Showing papers in "SIAM Journal on Discrete Mathematics in 2007"

On the Metric Dimension of Cartesian Products of Graphs

Abstract: A set of vertices $S$ resolves a graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The metric dimension of $G$ is the minimum cardinality of a resolving set of $G$. This paper studies the metric dimension of cartesian products $G\,\square\,H$. We prove that the metric dimension of $G\,\square\,G$ is tied in a strong sense to the minimum order of a so-called doubly resolving set in $G$. Using bounds on the order of doubly resolving sets, we establish bounds on $G\,\square\,H$ for many examples of $G$ and $H$. One of our main results is a family of graphs $G$ with bounded metric dimension for which the metric dimension of $G\,\square\,G$ is unbounded.

Adjacent Vertex Distinguishing Edge-Colorings

Abstract: An adjacent vertex distinguishing edge-coloring of a simple graph $G$ is a proper edge-coloring of $G$ such that no pair of adjacent vertices meets the same set of colors. The minimum number of colors $\chi^\prime_a(G)$ required to give $G$ an adjacent vertex distinguishing coloring is studied for graphs with no isolated edge. We prove $\chi^\prime_a(G)\le5$ for such graphs with maximum degree $\Delta(G)=3$ and prove $\chi^\prime_a(G)\le\Delta(G)+2$ for bipartite graphs. These bounds are tight. For $k$-chromatic graphs $G$ without isolated edges we prove a weaker result of the form $\chi^\prime_a(G)=\Delta(G)+O(\log k)$.

The Spectrum of the Corona of Two Graphs

Abstract: We consider only simple graphs. Given two graphs $G$ with vertices $1,\ldots,n$ and $H$, the corona $G\circ H$ is defined as the graph obtained by taking $n$ copies of $H$ and for each $i$ inserting edges between the $i$th vertex of $G$ and each vertex of the $i$th copy of $H$. For a connected graph $G$ and any $r$-regular graph $H$ we provide complete information about the spectrum of $G\circ H$ using the spectrum of $G$ and spectrum of $H$. Complete information about the Laplacian spectrum of $G\circ H$ is also provided even when $H$ is not regular. A graph $G$ is said to have the property (R) if $\frac{1}{\lambda}$ is an eigenvalue of $G$ whenever $\lambda$ is an eigenvalue of $G$. Further, if $\lambda$ and $\frac{1}{\lambda}$ have the same multiplicity, for each eigenvalue $\lambda$, then it is said to have the property (SR). We characterize all trees with property (SR) and show that such a tree is the corona product of some tree and an isolated vertex. We supply a family of bipartite graphs with property (R). As an application we construct infinitely many pairs of nonisomorphic graphs with the same spectrum and the same Laplacian spectrum. We prove some results about the eigenvector related to the second smallest eigenvalue of the Laplacian matrix of $G\circ H$ and give an application.

Correlation of Graph-Theoretical Indices

Abstract: The correlation of graph characteristics, such as the number of independent vertex or edge subsets, the number of connected subsets, or the sum of distances, which also play a role in combinatorial chemistry, is studied by a generating function approach and asymptotic analysis. It is shown how an asymptotic formula for the correlation coefficient can be obtained when simply generated families of trees are investigated. For rooted ordered trees, the calculations are done explicitly. Further feasible correlation measures are discussed.

Tura´n’s Theorem in the Hypercube

Abstract: We are motivated by the analogue of Tura´n’s theorem in the hypercube $Q_n$: How many edges can a $Q_d$-free subgraph of $Q_n$ have? We study this question through its Ramsey-type variant and obtain asymptotic results. We show that for every odd $d$ it is possible to color the edges of $Q_n$ with $\frac{(d+1)^2}{4}$ colors such that each subcube $Q_d$ is polychromatic, that is, contains an edge of each color. The number of colors is tight up to a constant factor, as it turns out that a similar coloring with ${d+1\choose 2} +1$ colors is not possible. The corresponding question for vertices is also considered. It is not possible to color the vertices of $Q_n$ with $d+2$ colors such that any $Q_d$ is polychromatic, but there is a simple $d+1$ coloring with this property. A relationship to anti-Ramsey colorings is also discussed. We discover much less about the Tura´n-type question which motivated our investigations. Numerous problems and conjectures are raised.

