2014

Generalised probabilistic theories and conic extensions of polytopes

Generalized probabilistic theories (GPT) provide a general framework that includes classical and quantum theories. It is described by a cone $C$ and its dual $C^*$. We show that whether some one-way communication complexity problems can be solved within a GPT is equivalent to the recently introduced cone factorisation of the corresponding communication matrix $M$. We also prove an analogue of Holevo's theorem: when the cone $C$ is contained in $\mathbb{R}^{n}$, the classical capacity of the channel realised by sending GPT states and measuring them is bounded by $\log n$. Polytopes and optimising functions over polytopes arise in many areas of discrete mathematics. A conic extension of a polytope is the intersection of a cone $C$ with an affine subspace whose projection onto the original space yields the desired polytope. Extensions of polytopes can sometimes be much simpler geometric objects than the polytope itself. The existence of a conic extension of a polytope is equivalent to that of a cone factorisation of the slack matrix of the polytope, on the same cone. We show that all $0/1$ polytopes whose vertices can be recognized by a polynomial size circuit, which includes as a special case the travelling salesman polytope and many other polytopes from combinatorial optimisation, have small conic extension complexity when the cone is the completely positive cone. Using recent exponential lower bounds on the linear extension complexity of polytopes, this provides an exponential gap between the communication complexity of GPT based on the completely positive cone and classical communication complexity, and a conjectured exponential gap with quantum communication complexity. Our work thus relates the communication complexity of generalisations of quantum theory to questions of mainstream interest in the area of combinatorial optimisation.

quant-ph↗
arXiv2014

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^Ω(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.

cs.CC↗