Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max k-cover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NP-hard. We prove that (1 \Gamma o(1)) ln n is a threshold below which set cover cannot be approximated efficiently, unless NP has slightly superpolynomial time algorithms. This closes the gap (up to low order terms) between the ratio of approximation achievable by the greedy algorithm (which is (1 \Gamma o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max k-cover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .

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 6= are considered. While Min RVLS with equations was known to be NP-hard in [27], we established in [2, 5] 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 [8]. In this paper we determine strong bou...

We survey some aspects of the computational complexity theory of discrete-time and discrete-state Hopfield networks. The emphasis is on topics that are not adequately covered by the existing survey literature, most significantly: 1. the known upper and lower bounds for the convergence times of Hopfield nets (here we consider mainly worst-case results); 2. the power of Hopfield nets as general computing devices (as opposed to their applications to associative memory and optimization); 3. the complexity of the synthesis ("learning") and analysis problems related to Hopfield nets as associative memories. Draft chapter for the forthcoming book The Computational and Learning Complexity of Neural Networks: Advanced Topics (ed. Ian Parberry).

We present a heuristic for finding a small independent dominating set, D, of cubic graphs. We analyse the performance of this heuristic, which is a random greedy algorithm, on random cubic graphs using differential equations and obtain an upper bound on the expected size of D. A corresponding lower bound is derived by means of a direct expectation argument. We prove that D asymptotically almost surely satisfies 0.2641n ≤ |D | ≤ 0.2794n. 1

A #-independent set S in a graph is parameterized by a set # of non-negative integers that constrains how the independent set S can dominate the remaining vertices (#v ## S : |N(v) # S| # #.) For all values of #, we classify as either NP-complete or polynomial-time solvable the problems of deciding if a given graph has a #-independent set. We complement this with approximation algorithms and inapproximability results, for all the corresponding optimization problems. These approximation results extend also to several related independence problems. In particular, we obtain a # m approximation of the Set Packing problem, where m is the number of base elements, as well as a # n approximation of the maximum independent set in power graphs G t , for t even. 1

The achromatic number ψ(G) of a graph G = (V, E) is the maximum k such that V has a partition V1, V2,..., Vk into independent sets, the union of no pair of which is independent. Here we show that ψ(G) can be viewed as the maximum over all minimal elements of a partial order defined on the set of all colourings of G. We introduce a natural refinement of this partial order, giving rise to a new parameter, which we call the b-chromatic number, ϕ(G), of G. We prove that determining ϕ(G) is NP-hard for general graphs, but polynomial-time solvable for trees.

The approximability of several NP maximization problems is investigated and strong lower bounds for the studied problems are proved. For some of the problems the bounds are the best that can be achieved, unless P = NP. For example we investigate the approximability of Max PB 0 \Gamma 1 Programming, the problem of finding a binary vector x that satisfies a set of linear relations such that the objective value P c i x i is maximized, where c i are binary numbers. We show that, unless P = NP, Max PB 0 \Gamma 1 Programming is not approximable within the factor n 1\Gamma" for any " ? 0, where n is the number of inequalities, and is not approximable within m 1=2\Gamma" for any " ? 0, where m is the number of variables. Similar hardness results are shown for other problems on binary linear systems, some problems on the satisfiability of boolean formulas and the longest induced cycle problem. 1 Introduction Approximation of NP-complete optimization problems is a very interesting and...

We consider the problem of finding an independent dominating set of minimum cardinality in bounded degree and regular graphs. We first give an approximation algorithm for at most cubic graphs, that achieves ratio 2, based on greedy and local search techniques. We then propose an heuristic based on an iterative application of the greedy technique on graphs of lower and lower degree. When the graph is at most cubic, a local search phase is executed in order to improve the performance. Keywords: Minimum Independent Dominating Set, Bounded Degree Graphs, Regular Graphs, Cubic Graphs, Greedy, Local Search. 1 Introduction It is widely known that many NP-complete graph problems remain NP-complete even if restricted to bounded degree and regular graphs [3]. On the other hand, variation in which the degree of the graph is bounded by a constant often allows to achieve different results with respect to the approximation properties. Namely, problems that for general graphs cannot be approximated ...

We investigate the computational complexity of the problem which consists, given a system of linear relations, of finding a solution violating as few relations as possible while satisfying all the others. This general combinatorial problem, referred to as Min ULR, is considered for the four basic types of relational operators =, , ? and 6=. We proved in [3] that Min ULR with =, or ? relations is NP-hard even when restricted to homogeneous systems with bipolar coefficients, whereas it is trivial for 6= relations. In this paper we determine strong bounds on the approximability of various intractable variants, including constrained ones where the variables are restricted to take bounded discrete values. The various NP-hard versions of Min ULR belong to different approximability classes depending on the type of relations and the additional constraints, but none of them can be approximated within any constant factor unless P=NP. In the process of studying Min ULR we also derive strong boun...

We investigate the computational complexity of two classes of combinatorial optimization problems related to linear systems and study the relationship between their approximability properties. 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 known to be NP-hard in [27], we established in [2, 6] 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 [8]. 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 bounded discrete 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. Two interesting special cases of Min RVLS and Min ULR that arise in discriminant analysis and machine learning are also discussed. In particular, we disprove a conjecture presented in [57] regarding the existence of a polynomial time algorithm to design linear classifiers (or perceptrons) that use a close-to-minimum number of features.