A Primal Barvinok Algorithm Based on Irrational Decompositions

Abstract: We introduce variants of Barvinok’s algorithm for counting lattice points in polyhedra. The new algorithms are based on irrational signed decomposition in the primal space and the construction of rational generating functions for cones with low index. We give computational results that show that the new algorithms are faster than the existing algorithms by a large factor.

On The Chromatic Number of Geometric Hypergraphs

Abstract: A finite family $\mathcal{R}$ of simple Jordan regions in the plane defines a hypergraph $H=H(\mathcal{R})$ where the vertex set of $H$ is $\mathcal{R}$ and the hyperedges are all subsets $S \subset \R$ for which there is a point $p$ such that $S = \{r \in \mathcal{R} | p \in r\}$. The chromatic number of $H(\mathcal{R})$ is the minimum number of colors needed to color the members of $\mathcal{R}$ such that no hyperedge is monochromatic. In this paper we initiate the study of the chromatic number of such hypergraphs and obtain the following results: (i) Any hypergraph that is induced by a family of $n$ simple Jordan regions such that the maximum union complexity of any $k$ of them (for $1\leq k \leq m$) is bounded by $\mathcal{U}(m)$ and $\frac{\mathcal{U}(m)}{m}$ is a nondecreasing function is $O(\frac{\mathcal{U}(n)}{n})$-colorable. Thus, for example, we prove that any finite family of pseudo-discs can be colored with a constant number of colors. (ii) Any hypergraph induced by a finite family of planar discs is four colorable. This bound is tight. In fact, we prove that this statement is equivalent to the four-color theorem. (iii) Any hypergraph induced by $n$ axis-parallel rectangles is $O(\log n)$-colorable. This bound is asymptotically tight. Our proofs are constructive. Namely, we provide deterministic polynomial-time algorithms for coloring such hypergraphs with only “few” colors (that is, the number of colors used by these algorithms is upper bounded by the same bounds we obtain on the chromatic number of the given hypergraphs). As an application of (i) and (ii) we obtain simple constructive proofs for the following: (iv) Any set of $n$ Jordan regions with near linear union complexity admits a conflict-free (CF) coloring with polylogarithmic number of colors. (v) Any set of $n$ axis-parallel rectangles admits a CF-coloring with $O(\log^2(n))$ colors.

Vertex-Magic Total Labelings of Regular Graphs

Abstract: In this paper it is shown that if a graph $G$ possesses a spanning subgraph $H$ with a strong vertex magic total labeling (VMTL) and $G-E(H)$ is even-regular, then $G$ also has a strong VMTL. Among other things, this is used to conclude that all Hamiltonian regular graphs of odd order possess strong VMTLs. A relationship is then demonstrated between regular graphs of even degree and sparse magic squares. We next consider cubic graphs of order $2n$ consisting of two 2-factors of order $n$, connected by a 1-factor (quasi-prisms). Based on McQuillan’s construction of VMTLs of such 3-regular graphs, VMTLs are derived for similar regular graphs of any odd degree. Finally, a construction is given for VMTLs of quartic graphs of order $4n+2$ consisting of two cycles of odd order $n$ connected by a 2-factor (simple quasi-anti-prisms), and based on this construction VMTLs are derived for similar regular graphs of any even degree.

Large Complete Bipartite Subgraphs In Incidence Graphs Of Points And Hyperplanes

Abstract: We show that if the number $I$ of incidences between $m$ points and $n$ planes in $\mathbb{R}^3$ is sufficiently large, then the incidence graph (which connects points to their incident planes) contains a large complete bipartite subgraph involving $r$ points and $s$ planes, so that $rs \ge \frac{I^2}{mn} - a(m+n)$, for some constant $a>0$. This is shown to be almost tight in the worst case because there are examples of arbitrarily large sets of points and planes where the largest complete bipartite incidence subgraph records only $\frac{I^2}{mn}-\frac{m+n}{16}$ incidences. We also take some steps towards generalizing this result to higher dimensions.

