The cycle roommates problem: a hard case of kidney exchange

Recently, a number of interesting algorithmic problems have arisen from the emergence, in a number of countries, of kidney exchange schemes, whereby live donors are matched with recipients according to compatibility and other considerations. One such problem can be modeled by a variant of the well-known stable roommates problem in which blocking cycles, as well as the normal blocking pairs, are significant. We show here that this variant of the stable roommates problem is NP-complete, thus solving an open question posed by Cechlárová and Lacko.     

Detecting directed 4-cycles still faster

We present a method to detect simple cycles of length 4 of a directed graph in O(n^1/ωe^2−2/ω) steps, where n denotes the number of nodes, e denotes the number of edges and ω is the exponent of matrix multiplication. This improves upon the currently fastest methods for α∈(2/(4−ω),(ω+1)/2), where e=n^α.     

Hashiwokakero is NP-complete

In a Hashiwokakero puzzle, one must build bridges to connect a set of islands. We show that deciding the solvability of such puzzles is NP-complete.     

On finding cycle bases and fundamental cycle bases with a shortest maximal cycle

An undirected biconnected graph G with nonnegative integer lengths on the edges is given. The problem we consider is that of finding a cycle basis B of G such that the length of the longest cycle included in B is the smallest among all cycle bases of G. We first observe that Horton's algorithm [SIAM J. Comput. 16 (2) (1987) 358-366] provides a fast solution of the problem that extends the one given by Chickering et al. [Inform. Process. Lett. 54 (1995) 55-58] for uniform graphs. On the other hand we show that, if the basis is required to be fundamental, then the problem is NP-hard and cannot be approximated within 2−?, ∀?>0, even with uniform lengths, unless P=NP. This problem remains NP-hard even restricted to the class of complete graphs; in this case it cannot be approximated within 13/11−?, ∀?>0, unless P=NP; it is instead approximable within 2 in general, and within 3/2 if the triangle inequality holds.     

Aspect-ratio Voronoi diagram and its complexity bounds

This Letter first defines an aspect ratio of a triangle by the ratio of the longest side over the minimal height. Given a set of line segments, any point p in the plane is associated with the worst aspect ratio for all the triangles defined by the point and the line segments. When a line segment si gives the worst ratio, we say that p is dominated by si. Now, an aspect-ratio Voronoi diagram for a set of line segments is a partition of the plane by this dominance relation. We first give a formal definition of the Voronoi diagram and give O(n2+ε) upper bound and Ω(n²) lower bound on the complexity, where ε is any small positive number. The Voronoi diagram is interesting in itself, and it also has an application to a problem of finding an optimal point to insert into a simple polygon to have a triangulation that is optimal in the sense of the aspect ratio.     

The complexity of changing colourings with bounded maximum degree

Let G be a graph, x,y∈V(G), and ?:V(G)→[k] a k-colouring of G such that ?(x)=?(y). If then the following question is NP-complete: Does there exist a k-colouring ?^′ of G such that ?^′(x)≠?^′(y)? Conversely, if then the problem is polynomial time.     

Rigidity of a simple extended lower triangular matrix

For the all-ones lower triangular matrices, the upper and lower bounds on rigidity are known to match [P. Pudlak, Z. Vavrin, Computation of rigidity of order n²/r for one simple matrix, Comment Math. Univ. Carolin. 32 (2) (1991) 213-218]. In this short note, we apply these techniques to the all-ones extended lower triangular matrices, to obtain upper and lower bounds with a small gap between the two; we show that the rigidity is .     

Computing Vision Points in Polygons

We consider a restricted version of the art gallery problem within simple polygons in which the guards are required to lie on a given one-dimensional object, a watchman route. We call this problem the vision point problem . We prove the following: • The original art gallery problem is NP-hard for the very restricted class of street polygons. • The vision point problem can be solved efficiently for the class of street polygons. • A linear time algorithm for the vision point problem exists for the subclass of street polygons called straight walkable polygons. Received June 6, 1996; revised September 12, 1997.     

