We study the natural extended-variable formulation for the disjunction of n+1 polytopes in ℝ^d . We demonstrate that the convex hull 𝒟 in the natural extended-variable space ℝ^d+n is given by full optimal big-M lifting (i) when d≤ 2 (and that it is not generally true for d≥ 3 ), and also (ii) when the polytopes are all axis-aligned hyper-rectangles. We give further results on the polyhedral structure of 𝒟 , emphasizing the role of full optimal big-M lifting.
We present a family of integer programming formulations for the maximum cut problem. These formulations encode the incidence vectors of the cuts of a connected graph by employing a subset of the odd-cycle inequalities that relate to a spanning tree, and they require only the corresponding edge variables to be integral explicitly. They so describe sufficient restrictions of the classic integer linear program by Barahona and Mahjoub. In addition, we characterize according formulations comprising facet-defining inequalities only. Trade-offs and comparisons to prevalent formulations concerning size and relaxation strength are subject to an experimental study.
We study the single pair capacitated network design problem and the budget constrained max flow problem on undirected series-parallel graphs. These problems were well studied on directed series-parallel graphs, but little is known in the context of undirected graphs. The major difference between the cases is that the source and sink of the problem instance do not necessarily coincide with the terminals of the underlying series-parallel graph in the undirected case, thus creating certain complications. We provide pseudopolynomial time algorithms to solve both of the problems and provide an FPTAS for the budget constrained max flow problem. We also provide some extensions, arguing important cases when the problems are polynomial-time solvable, and describing a series-parallel gadget that captures an edge upgrade version of the problems.
In this paper we study formulations and algorithms for the cycle clustering problem, a partitioning problem over the vertex set of a directed graph with nonnegative arc weights that is used to identify cyclic behavior in simulation data generated from nonreversible Markov state models. Here, in addition to partitioning the vertices into a set of coherent clusters, the resulting clusters must be ordered into a cycle such as to maximize the total net flow in the forward direction of the cycle. We provide a problem-specific binary programming formulation and compare it to a formulation based on the reformulation-linearization technique (RLT). We present theoretical results on the polytope associated with our custom formulation and develop primal heuristics and separation routines for both formulations. In computational experiments on simulation data from biology we find that branch and cut based on the problem-specific formulation outperforms the one based on RLT.
Given a connected undirected graph G = ( V , E ) , let G [ S ] be the subgraph of G induced by the set of vertices S ⊆ V . The Chordless Cycle Problem (CCP) consists in finding a subset S ⊆ V of maximum cardinality such that G [ S ] is a chordless cycle. We present a Quadratically Constrained reformulation for the CCP, derive a Semidefinite Programming (SDP) relaxation for it and solve that relaxation by Lagrangian Relaxation (LR). Compared to previously available dual bounds, our SDP bounds resulted to be quite strong. We then introduce a hybrid algorithm involving two combined phases: the LR scheme, which acts as a warm starter for a Branch-and-cut (BC) algorithm that follows it. In short, the LR algorithm allows us to formulate a finite set of SDP cuts that can be used to retrieve the SDP bounds in a Linear Programming relaxation for the CCP. Such cuts are not ready to be used by the BC as they are formulated in an extended variable space. Thus, the BC projects them back onto the original space of variables and separates them by solving a Linear Program. On dense input graphs, our proposed BC algorithm, in its current preliminary state of development, already outperforms its competitors in the literature.
We introduce learning augmented algorithms to the online graph coloring problem. Although the simple greedy algorithm FirstFit is known to perform poorly in the worst case, we are able to establish a relationship between the structure of any input graph G that is revealed online and the number of colors that FirstFit uses for G. Based on this relationship, we propose an online coloring algorithm FirstFitPredictions that extends FirstFit while making use of machine learned predictions. We show that FirstFitPredictions is both consistent and smooth. Moreover, we develop a novel framework for combining online algorithms at runtime specifically for the online graph coloring problem. Finally, we show how this framework can be used to robustify FirstFitPredictions by combining it with any classical online coloring algorithm (that disregards the predictions).