On Nearly Orthogonal Lattice Bases and Random Lattices

Abstract: We study lattice bases where the angle between any basis vector and the linear subspace spanned by the other basis vectors is at least $\frac{\pi}{3}$ radians; we denote such bases as “nearly orthogonal.” We show that a nearly orthogonal lattice basis always contains a shortest lattice vector. Moreover, we prove that if the basis vector lengths are “nearly equal,” then the basis is the unique nearly orthogonal lattice basis up to multiplication of basis vectors by $\pm 1$. We also study random lattices generated by the columns of random matrices with $n$ rows and $m \leq n$ columns. We show that if $m \leq c\,n$, with $c \approx 0.071$, then the random matrix forms a nearly orthogonal basis for the random lattice with high probability for large $n$ and almost surely as $n$ tends to infinity. Consequently, the columns of such a random matrix contain the shortest vector in the random lattice. Finally, we discuss an interesting JPEG image compression application where nearly orthogonal lattice bases play an important role.

Ramsey Properties of Random k-Partite, k-Uniform Hypergraphs

Abstract: We investigate the threshold probability for the property that every $r$-coloring of the edges of a random binomial $k$-uniform hypergraph ${\mathbb G }^{(k)}(n,p)$ yields a monochromatic copy of some fixed hypergraph $G$. In this paper we solve the problem for arbitrary $k\geq 3$ and $k$-partite, $k$-uniform hypergraphs $G$.

Virtual Private Network Design: A Proof of the Tree Routing Conjecture on Ring Networks

Abstract: A basic question in virtual private network (VPN) design is if the symmetric version of the problem always has an optimal solution which is a tree network. An affirmative answer would imply that the symmetric VPN problem is solvable in polynomial time. We give an affirmative answer in case the communication network, within which the VPN must be created, is a circuit. This seems to be an important step towards answering the general question. The proof relies on a dual pair of linear programs and actually implies an even stronger property of VPNs. We show that this property also holds for some other special cases of the problem, in particular when the network is a tree of rings.

Operations on M-Convex Functions on Jump Systems

Abstract: A jump system is a set of integer points with an exchange property, which is a generalization of a matroid, a delta-matroid, and a base polyhedron of an integral polymatroid (or a submodular system). Recently, the concept of M-convex functions on constant-parity jump systems was introduced by Murota as a class of discrete convex functions that admit a local criterion for global minimality. M-convex functions on constant-parity jump systems generalize valuated matroids, valuated delta-matroids, and M-convex functions on base polyhedra. This paper reveals that the class of M-convex functions on constant-parity jump systems is closed under a number of natural operations such as splitting, aggregation, convolution, composition, and transformation by networks. The present results generalize hitherto-known similar constructions for matroids, delta-matroids, valuated matroids, valuated delta-matroids, and M-convex functions on base polyhedra.

A New Min-Cut Max-Flow Ratio for Multicommodity Flows

Abstract: In this paper we present a new bound on the min-cut max-flow ratio for multicommodity flow problems with specified demands. For multicommodity flows, this is a generalization of the well-known relationship between the capacity of a minimum cut and the value of the maximum flow of a single commodity flow problem. For multicommodity flows, capacity of a cut is scaled by the demand that has to cross the cut to obtain the numerator of this ratio. In the denominator, the maximum concurrent flow value is used. Currently, the best known bound for this ratio is proportional to $\log(k)$, where $k$ is the number of origin-destination pairs with positive demand. Our new bound is proportional to $\log(k^*)$, where $k^*$ is the cardinality of the minimum cardinality vertex cover of the demand graph. To obtain this bound, we start with a so-called aggregated commodity formulation of the maximum concurrent flow problem with $k^*$ commodities. We also show a similar bound for the maximum multicommodity flow problem. The new bound is proportional to $\min\{\log(k^*), k^{**}\}$, where $k^{**}$ denotes the size of the of the demand graph.

The Complexity of Combinatorial Optimization Problems on $d$-Dimensional Boxes

Abstract: The MAXIMUM INDEPENDENT SET problem in $d$-box graphs, i.e., in intersection graphs of axis-parallel rectangles in $\mathbb{R}^d$, is known to be NP-hard for any fixed $d\geq 2$. A challenging open problem is that of how closely the solution can be approximated by a polynomial time algorithm. For the restricted case of $d$-boxes with bounded aspect ratio a PTAS exists [T. Erlebach, K. Jansen, and E. Seidel, SIAM J. Comput., 34 (2005), pp. 1302-1323]. In the general case no polynomial time algorithm with approximation ratio $o(\log^{d-1} n)$ for a set of $n$ $d$-boxes is known. In this paper we prove APX-hardness of the MAXIMUM INDEPENDENT SET problem in $d$-box graphs for any fixed $d\geq 3$. We give an explicit lower bound $\frac{245}{244}$ on efficient approximability for this problem unless $\PP=\text{\rm NP}$. Additionally, we provide a generic method how to prove APX-hardness for other graph optimization problems in $d$-box graphs for any fixed $d\geq 3$.

