On the Power and Limitations of Branch and Cut

The Stabbing Planes proof system [Paul Beame et al., 2018] was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas - certain unsatisfiable systems of linear equations od 2 - which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari [Daniel Dadush and Samarth Tiwari, 2020] showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes. In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients (SP*) can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas

Generating general-purpose cutting planes for mixed-integer programs

Franz WesselmannPaderborn, Univ., Diss., 201

Robust Design of Single-Commodity Networks

The results in the present work were obtained in a collaboration with Eduardo Álvarez- Miranda, Valentina Cacchiani, Tim Dorneth, Michael Jünger, Frauke Liers, Andrea Lodi and Tiziano Parriani. The subject of this thesis is a robust network design problem, i.e., a problem of the type “dimension a network such that it has sufficient capacity in all likely scenarios.” In our case, we model the network with an undirected graph in which each scenario defines a supply or demand for each node. We say that a flow in the network is feasible for a scenario if it can balance out its supplies and demands. A scenario polytope B defines which scenarios are relevant. The task is now to find integer capacities that minimize the total installation costs while allowing for a feasible flow in each scenario. This problem is called Single-Commodity Robust Network Design Problem (sRND) and was introduced by Buchheim, Liers and Sanità (INOC 2011). The problem contains the Steiner Tree Problem (given an undirected graph and a terminal set, find a minimum cost subtree that connects all terminals) and therefore is N P-hard. The problem is also a natural extension of minimum cost flows. The network design literature treats the case that the scenario polytope B is given as the finite set of its extreme points (finite case) and that it is given as the feasible region of finitely many linear inequalities (polyhedral case). Both descriptions are equivalent, however, an efficient transformation is not possible in general. Buchheim, Liers and Sanità (INOC 2011) propose a Branch-and-Cut algorithm for the finite case. In this case, there exists a canonical problem formulation as a mixed integer linear program (MIP). It contains a set of flow variables for every scenario. Buchheim, Liers and Sanità enhance the formulation with general cutting planes that are called target cuts. The first part of the dissertation considers the problem variant where every scenario has exactly two terminal nodes. If the underlying network is a complete, unweighted graph, then this problem is the Network Synthesis Problem as defined by Chien (IBM Journal of R&D 1960). There exist polynomial time algorithms by Gomory and Hu (SIAM J. of Appl. Math 1961) and by Kabadi, Yan, Du and Nair (SIAM J. on Discr. Math.) for this special case. However, these algorithms are based on the fact that complete graphs are Hamiltonian. The result of this part is a similar algorithm for hypercube graphs that assumes a special distribution of the supplies and demands. These graphs are also Hamiltonian. The second part of the thesis discusses the structure of the polyhedron of feasible sRND solutions. Here, the first result is a new MIP-based capacity formulation for the sRND problem. The size of this formulation is independent of the number of extreme points of B and therefore, it is also suited for the polyhedral case. The formulation uses so-called cut-set inequalities that are known in similar form from other network design problems. By adapting a proof by Mattia (Computational Optimization and Applications 2013), we show that cut-set inequalities induce facets of the sRND polyhedron. To obtain a better linear programming relaxation of the capacity formulation, we interpret certain general mixed integer cuts as 3-partition inequalities and show that these inequalities induce facets as well. The capacity formulation has exponential size and we therefore need a separation algorithm for cut-set inequalities. In the finite case, we reduce the cut-set separation problem to a minimum cut problem that can be solved in polynomial time. In the polyhedral case, however, the separation problem is N P-hard, even if we assume that the scenario polytope is basically a cube. Such a scenario polytope is called Hose polytope. Nonetheless, we can solve the separation problem in practice: We show a MIP based separation procedure for the Hose scenario polytope. Additionally, the thesis presents two separation methods for 3-partition inequalities. These methods are independent of the encoding of the scenario polytope. Additionally, we present several rounding heuristics. The result is a Branch-and-Cut algorithm for the capacity formulation. We analyze the algorithm in the last part of the thesis. There, we show experimentally that the algorithm works in practice, both in the finite and in the polyhedral case. As a reference point, we use a CPLEX implementation of the flow based formulation and the computational results by Buchheim, Liers and Sanità. Our experiments show that the new Branch-and-Cut algorithm is an improvement over the existing approach. Here, the algorithm excels on problem instances with many scenarios. In particular, we can show that the MIP separation of the cut-set inequalities is practical

Cutting Planes Width and the Complexity of Graph Isomorphism Refutations

The width complexity measure plays a central role in Resolution and other propositional proof systems like Polynomial Calculus (under the name of degree). The study of width lower bounds is the most extended method for proving size lower bounds, and it is known that for these systems, proofs with small width also imply the existence of proofs with small size. Not much has been studied, however, about the width parameter in the Cutting Planes (CP) proof system, a measure that was introduced by Dantchev and Martin in 2011 under the name of CP cutwidth. In this paper, we study the width complexity of CP refutations of graph isomorphism formulas. For a pair of non-isomorphic graphs G and H, we show a direct connection between the Weisfeiler-Leman differentiation number WL(G, H) of the graphs and the width of a CP refutation for the corresponding isomorphism formula Iso(G, H). In particular, we show that if WL(G, H) ? k, then there is a CP refutation of Iso(G, H) with width k, and if WL(G, H) > k, then there are no CP refutations of Iso(G, H) with width k-2. Similar results are known for other proof systems, like Resolution, Sherali-Adams, or Polynomial Calculus. We also obtain polynomial-size CP refutations from our width bound for isomorphism formulas for graphs with constant WL-dimension