In this paper, we consider a generalization of the classical vertex coloring problem of a graph, where the edge set of the graph is partitioned into strong and weak edges ; the endpoints of a weak edge can be assigned to the same color and the minimum chromatic violation problem (MCVP) asks for a coloring of the graph minimizing the number of weak edges having its endpoints assigned to the same color. Previous works in the literature on MCVP focus on defining integer programming formulations and performing polyhedral studies on the associated polytopes but, to the best of our knowledge, very few computational complexity studies exist for MCVP. In this work, we focus on the computational complexity of this problem over several graph families such as interval and unit interval graphs, among others. We show that MCVP is NP-hard for general graphs and it remains NP-hard when the graph induced by the strong edges is unit interval or distance hereditary. On the other side, we provide a polynomial algorithm that properly solves MCVP when the graph is a unit interval graph without triangles with two or more weak edges.
This paper deals with scheduling the operations in systems with storage modeled as a mixed integer nonlinear program (MINLP). Due to time interdependency induced by storage, discrete control, and nonlinear operational conditions, computing even a feasible solution may require an unaffordable computational burden. We exploit a property common to a broad class of these problems to devise a decomposition algorithm related to alternating direction methods, which progressively adjusts the operations to the storage state profile. We also design a deep learning model to predict the continuous storage states to start the algorithm instead of the discrete decisions, as commonly done in the literature. This enables search diversification through a multi-start mechanism and prediction using scaling in the absence of a training set. Numerical experiments on the pump scheduling problem in water networks show the effectiveness of this hybrid learning/decomposition algorithm in computing near-optimal strict-feasible solutions in more reasonable times than other approaches.
In the total matching problem, one is given a graph $G$ with weights on the vertices and edges. The goal is to find a maximum weight set of vertices and edges that is the non-incident union of a stable set and a matching. We consider the natural formulation of the problem as an integer program (IP), with variables corresponding to vertices and edges. Let $M = M(G)$ denote the constraint matrix of this IP. We define $\Delta(G)$ as the maximum absolute value of the determinant of a square submatrix of $M$. We show that the total matching problem can be solved in strongly polynomial time provided $\Delta(G) \leq \Delta$ for some constant $\Delta \in \mathbb{Z}_{\ge 1}$. We also show that the problem of computing $\Delta(G)$ admits an FPT algorithm. We also establish further results on $\Delta(G)$ when $G$ is a forest.
In the context of classification, robustness verification of a neural network is the problem which consists in determining if small changes of inputs lead to a change of their assigned classes. We investigate such a problem on binarized neural networks via an integer linear programming perspective. We namely present a constraint generation framework based on disjunctive programming and complete descriptions of polytopes related to outputs of neuron pairs. We also introduce an alternative relying on specific families of facet defining inequalities. Preliminary experiments assess the performance of the latter approach against recent single neuron convexification results.
A polytope P⊂ [0,1]^n is said to have the persistency property if for every vector c∈ℝ^n and every c-optimal point x∈ P , there exists a c-optimal integer point y∈ P∩{0,1}^n such that x_i = y_i for each i ∈{1,… ,n} with x_i ∈{0,1} . In this paper, we consider a relaxation of the persistency property, called 1-persistency, over the clique relaxation of the stable set polytope in graphs. In particular, we study the family 𝒬 of graphs whose clique relaxation of the stable set polytope has 1-persistency. The main objective of this contribution is to analyze forbidden structures for a given graph to belong to 𝒬 . The graphs given by these structures are denoted here as mn𝒬 . On one hand, we provide sufficient conditions for a graph to belong to 𝒬 , and identify several graph classes of this family. On the other hand, we give two different infinite families of forbidden minimal structures for this class of graphs. We conclude the paper by suggesting an interesting future line of work about the persistency-preservation property of valid inequalities and its potential practical applications.
