Representations of Monotone Boolean Functions by Linear Programs

We introduce the notion of monotone linear-programming circuits (MLP circuits), a model of computation for partial Boolean functions. Using this model, we prove the following results. 1. MLP circuits are superpolynomially stronger than monotone Boolean circuits. 2. MLP circuits are exponentially stronger than monotone span programs. 3. MLP circuits can be used to provide monotone feasibility interpolation theorems for Lovasz-Schrijver proof systems, and for mixed Lovasz-Schrijver proof systems. 4. The Lovasz-Schrijver proof system cannot be polynomially simulated by the cutting planes proof system. This is the first result showing a separation between these two proof systems. Finally, we discuss connections between the problem of proving lower bounds on the size of MLPs and the problem of proving lower bounds on extended formulations of polytopes

On the Integrality Gap of Directed Steiner Tree Problem

In the Directed Steiner Tree problem, we are given a directed graph G = (V,E) with edge costs, a root vertex r ∈ V, and a terminal set X ⊆ V . The goal is to find the cheapest subset of edges that contains an r-t path for every terminal t ∈ X. The only known polylogarithmic approximations for Directed Steiner Tree run in quasi-polynomial time and the best polynomial time approximations only achieve a guarantee of O(|X|^ε) for any constant ε > 0. Furthermore, the integrality gap of a natural LP relaxation can be as bad as Ω(√|X|).  We demonstrate that l rounds of the Sherali-Adams hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in l-layered graphs from Ω( k) to O(l · log k) where k is the number of terminals. This is an improvement over Rothvoss’ result that 2l rounds of the considerably stronger Lasserre SDP hierarchy reduce the integrality gap of a similar formulation to O(l · log k). We also observe that Directed Steiner Tree instances with 3 layers of edges have only an O(logk) integrality gap bound in the standard LP relaxation, complementing the fact that the gap can be as large as Ω(√k) in graphs with 4 layers. Finally, we consider quasi-bipartite instances of Directed Steiner Tree meaning no edge in E connects two Steiner nodes V − (X ∪ {r}). By a simple reduction from Set Cover, it is still NP-hard to approximate quasi-bipartite instances within a ratio better than O(log|X|). We present a polynomial-time O(log |X|)-approximation for quasi-bipartite instances of Directed Steiner Tree. Our approach also bounds the integrality gap of the natural LP relaxation by the same quantity. A novel feature of our algorithm is that it is based on the primal-dual framework, which typically does not result in good approximations for network design problems in directed graphs

Lower Bounds on Query Complexity for Testing Bounded-Degree CSPs

In this paper, we consider lower bounds on the query complexity for testing CSPs in the bounded-degree model. First, for any ``symmetric'' predicate $P:{0,1}^{k} \to {0,1}$ except \equ where $k\geq 3$, we show that every (randomized) algorithm that distinguishes satisfiable instances of CSP(P) from instances $(|P^{-1}(0)|/2^k-\epsilon)$-far from satisfiability requires $\Omega(n^{1/2+\delta})$ queries where $n$ is the number of variables and $\delta>0$ is a constant that depends on $P$ and $\epsilon$. This breaks a natural lower bound $\Omega(n^{1/2})$, which is obtained by the birthday paradox. We also show that every one-sided error tester requires $\Omega(n)$ queries for such $P$. These results are hereditary in the sense that the same results hold for any predicate $Q$ such that $P^{-1}(1) \subseteq Q^{-1}(1)$. For EQU, we give a one-sided error tester whose query complexity is $\tilde{O}(n^{1/2})$. Also, for 2-XOR (or, equivalently E2LIN2), we show an $\Omega(n^{1/2+\delta})$ lower bound for distinguishing instances between $\epsilon$-close to and $(1/2-\epsilon)$-far from satisfiability. Next, for the general k-CSP over the binary domain, we show that every algorithm that distinguishes satisfiable instances from instances $(1-2k/2^k-\epsilon)$-far from satisfiability requires $\Omega(n)$ queries. The matching NP-hardness is not known, even assuming the Unique Games Conjecture or the $d$-to-$1$ Conjecture. As a corollary, for Maximum Independent Set on graphs with $n$ vertices and a degree bound $d$, we show that every approximation algorithm within a factor d/\poly\log d and an additive error of $\epsilon n$ requires $\Omega(n)$ queries. Previously, only super-constant lower bounds were known

Rank Lower Bounds in Propositional Proof Systems Based on Integer Linear Programming Methods

The work of this thesis is in the area of proof complexity, an area which looks to uncover the limitations of proof systems. In this thesis we investigate the rank complexity of tautologies for several of the most important proof systems based on integer linear programming methods. The three main contributions of this thesis are as follows: Firstly we develop the first rank lower bounds for the proof system based on the Sherali-Adams operator and show that both the Pigeonhole and Least Number Principles require linear rank in this system. We also demonstrate a link between the complexity measures of Sherali-Adams rank and Resolution width. Secondly we present a novel method for deriving rank lower bounds in the well-studied Cutting Planes proof system. We use this technique to show that the Cutting Plane rank of the Pigeonhole Principle is logarithmic. Finally we separate the complexity measures of Resolution width and Sherali-Adams rank from the complexity measures of Lovasz and Schrijver rank and Cutting Planes rank