A Branch-and-Cut based Pricer for the Capacitated Vehicle Routing Problem

openIl Capacitated Vehicle Routing Problem, abbreviato come CVRP, è un problema di ottimizzazione combinatoria d'instradamento nel quale, un insieme geograficamente sparso di clienti con richieste note deve essere servito da una flotta di veicoli stazionati in una struttura centrale. Negli ultimi due decenni, tecniche di Column generation incorporate all'interno di frameworks branch-price-and-cut sono state infatti l'approccio stato dell'arte dominante per la costruzione di algoritmi esatti per il CVRP. Il pricer, un componente critico nella column generation, deve risolvere il Pricing Problem (PP) che richiede la risoluzione di un Elementary Shortest Path Problem with Resource Constraints (ESPPRC) in una rete di costo ridotto. Pochi sforzi scientifici sono stati dedicati allo studio di approcci branch-and-cut per affrontare il PP. L'ESPPRC è stato tradizionalmente rilassato e risolto attraverso algoritmi di programmazione dinamica. Questo approccio, tuttavia, ha due principali svantaggi. Per cominciare, peggiora i dual bounds ottenuti. Inoltre, il tempo di esecuzione diminuisce all'aumentare della lunghezza dei percorsi generati. Per valutare la performance dei loro contributi, la comunità di ricerca operativa ha tradizionalmente utilizzato una serie d'istanze di test storiche e artificiali. Tuttavia, queste istanze di benchmark non catturano le caratteristiche chiave dei moderni problemi di distribuzione del mondo reale, che sono tipicamente caratterizzati da lunghi percorsi. In questa tesi sviluppiamo uno schema basato su un approccio branch-and-cut per risolvere il pricing problem. Studiamo il comportamento e l'efficacia della nostra implementazione nel produrre percorsi più lunghi comparandola con soluzioni all'avanguardia basate su programmazione dinamica. I nostri risultati suggeriscono che gli approcci branch-and-cut possono supplementare il tradizionale algoritmo di etichettatura, indicando che ulteriore ricerca in quest'area possa portare benefici ai risolutori CVRP.The Capacitated Vehicle Routing Problem, CVRP for short, is a combinatorial optimization routing problem in which, a geographically dispersed set of customers with known demands must be served by a fleet of vehicles stationed at a central facility. Column generation techniques embedded within branch-price-and-cut frameworks have been the de facto state-of-the-art dominant approach for building exact algorithms for the CVRP over the last two decades. The pricer, a critical component in column generation, must solve the Pricing Problem (PP), which asks for an Elementary Shortest Path Problem with Resource Constraints (ESPPRC) in a reduced-cost network. Little scientific efforts have been dedicated to studying branch-and-cut based approaches for tackling the PP. The ESPPRC has been traditionally relaxed and solved through dynamic programming algorithms. This approach, however, has two major drawbacks. For starters, it worsens the obtained dual bounds. Furthermore, the running time degrades as the length of the generated routes increases. To evaluate the performance of their contributions, the operations research community has traditionally used a set of historical and artificial test instances. However, these benchmark instances do not capture the key characteristics of modern real-world distribution problems, which are usually characterized by longer routes. In this thesis, we develop a scheme based on a branch-and-cut approach for solving the pricing problem. We study the behavior and effectiveness of our implementation in producing longer routes by comparing it with state-of-the-art solutions based on dynamic programming. Our results suggest that branch-and-cut approaches may supplement the traditional labeling algorithm, indicating that further research in this area may bring benefits to CVRP solvers

Extremely Deep Proofs

We further the study of supercritical tradeoffs in proof and circuit complexity, which is a type of tradeoff between complexity parameters where restricting one complexity parameter forces another to exceed its worst-case upper bound. In particular, we prove a new family of supercritical tradeoffs between depth and size for Resolution, Res(k), and Cutting Planes proofs. For each of these proof systems we construct, for each c ? n^{1-?}, a formula with n^{O(c)} clauses and n variables that has a proof of size n^{O(c)} but in which any proof of size no more than roughly exponential in n^{1-?}/c must necessarily have depth ? n^c. By setting c = o(n^{1-?}) we therefore obtain exponential lower bounds on proof depth; this far exceeds the trivial worst-case upper bound of n. In doing so we give a simplified proof of a supercritical depth/width tradeoff for tree-like Resolution from [Alexander A. Razborov, 2016]. Finally, we outline several conjectures that would imply similar supercritical tradeoffs between size and depth in circuit complexity via lifting theorems

An Integer Programming approach to Bayesian Network Structure Learning

We study the problem of learning a Bayesian Network structure from data using an Integer Programming approach. We study the existing approaches, an in particular some recent works that formulate the problem as an Integer Programming model. By discussing some weaknesses of the existing approaches, we propose an alternative solution, based on a statistical sparsification of the search space. Results show how our approach can lead to promising results, especially for large network