On Rota's Basis Conjecture

Abstract: Rota conjectured that if $(B_1,\ldots,B_n)$ are disjoint bases in a rank-$n$ matroid $M$, then there are $n$ disjoint transversals of $(B_1,\ldots,B_n)$ that are bases of $M$. We prove the weaker result that there are $O(\sqrt n)$ disjoint transversals of $(B_1,\ldots,B_n)$ that are bases. We also prove that if $(B_1,\ldots,B_k)$ are disjoint bases of a rank-$n$ matroid with $n> \binom{k+1}{2}$, then there are $n$ disjoint independent transversals of $(B_1,\ldots,B_k)$.

On Maximum Cost $K_{t,t}$-Free $t$-Matchings of Bipartite Graphs

Abstract: Frank examined the maximum $K_{t,t}$-free $t$-matching problem of simple bipartite graphs. As the $C_6$-free $2$-matching problem is NP-hard (Geelen), this is a promising generalization of restricted $2$-matchings. Given an arbitrary family $\mathcal{T}$ of $K_{t,t}$-subgraphs of the underlying graph, a $\mathcal{T}$-free $t$-matching is a subgraph of maximum degree at most $t$ that contains no member of $\mathcal{T}$. We show that the maximum size $\mathcal{T}$-free $t$-matching problem also admits a nice min-max formula. Given an integer cost function on the edge-set which is vertex-induced on any member of $\mathcal{T}$, we also show an integer min-max formula for the maximum cost of $\mathcal{T}$-free $t$-matchings. As the maximum cost $C_4$-free 2-matching problem is NP-hard (Kira´ly), we cannot expect a nice characterization in general.

Sharp Threshold for Hamiltonicity of Random Geometric Graphs

Abstract: We show for an arbitrary $\ell_p$ norm that the property that a random geometric graph $\mathcal G(n,r)$ contains a Hamiltonian cycle exhibits a sharp threshold at $r=r(n)=\sqrt{\frac{\log n}{\alpha_p n}}$, where $\alpha_p$ is the area of the unit disk in the $\ell_p$ norm. The proof is constructive and yields a linear time algorithm for finding a Hamiltonian cycle of $\mathcal{G}(n,r)$ asymptotically almost surely, provided $r=r(n)\ge\sqrt{\frac{\log n}{(\alpha_p -\epsilon)n}}$ for some fixed $\epsilon>0$.

Every Monotone $3$-Graph Property is Testable

Abstract: Recently Alon and Shapira [Every monotone graph property is testable, New York, Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, ACM Press, 2005, pp. 128-137] have established that every monotone graph property is testable. They raised the question whether their results can be extended to hypergraphs. The aim of this paper is to address this problem. Based on the recent regularity lemma of Ro¨dl and Schacht [Regular partitions of hypergraphs, Combin. Probab. Comput., to appear], we prove that any monotone property of 3-uniform hypergraphs is testable answering in part the question of Alon and Shapira. Our approach is similar to the one developed by Alon and Shapira for graphs. We believe that based on the general version of the hypergraph regularity lemma the proof presented in this article extends to $k$-uniform hypergraphs.

Recognizing Chordal Probe Graphs and Cycle-Bicolorable Graphs

Abstract: A graph $G=(V,E)$ is a chordal probe graph if its vertices can be partitioned into two sets, $P$ (probes) and $N$ (non-probes), where $N$ is a stable set and such that $G$ can be extended to a chordal graph by adding edges between non-probes. We give several characterizations of chordal probe graphs, first, in the case of a fixed given partition of the vertices into probes and non-probes, and second, in the more general case where no partition is given. In both of these cases, our results are obtained by introducing new classes, namely, $N$-triangulatable graphs and cycle-bicolorable graphs. We give polynomial time recognition algorithms for each class. $N$-triangulatable graphs have properties similar to chordal graphs, and we characterize them using graph separators and using a vertex elimination ordering. For cycle-bicolorable graphs, which are shown to be perfect, we prove that any cycle-bicoloring of a graph renders it $N$-triangulatable. The corresponding recognition complexity for chordal probe graphs, given a partition of the vertices into probes and non-probes, is $O(|P||E|)$, thus also providing an interesting tractable subcase of the chordal graph sandwich problem. If no partition is given in advance, the complexity of our recognition algorithm is $O(|E|^2)$.