In the thief orienteering problem an agent called athiefcarries a knapsack of capacity $W$ and has a time limit $T$ to collect a set of items of total weight at most $W$ and maximum profit along a simple path in a weighted graph $G = (V, E)$ from a start vertex $s$ to an end vertex $t$. There is a set $I$ of items each with weight $w_{i}$ and profit $p_{i}$ that are distributed among $V \setminus \{s,t\}$. The time needed by the thief to travel an edge depends on the length of the edge and the weight of the items in the knapsack at the moment when the edge is traversed.There is a polynomial-time approximation scheme for the thief orienteering problem on directed acyclic graphs. We give a polynomial-time algorithm for transforming instances of the problem on series-parallel graphs into equivalent instances of the thief orienteering problem on directed acyclic graphs; therefore, yielding a polynomial-time approximation scheme for the thief orienteering problem on this graph class.
Computing the edge expansion of a graph is a famously hard combinatorial problem for which there have been many approximation studies. We present two versions of an exact algorithm using semidefinite programming (SDP) to compute this constant for any graph. The SDP relaxation is used to first reduce the search space considerably. One version applies then an SDP-based branch-and-bound algorithm, along with heuristic search. The other version transforms the problem into an instance of a max-cut problem and solves this using a state-of-the-art solver. Numerical results demonstrate that we clearly outperform mixed-integer quadratic solvers as well as another SDP-based algorithm from the literature.
For an undirected and edge-weighted graph G=(V, E) and a vertex subset S⊆ V , we define a function φ _G(S) := (1-α )· w(S) + α· w(S, V∖ S) , where α∈ [0, 1] is a real number, w(S) is the sum of weights of edges having two endpoints in S, and w(S, V∖ S) is the sum of weights of edges having one endpoint in S and the other in V∖ S . Then, given a graph G=(V, E) and a positive integer k, Max (Min) α -Fixed Cardinality Graph Partitioning (Max (Min) α -FCGP) is the problem to find a vertex subset S⊆ V of size k that maximizes (minimizes) φ _G(S) . In this paper, we first show that Max α -FCGP with α∈ [1/3,1] and Min α -FCGP with α∈ [0,1/3] can be solved in time 2^o(kd+k)(e+ed)^k n^O(1) where k is the solution size, d is the degeneracy of an input graph, and e is Napier’s constant.Then we consider Max (Min) Connected α -FCGP, which additionally requires the connectivity of a solution. For Max (Min) Connected α -FCGP, we give an (e ( -1))^k-1n^O(1) -time algorithm on general graphs and a 2^O(√(k)log ^2 k)n^O(1) -time randomized algorithm on apex-minor-free graphs. Moreover, for Max α -FCGP with α∈ [1/3,1] and Min α -FCGP with α∈ [0,1/3] , we propose an (1+d)^k 2^o(kd)+O(k) n^O(1) -time algorithm. Finally, we show that they admit FPT-ASs when edge weights are constant.
We consider a scheduling problem with reconfigurable resources. Several types of jobs have to be processed by a set of identical resources (e.g. robots, workers, processors) over a discrete time horizon. In each time period, teams of resources must be formed to process jobs. During a given time period, a team handles one type of job and the number of jobs that can be processed depends on the team size. A resource which is used to perform some job type in a given period may be employed for another job type in the next period. The objective is to determine the minimum number of resources needed to meet a given demand for each job type. We provide a polynomial-time 4/3-approximation algorithm for this strongly NP-hard problem.
This paper revisits the single machine scheduling problem to minimize total weighted completion times. The twist is that job sizes are stochastic from unknown distributions, and the scheduler has access to only a single sample from the distributions. For this restricted information regime, we analyze the simplest and probably only reasonable scheduling algorithm, namely to schedule by ordering the jobs by weight over sampled processing times. In general, this algorithm can be tricked by adversarial input distributions, performing in expectation arbitrarily worse even in comparison to choosing a random schedule. The paper suggests notions to capture the idea that this algorithm, on reasonable inputs, should exhibit a provably good expected performance.
