Viggo Kann

On the approximability of minimizing nonzero variables or unsatisfied relations in linear systems

We investigate the computational complexity of two closely related classes of combinatorial optimization problems for linear systems which arise in various fields such as machine learning, operations research and pattern recognition. In the first class (Min ULR) one wishes, given a possibly infeasible system of linear relations, to find a solution that violates as few relations as possible while satisfying all the others. In the second class (Min RVLS) the linear system is supposed to be feasible and one looks for a solution with as few nonzero variables as possible. For both Min ULR and Min RVLS the four basic types of relational operators =, ⩾, > and ≠ are considered. While Min RVLS with equations was mentioned to be NP-hard in (Garey and Johnson, 1979), we established in (Amaldi; 1992; Amaldi and Kann, 1995) that min ULR with equalities and inequalities are NP-hard even when restricted to homogeneous systems with bipolar coefficients. The latter problems have been shown hard to approximate in (Arora et al., 1993). In this paper we determine strong bounds on the approximability of various variants of Min RVLS and min ULR, including constrained ones where the variables are restricted to take binary values or where some relations are mandatory while others are optional. The various NP-hard versions turn out to have different approximability properties depending on the type of relations and the additional constraints, but none of them can be approximated within any constant factor, unless P = NP. Particular attention is devoted to two interesting special cases that occur in discriminant analysis and machine learning. In particular, we disprove a conjecture of van Horn and Martinez (1992) regarding the existence of a polynomial-time algorithm to design linear classifiers (or perceptrons) that involve a close-to-minimum number of features.

Some APX-completeness results for cubic graphs

Four fundamental graph problems, Minimum vertex cover, Maximum independent set, Minimum dominating set and Maximum cut, are shown to be APX-complete even for cubic graphs. Therefore, unless P = NP, these problems do not admit any polynomial time approximation scheme on input graphs of degree bounded by three.

Maximum bounded 3-dimensional matching is MAX SNP-complete

Abstract We prove that maximum 3-dimensional matching is a MAX SNP-hard problem. If the number of occurrences of elements in triples is bounded by a constant the problem is MAX SNP-complete. As corollaries we prove that maximum covering by 3-sets and maximum covering of a graph by triangles are MAX SNP-hard. The problems are MAX SNP-complete if the number of occurrences of the elements and the degree of the nodes respectively are bounded by a constant.

The complexity and approximability of finding maximum feasible subsystems of linear relations

We study the combinatorial problem which consists, given a system of linear relations, of finding a maximum feasible subsystem, that is a solution satisfying as many relations as possible. The computational complexity of this general problem, named Max FLS, is investigated for the four types of relations =, ⩾, > and ≠. Various constrained versions of Max FLS, where a subset of relations must be satisfied or where the variables take bounded discrete values, are also considered. We establish the complexity of solving these problems optimally and, whenever they are intractable, we determine their degree of approximability. Max FLS with =, ⩾ or > relations is NP-hard even when restricted to homogeneous systems with bipolar coefficients, whereas it can be solved in polynomial time for ≠ relations with real coefficients. The various NP-hard versions of Max FLS belong to different approximability classes depending on the type of relations and the additional constraints. We show that the range of approximability stretches from Apx-complete problems which can be approximated within a constant but not within every constant unless P = NP, to NPO PB-complete ones that are as hard to approximate as all NP optimization problems with polynomially bounded objective functions. While Max FLS with equations and integer coefficients cannot be approximated within pe for some e > 0, where p is the number of relations, the same problem over GF(q) for a prime q can be approximated within q but not within qe for some e > 0. Max FLS with strict or nonstrict inequalities can be approximated within 2 but not within every constant factor. Our results also provide strong bounds on the approximability of two variants of Max FLS with ⩾ and > relations that arise when training perceptrons, which are the building blocks of artificial neural networks, and when designing linear classifiers.

Hardness of Approximating Problems on Cubic Graphs

Four fundamental graph problems, Minimum vertex cover, Maximum independent set, Minimum dominating set and Maximum cut, are shown to be APX-complete even for cubic graphs. This means that unless P=NP these problems do not admit any polynomial time approximation scheme on input graphs of degree bounded by three.

Polynomially bounded minimization problems that are hard to approximate

Min PB is the class of minimization problems whose objective functions are bounded by a polynomial in the size of the input. We show that there exist several problems which are Min PB-complete with respect to an approximation preserving reduction. These problems are very hard to approximate; in polynomial time they cannot be approximated within ne for some e>0, where n is the size of the input, provided that P≠NP. In particular, the problem of finding the minimum independent dominating set in a graph, the problem of satisfying a 3-SAT formula setting the least number of variables to one, and the minimum bounded 0–1 programming problem are shown to be Min PB-complete. We also present a new type of approximation preserving reduction that is designed for problems whose approximability is expressed as a function in some size parameter. Using this reduction we obtain good lower bounds on the approximability of the treated problems.

Implementing an efficient part-of-speech tagger

An efficient implementation of a part‐of‐speech tagger for Swedish is described. The stochastic tagger uses a well‐established Markov model of the language. The tagger tags 92 per cent of unknown words correctly and up to 97 per cent of all words. Several implementation and optimization considerations are discussed. The main contribution of this paper is the thorough description of the tagging algorithm and the addition of a number of improvements. The paper contains enough detail for the reader to construct a tagger for his own language. Copyright

On the Approximability of the Maximum Common Subgraph Problem

Some versions of the maximum common subgraph problem are studied and approximation algorithms are given. The maximum bounded common induced subgraph problem is shown to be Max SNP-hard and the maximum unbounded common induced subgraph problem is shown to be as hard to approximate as the maximum independent set problem. The maximum common induced connected subgraph problem is still harder to approximate and is shown to be NPO PB-complete, i.e. complete in the class of optimization problems with optimal value bounded by a polynomial.

Maximum bounded H-matching is Max SNP-complete

Abstract We prove that maximum H-matching (the problem of determining the maximum number of node-disjoint copies of the fixed graph H contained in a variable graph) is a M AX SNP-hard problem for any graph H that has three or more nodes in some connected component. If H is connected and the degrees of the nodes in H are bounded by a constant the problem is M AX SNP-complete.

Approximability of maximum splitting of k -sets and some other Apx -complete problems

In this paper we shrink the gap for the Max 3-Set Splitting problem and its generalization Max k-Set Splitting from both ends, finding new lower bounds and new upper bounds