Approximation Algorithms for Network Design with Metric Costs

Abstract: We study undirected networks with edge costs that satisfy the triangle inequality. Let $n$ denote the number of nodes. We present an $O(1)$-approximation algorithm for a generalization of the metric-cost subset $k$-node-connectivity problem. Our approximation guarantee is proved via lower bounds that apply to the simple edge-connectivity version of the problem, where the requirements are for edge-disjoint paths rather than for openly node-disjoint paths. A corollary is that, for metric costs and for each $k=1,2,\dots,n-1$, there exists a $k$-node connected graph whose cost is within a factor of ${ 22\/}$ of the cost of any simple $k$-edge connected graph. Based on our $O(1)$-approximation algorithm, we present an $O(\log r_{\max})$-approximation algorithm for the metric-cost node-connectivity survivable network design problem, where $r_{\max}$ denotes the maximum requirement over all pairs of nodes. Our results contrast with the case of edge costs of 0 or 1, where Kortsarz, Krauthgamer, and Lee. [SIAM J. Comput., 33 (2004), pp. 704-720] recently proved, assuming NP$ subseteq\;$DTIME($n^{polylog(n)}$), a hardness-of-approximation lower bound of $2^{\log^{1-\epsilon}n}$ for the subset $k$-node-connectivity problem, where $\epsilon$ denotes a small positive number.

Polynomial-Time Algorithms for Linear and Convex Optimization on Jump Systems

Abstract: The concept of a jump system, introduced by Bouchet and Cunningham [SIAM J. Discrete Math., 8 (1995), pp. 17-32], is a set of integer points with a certain exchange property. In this paper, we discuss several linear and convex optimization problems on jump systems and show that these problems can be solved in polynomial time under the assumption that a membership oracle for a jump system is available. We first present a polynomial-time implementation of the greedy algorithm for the minimization of a linear function. We then consider the minimization of a separable-convex function on a jump system and propose the first polynomial-time algorithm for this problem. The algorithm is based on the domain reduction approach developed in Shioura [Discrete Appl. Math., 84 (1998), pp. 215-220]. We finally consider the concept of M-convex functions on constant-parity jump systems which has been recently proposed by Murota [SIAM J. Discrete Math., 20 (2006), pp. 213-226]. It is shown that the minimization of an M-convex function can be solved in polynomial time by the domain reduction approach.

Further Results on Bar k-Visibility Graphs

Abstract: A bar visibility representation of a graph $G$ is a collection of horizontal bars in the plane corresponding to the vertices of $G$ such that two vertices are adjacent if and only if the corresponding bars can be joined by an unobstructed vertical line segment. In a bar $k$-visibility graph, two vertices are adjacent if and only if the corresponding bars can be joined by a vertical line segment that intersects at most $k$ other bars. Bar $k$-visibility graphs were introduced by Dean et al. [J. Graph Algorithms Appl., 11 (2007), pp. 45-59]. In this paper, we present sharp upper bounds on the maximum number of edges in a bar $k$-visibility graph on $n$ vertices and the largest order of a complete bar $k$-visibility graph. We also discuss regular bar $k$-visibility graphs and forbidden induced subgraphs of bar $k$-visibility graphs.

Two New Bounds for the Random-Edge Simplex-Algorithm

Abstract: We prove that the RANDOM-EDGE simplex-algorithm requires an expected number of at most $13n/\sqrt{d}$ pivot steps on any simple $d$-polytope with $n$ vertices. This is the first nontrivial upper bound for general polytopes. We also describe a refined analysis that potentially yields much better bounds for specific classes of polytopes. As one application, we show that for combinatorial $d$-cubes the trivial upper bound of $2^d$ on the performance of RANDOM-EDGE can asymptotically be improved by the factor $1/d^{(1-\varepsilon)\log d}$ for every $\varepsilon>0$.