This work introduces the class of crystal trees and their mathematical modeling. Consider a simple undirected weighted graph G=(V,E) of edge weights c_e>0 , for all e∈ E . Let T_k be a spanning tree of G rooted at vertex k∈ V , and 𝒫^T_k_ij denote the edge set of the path between vertices i and j in T_k . Associate with every vertex v∈ V of T_k a potential ^T_k_v ={c_e | e∈𝒫^T_k_kv} , with ^T_k_k=∅ . Let _uv= ^T_k_v ^T_k_u be the multiset symmetric difference between the potentials of the extremities of an edge {u,v}∈ E . We say that these potentials are: (i) in equilibrium if max {c_e| e∈ ^T_k_v ^T_k_u}≤ c_uv ; or (ii) in non-equilibrium, otherwise. If, for all edges in E, the potentials of their extremities are in equilibrium, then T_k is a crystal tree. We show that this new class of trees generalizes the Minimum Spanning Trees (MSTs) of a graph. We present theoretical results for crystal trees and an algebraic representation allowing us to describe MSTs by a system of linear inequalities. This opens up new possibilities for solving optimization problems with optimal tree structure in the set of constraints.
Tree decompositions are a powerful tool to obtain parameterized algorithms, in particular to solve different variants of the satisfiability problem. Most algorithms are based on a tree decomposition of the so called primal graph. Variants of the satisfiability problem that allow parameterized algorithms in the treewidth of the primal graph are for example Model Counting, MaxSat or QBF. To obtain efficient algorithms in practice, reducing the size of the instance by preprocessing is a very important technique and hence is highly investigated. In this paper, we investigate how preprocessing techniques can be used to reduce the parameter of a parameterized algorithm other than the size of the instance. In particular, we look at satisfiability and related problems and try to preprocess the formula in order to reduce the treewidth of the resulting primal graph. To the best of our knowledge, this is the first such approach. We show how to compute a set of auxiliary variables and an equisatisfiable (w.r.t. the original variables) formula using those such that the treewidth of the resulting primal graph is minimal under all sets of auxiliary variables. To reach this goal, we restrict our attention to auxiliary variables such that their value has to be the value of a subclause of the formula for each satisfying truth assignment. We implemented our approach and evaluated it on standard benchmark instances. While our approach is able to reduce the treewidth of around 10% of the instances, there is no clear improvement in the running time when solving the formula, due to the dependence of the practical efficiency of the solver on the structure of the formula.
In this paper, we conduct numerical experiments to test the effectiveness of several integer programming formulations of the cycle selection problem. Specifically, we carry out experiments to identify a maximum weighted cycle selection in random or in structured digraphs. The results show that random instances are relatively easy and that two formulations outperform the other ones in terms of total running time. We also examine variants of the problem obtained by adding a budget constraint and/or a maximum cycle length constraint. These variants are more challenging, especially when a budget constraint is imposed. To investigate the cycle selection problem with a maximum cycle length equal to 3, we provide an arc-based formulation with an exponential number of constraints that can be separated in polynomial time. All inequalities in the formulation are facet-defining for complete digraphs.
In this paper we study the single machine scheduling problem with release dates, deadlines and precedence relations where the objective is to minimize the makespan. This is a well-known strongly NP -hard scheduling problem [ 18 ]. We analyze the problem from the parameterized complexity point of view. We propose parameter q which is the maximum number of time windows [ r j , d j ) that can strictly include a time window [ r i , d i ) on both ends. We show that problems 1 | p r e c , r j , d j | C max and 1 | p r e c , r j | L max are fixed-parameter tractable parameterized by q . We use a dynamic programming approach and define a new dominance rule, which we call the weak earliest deadline rule. This rule narrows down the number of relevant scheduling prefixes enough to complete the search via a fixed-parameter tractable number of dynamic programming states.
