Series and parallel reductions for the Tutte polynomial

We discuss reducing the number of steps involved in computing the Tutte polynomial of a matroid by using series and parallel reductions in conjunction with the usual deletion and contraction operations.     

An analysis of neighborhood functions on generic solution spaces

The effectiveness of local search algorithms on discrete optimization problems is influenced by the choice of the neighborhood function. A neighborhood function that results in all local minima being global minima is said to have zero L-locals. A polynomially sized neighborhood function with zero L-locals would ensure that at each iteration, a local search algorithm would be able to find an improving solution or conclude that the current solution is a global minimum. This paper presents a recursive relationship for computing the number of neighborhood functions over a generic solution space that results in zero L-locals. Expressions are also given for the number of tree neighborhood functions with zero L-locals. These results provide a first step towards developing expressions that are applicable to discrete optimization problems, as well as providing results that add to the collection of solved graphical enumeration problems.     

A generic approach to proving NP-hardness of partition type problems

This note presents a generic approach to proving NP-hardness of unconstrained partition type problems, namely partitioning a given set of entities into several subsets such that a certain objective function of the partition is optimized. The idea is to represent the objective function of the problem as a function of aggregate variables, whose optimum is achieved only at the points where problem Partition (if proving ordinary NP-hardness), or problem 3-Partition or Product Partition (if proving strong NP-hardness) has a solution. The approach is demonstrated on a number of discrete optimization and scheduling problems.     

Satisfiability of algebraic circuits over sets of natural numbers

We investigate the complexity of satisfiability problems for {∪,∩,⁻,+,×}-circuits computing sets of natural numbers. These problems are a natural generalization of membership problems for expressions and circuits studied by Stockmeyer and Meyer (1973) [10] and McKenzie and Wagner (2003) [8].Our work shows that satisfiability problems capture a wide range of complexity classes such as NL, P, NP, PSPACE, and beyond. We show that in several cases, satisfiability problems are harder than membership problems. In particular, we prove that testing satisfiability for {∩,+,×}-circuits is already undecidable. In contrast to this, the satisfiability for {∪,+,×}-circuits is decidable in PSPACE.     

An improved semidefinite programming relaxation for the satisfiability problem

The satisfiability (SAT) problem is a central problem in mathematical logic, computing theory, and artificial intelligence. An instance of SAT is specified by a set of boolean variables and a propositional formula in conjunctive normal form. Given such an instance, the SAT problem asks whether there is a truth assignment to the variables such that the formula is satisfied. It is well known that SAT is in general NP-complete, although several important special cases can be solved in polynomial time. Semidefinite programming (SDP) refers to the class of optimization problems where a linear function of a matrix variable X is maximized (or minimized) subject to linear constraints on the elements of X and the additional constraint that X be positive semidefinite. We are interested in the application of SDP to satisfiability problems, and in particular in how SDP can be used to detect unsatisfiability. In this paper we introduce a new SDP relaxation for the satisfiability problem. This SDP relaxation arises from the recently introduced paradigm of higher liftings for constructing semidefinite programming relaxations of discrete optimization problems. To derive the SDP relaxation, we first formulate SAT as an optimization problem involving matrices. Relaxing this formulation yields an SDP which significantly improves on the previous relaxations in the literature. The important characteristics of the SDP relaxation are its ability to prove that a given SAT formula is unsatisfiable independently of the lengths of the clauses in the formula, its potential to yield truth assignments satisfying the SAT instance if a feasible matrix of sufficiently low rank is computed, and the fact that it is more amenable to practical computation than previous SDPs arising from higher liftings. We present theoretical and computational results that support these claims.Mathematics Subject Classification (2000): 20E28, 20G40, 20C20     

Open problems around exact algorithms

We list a number of open questions around worst case time bounds and worst case space bounds for NP-hard problems. We are interested in exponential time solutions for these problems with a relatively good worst case behavior. We summarize what is known on these problems, we discuss related results, and we provide pointers to the literature.     

Beyond Convex? Global Optimization is Feasible Only for Convex Objective Functions: A Theorem

It is known that there are feasible algorithms for minimizing convex functions, and that for general functions, global minimization is a difficult (NP-hard) problem. It is reasonable to ask whether there exists a class of functions that is larger than the class of all convex functions for which we can still solve the corresponding minimization problems feasibly. In this paper, we prove, in essence, that no such more general class exists. In other words, we prove that global optimization is always feasible only for convex objective functions.     