Packing Odd Circuits

Abstract: We determine the structure of a class of graphs that do not contain the complete graph on five vertices as a “signed minor.” The result says that each graph in this class can be decomposed into elementary building blocks in which maximum packings by odd circuits can be found by flow or matching techniques. This allows us to actually find a largest collection of pairwise edge disjoint odd circuits in polynomial time (for general graphs this is NP-hard). Furthermore it provides an algorithm to test membership of our class of graphs.

Quadratically Many Colorful Simplices

Abstract: The colorful Carathe´odory theorem asserts that if $X_1,X_2,\ldots,X_{d+1}$ are sets in ${\bf R}^d$, each containing the origin 0 in its convex hull, then there exists a set $S \subseteq X_1 \cup \cdots \cup X_{d+1}$ with $|S \cap X_i| = 1$ for all $i=1,2,\ldots,d+1$ and $0 \in conv(S)$ (we call $conv(S)$ a colorful covering simplex). Deza e [Discrete Comput. Geom., 35 (2006), pp. 597-615] proved that if the $X_i$ are in general position with respect to 0 (consequently, each $X_i$ has at least $d+1$ points), then there are at least $2d$ colorful covering simplices, and they constructed an example with no more than $d^2+1$ such simplices. Under the same assumption, we show that there are at least $\frac{1}{5}d(d+1)$ colorful covering simplices, thus determining the order of magnitude. A similar result was proved independently by Stephen and Thomas [http://www.arxiv.org/abs/math.CO/0512400 (2005)]. We also obtain a lower bound of $3d$ for $d \geq 3$, which is better for small $d$ and, in particular, together with a parity argument it settles the case $d=3$, where the minimum possible number of colorful covering simplices is 10.

The Mixing Set with Flows

Abstract: We consider the mixing set with flows: $s+x_t \geq b_t, x_t \leq y_t {\rm for} 1 \leq t \leq n; s \in \R^1_+, x \in \R^n_+, y \in \Z^n_+.$ It models a “flow version” of the basic mixing set introduced and studied by Gu¨nlu¨k and Pochet [Math. Program., 90 (2001), pp. 429-457], as well as the most simple stochastic lot-sizing problem with recourse. More generally it is a relaxation of certain mixed integer sets that arise in the study of production planning problems. We study the polyhedron defined as the convex hull of the above set. Specifically we provide an inequality description, and we also characterize its vertices and rays.

Graphs Having Small Number of Sizes on Induced k-Subgraphs

Abstract: Let $\ell$ be any positive integer, let $n$ be a sufficiently large number, and let $G$ be a graph on $n$ vertices. Define, for any $k$, $ u_k(G)= | \{ |E(H)| : H$ is an induced subgraph of $G$ on $k$ vertices$\} |$. We show that if there exists a $k$, $2\ell \leq k \leq n-2\ell$, such that $ u_k(G) \le \ell$, then $G$ has a complete or an empty subgraph on at least $n-\ell+1$ vertices and a homogeneous set of size at least $n-2\ell+2$. These results are sharp.

Crossing Stars in Topological Graphs

Abstract: Let $G$ be a graph without loops or multiple edges drawn in the plane. It is shown that, for any $k$, if $G$ has at least $C_k n$ edges and $n$ vertices, then it contains three sets of $k$ edges, such that every edge in any of the sets crosses all edges in the other two sets. Furthermore, two of the three sets can be chosen such that all $k$ edges in the set have a common vertex.

Prefix Reversals on Binary and Ternary Strings

Abstract: Given a permutation $\pi$, the application of prefix reversal $f^{(i)}$ to $\pi$ reverses the order of the first $i$ elements of $\pi$. The problem of sorting by prefix reversals (also known as pancake flipping), made famous by Gates and Papadimitriou (Discrete Math., 27 (1979), pp. 47-57), asks for the minimum number of prefix reversals required to sort the elements of a given permutation. In this paper we study a variant of this problem where the prefix reversals act not on permutations but on strings over a fixed size alphabet. We determine the minimum number of prefix reversals required to sort binary and ternary strings, with polynomial-time algorithms for these sorting problems as a result; demonstrate that computing the minimum prefix reversal distance between two binary strings is NP-hard; give an exact expression for the prefix reversal diameter of binary strings; and give bounds on the prefix reversal diameter of ternary strings. We also consider a weaker form of sorting called grouping (of identical symbols) and give polynomial-time algorithms for optimally grouping binary and ternary strings. A number of intriguing open problems are also discussed.