The complexity of the certification of properties of Stable Marriage

We give some lower bounds on the certificate complexity of some problems concerning stable marriage, answering a question of Gusfield and Irving.     

Improved upper bounds on synchronizing nondeterministic automata

We show that i-directable nondeterministic automata can be i-directed with a word of length O(2ⁿ) for i=1,2, where n stands for the number of states. Since for i=1,2 there exist i-directable automata having i-directing words of length Ω(2ⁿ), these upper bounds are asymptotically optimal. We also show that a 3-directable nondeterministic automaton with n states can be 3-directed with a word of length , improving the previously known upper bound O(2ⁿ). Here the best known lower bound is .     

Sensitivity versus block sensitivity of Boolean functions

Determining the maximal separation between sensitivity and block sensitivity of Boolean functions is of interest for computational complexity theory. We construct a sequence of Boolean functions with . The best known separation previously was due to Rubinstein. We also report results of computer search for functions with at most 13 variables.     

Nearly-perfect hypergraph packing is in NC

Answering a question of Rödl and Thoma, we show that the Nibble Algorithm for finding a collection of disjoint edges covering almost all vertices in an almost regular, uniform hypergraph with negligible pair degrees can be derandomized and parallelized to run in polylog time on polynomially many parallel processors. In other words, the nearly-perfect packing problem on this class of hypergraphs is in the complexity class NC.     

A polynomial time algorithm for obtaining minimum edge ranking on two-connected outerplanar graphs

An edge ranking of a graph G is a labeling r of its edges with positive integers such that every path between two different edges eu, ev with the same rank r(eu)=r(ev) contains an intermediate edge ew with rank r(ew)>r(eu). An edge ranking of G is minimum if the largest rank k assigned is the smallest among all rankings of G. The edge ranking problem is to find a minimum edge ranking of given graph G. This problem is NP-hard and no polynomial time algorithm for solving it is known for non-trivial classes of graphs other than the class of trees. In this paper, we first show, on a general graph G, a relation between a minimum edge ranking of G and its minimal cuts, which ensures that we can obtain a polynomial time algorithm for obtaining minimum edge ranking of a given graph G if minimal cuts for each subgraph of G can be found in polynomial time and the number of those is polynomial. Based on this relation, we develop a polynomial time algorithm for finding a minimum edge ranking on a 2-connected outerplanar graph.     

Bases for Boolean co-clones

The complexity of various problems in connection with Boolean constraints, like, for example, quantified Boolean constraint satisfaction, have been studied recently. Depending on what types of constraints may be used, the complexity of such problems varies. A very interesting observation of the recent past has been that the thus derived classification of constraints can be explained with the help of universal algebra. More precisely, the difficulty of such a constraint problem often depends on the co-clone the constraints are from. A co-clone is a set of Boolean relations that is closed under very natural closure operations. Nearly all these co-clones can be generated by said operators out of a finite set of relations, a so-called base. Knowing a, preferably simple, base for each co-clone can therefore be of great value when studying the complexity of Boolean constraint problems, since this knowledge reduces the infinitely many cases of equivalent problems to a single one—the constraint satisfaction problem for this base. In this paper we give a finite and simple base for every Boolean co-clone, where this is possible. We give evidence that the presented bases are as easy as possible.     

On the approximability of minmax (regret) network optimization problems

In this paper the minmax (regret) versions of some basic polynomially solvable deterministic network problems are discussed. It is shown that if the number of scenarios is unbounded, then the problems under consideration are not approximable within log1−?K for any ?>0 unless NP⊆DTIME(npolylogn), where K is the number of scenarios.     

An FPT algorithm in polynomial space for the Directed Steiner Tree problem with Limited number of Diffusing nodes

Given a directed graph with n nodes, a root r, a set X of k nodes called terminals and non-negative weights ω   over the arcs, the Directed Steiner Tree problem (DST) asks for a directed tree T^?$T^{?}$ of minimum cost ω(T^?)$\omega (T^{?})$ rooted at r and spanning X.