Multigraph realizations of degree sequences: Maximization is easy, minimization is hard

The following minimization problem is shown to be NP-hard: Given a graphic degree sequence, find a realization of this degree sequence as loopless multigraph that minimizes the number of edges in the underlying support graph. The corresponding maximization problem is known to be solvable in polynomial time.     

Pinpointing the complexity of the interval min–max regret knapsack problem

We show that a natural robust optimization variant of the knapsack problem is complete for the second level of the polynomial hierarchy.     

On the robust assignment problem under a fixed number of cost scenarios

We investigate the complexity of the min-max assignment problem under a fixed number of scenarios. We prove that this problem is polynomial-time equivalent to the exact perfect matching problem in bipartite graphs, an infamous combinatorial optimization problem of unknown computational complexity.     

Combinatorial optimization with interaction costs: Complexity and solvable cases

We introduce and study the combinatorial optimization problem with interaction costs (COPIC). COPIC is the problem of finding two combinatorial structures, one from each of two given families, such that the sum of their independent linear costs and the interaction costs between elements of the two selected structures is minimized. COPIC generalizes the quadratic assignment problem and many other well studied combinatorial optimization problems, and hence covers many real world applications. We show how various topics from different areas in the literature can be formulated as special cases of COPIC. The main contributions of this paper are results on the computational complexity and approximability of COPIC for different families of combinatorial structures (e.g. spanning trees, paths, matroids), and special structures of the interaction costs. More specifically, we analyze the complexity if the interaction cost matrix is parameterized by its rank and if it is a diagonal matrix. Also, we determine the structure of the intersection cost matrix, such that COPIC is equivalent to independently solving linear optimization problems for the two given families of combinatorial structures.     

Unconstrained and constrained global optimization of polynomial functions in one variable

In Floudas and Visweswaran (1990), a new global optimization algorithm (GOP) was proposed for solving constrained nonconvex problems involving quadratic and polynomial functions in the objective function and/or constraints. In this paper, the application of this algorithm to the special case of polynomial functions of one variable is discussed. The special nature of polynomial functions enables considerable simplification of the GOP algorithm. The primal problem is shown to reduce to a simple function evaluation, while the relaxed dual problem is equivalent to the simultaneous solution of two linear equations in two variables. In addition, the one-to-one correspondence between the x and y variables in the problem enables the iterative improvement of the bounds used in the relaxed dual problem. The simplified approach is illustrated through a simple example that shows the significant improvement in the underestimating function obtained from the application of the modified algorithm. The application of the algorithm to several unconstrained and constrained polynomial function problems is demonstrated.     

The hardness of train rearrangements

We derive several results on the computational complexity of train rearrangement problems in railway optimization. Our main result states that arranging a departing train in a depot is NP-complete, even if each track in the depot contains only two cars.     

Bottleneck combinatorial optimization problems with uncertain costs and the OWA criterion

In this paper a class of bottleneck combinatorial optimization problems with uncertain costs is discussed. The uncertainty is modeled by specifying a discrete scenario set containing a finite number of cost vectors, called scenarios. In order to choose a solution the Ordered Weighted Averaging aggregation operator (OWA for short) is applied. The OWA operator generalizes traditional criteria in decision making under uncertainty such as the maximum, minimum, average, median, or Hurwicz criterion. New complexity and approximation results in this area are provided. These results are general and remain valid for many problems, in particular for a wide class of network problems.     

Integer equal flows

The integer equal flow problem is an NP-hard network flow problem, in which all arcs in given sets R₁,…,R_ℓ must carry equal flow. We show that this problem is effectively inapproximable, even if the cardinality of each set R_k is two. When ℓ is fixed, it is solvable in polynomial time.     

An inverse model for the most uniform problem

In this paper, we consider a kind of inverse model for the most uniform problem. This model has some practical background. It is shown that the model can be solved in polynomial time whenever an associated min-sum problem can be solved in polynomial time.     

Nonintersecting paths and the Hahn orthogonal polynomial ensemble

We compute the bulk limit of the correlation functions for the uniform measure on lozenge tilings of a hexagon. The limiting determinantal process is a translation-invariant extension of the discrete sine process, which can also be described by an ergodic Gibbs measure with appropriate parameters.