Size bounds for algebraic and semialgebraic proof systems

This thesis concerns the proof complexity of algebraic and semialgebraic proof systems Polynomial Calculus, Sums-of-Squares and Sherali-Adams. The most studied complexity measure for these systems is the degree of the proofs. This thesis concentrates on other possible complexity measures of interest to proof complexity, monomial-size and bit-complexity. We aim to showcase that there is a reasonably well-behaved theory for these measures also. Firstly we tie the complexity measures of degree and monomial size together by proving a size-degree trade-off for Sums-of-Squares and Sherali-Adams. We show that if there is a refutation with at most s many monomials, then there is a refutation whose degree is of order square root of n log s plus k, where k is the maximum degree of the constraints and n is the number of variables. For Polynomial Calculus similar trade-off was obtained earlier by Impagliazzo, Pudlák and Sgall. Secondly we prove a feasible interpolation property for all three systems. We show that for each system there is a polynomial time algorithm that given two sets P(x,z) and Q(y,z) of polynomial constraints in disjoint sequences x,y and z of variables, a refutation of the union of P(x,z) and Q(y,z), and an assignment a to the z-variables, finds either a refutation of P(x,a) or a refutation of Q(y,a). Finally we consider the relation between monomial-size and bit-complexity in Polynomial Calculus and Sums-of-Squares. We show that there is an unsatisfiable set of polynomial constraints that has both Polynomial Calculus and Sums-of-Squares refutations of polynomial monomial-size, but for which any Polynomial Calculus or Sums-of-Squares refutation requires exponential bit-complexity. Besides the emphasis on complexity measures other than degree, another unifying theme in all the three results is the use of semantic characterizations of resource-bounded proofs and refutations. All results make heavy use of the completeness properties of such characterizations. All in all, the work on these semantic characterizations presents itself as the fourth central contribution of this thesis.Aquesta tesi tracta de la complexitat de les proves en els sistemes de prova algebraics i semialgebraics Càlcul Polinomial (Polynomial Calculus), Sumes de Quadrats (Sums of Squares), i Sherali-Adams. La mesura de complexitat més estudiada per a aquests sistemes és el grau dels polinomis. Aquesta tesi se centra en altres possibles mesures de complexitat d'interès per a la complexitat de proves: el nombre de monomis i la longitud de representació en nombre de bits. Pretenem demostrar que aquestes mesures admeten una teoria comparable i complementària a la teoria del grau com a mesura de complexitat. En primer lloc, establim una relació entre les mesures de grau i de nombre de monomis demostrant una propietat d'intercanvi (trade-off) entre les dues mesures per als sistemes Sumes de Quadrats i Sherali-Adams. Demostrem que si hi ha una refutació amb com a màxim s monomis, aleshores hi ha una refutació el grau de la qual és d'ordre de l'arrel quadrada de n.log(s) més k, on k és el grau màxim de les restriccions i n és el nombre de variables. Per al Càlcul Polinomial, una propietat d'intercanvi similar va ser obtinguda per Impagliazzo, Pudlák i Sgall. En segon lloc, demostrem que els tres sistemes admeten la propietat d'interpolació eficient. Mostrem que, per a cadascun dels sistemes, hi ha un algorisme de temps polinomial que, donat dos conjunts P(x,z) i Q(y,z) de restriccions polinomials en successions disjuntes de variables x, y i z, donada una refutació de la unió de les restriccions de P(x,z) i Q(y,z), i donada una assignació per a les variables z, troba una refutació de P(x,a) o una refutació de Q(y,a). Finalment considerem la relació entre el nombre de monomis i la longitud de representació en bits per al Càlcul Polinomial i per a Sumes de Quadrats. Mostrem que hi ha un conjunt insatisfactible de restriccions polinomials que admet refutacions tant en Càlcul Polinomial com en Sumes de Quadrats amb un nombre polinòmic de monomis, però per a les quals qualsevol refutació en Càlcul Polinomial o en Sumes de Quadrats requereix complexitat en nombre de bits exponencial. A més de l'èmfasi en les mesures de complexitat diferents del grau, un altre tema unificador en els tres resultats és l'ús de certes caracteritzacions semàntiques de proves i refutacions limitades en recursos. Tots els resultats fan un ús clau de la propietat de completesa d'aquestes caracteritzacions. Amb tot, el treball sobre aquestes caracteritzacions semàntiques es presenta com la quarta aportació central d'aquesta tesi.Postprint (published version

Facets and Algorithms for Capacitated Lot Sizing

The dynamic economic lot sizing model, which lies at the core of numerous production planning applications, is one of the most highly studied models in all of operations research. And yet, capacitated multi-item versions of this problem remain computationally elusive. We study the polyhedral structure of an integer programming formulation of a single-item capacitated version of this problem, and use these results to develop solution methods for multi-item applications. In particular, we introduce a set of valid inequalities for the problem and show that they define facets of the underlying integer programming polyhedron. Computational results on several single and multiple product examples show that these inequalities can be used quite effectively to develop an efficient cutting plane/branch and bound procedure. Moreover, our results show that in many instances adding certain of these inequalities a priori to the problem formulation, and avoiding the generation of cutting planes, can be equally effective