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Given a directed graph G and an arc weight function w : E(G) → R+, the maximum directed cut problem (max dicut) is that of finding a directed cut δ(X) with maximum total weight. In this paper we consider a version of max dicut--max dicut with given sizes of parts or max dicut with gsp--whose instance is that of max dicut plus a positive integer p, and it is required to find a directed cut δ(X) having maximum weight over all cuts δ(X) with |X| = p. Our main result is a 0.5-approximation algorithm for solving the problem. The algorithm is based on a tricky application of the pipage rounding technique developed in some earlier papers by two of the authors and a remarkable structural property of basic solutions to a linear relaxation. The property is that each component of any basic solution is an element of a set {0,δ, 1/2, 1 - δ, 1}, where δ is a constant that satisfies 0 < δ < 1/2 and is the same for all components. [ABSTRACT FROM AUTHOR]

It is known that the First-Fit algorithm for partitioning a poset P into chains uses relatively few chains when P does not have two incomparable chains each of size k. In particular, if P has width w, then Bosek, Krawczyk, and Szczypka [SIAM J. Discrete Math., 23 (2010), pp. 1992-1999], proved an upper bound of ckw² on the number of chains used by First-Fit for some constant c, while Joret and Milans [Order, 28 (2011), pp. 455-464] gave one of ck²w. In this paper we prove an upper bound of the form ckw. This is most possible up to the value of c. [ABSTRACT FROM AUTHOR]

Cops and robber is a classical pursuit-evasion game on undirected graphs, where the task is to identify the minimum number of cops sufficient to catch the robber. In this paper, we investigate the changes in problem's complexity and combinatorial properties with constraining the following natural game parameters: fuel, the number of steps each cop can make; cost, the total sum of steps along edges all cops can make; and time, the number of rounds of the game. [ABSTRACT FROM AUTHOR]

In this paper, we consider the half-integral disjoint paths packing. For a graph G and k pairs of vertices (s1, t1), (s2, t2), …, (sk, tk) in G, the objective is to find paths P1, …, Pk in G such that Pi joins si and ti for i = 1, 2, …, k, and in addition, each vertex is on at most two of these paths. We give a polynomial-time algorithm to decide the feasibility of this problem with k = O((log n/log log n)1/7). This improves a result by Kleinberg [Proceedings of the 30th ACM Symposium on Theory of Computing, 1998, pp 530-539] who proved the same conclusion when k = O((log log n)2/15). Our algorithm still works for several problems related to the bounded unsplittable flow. These results can all carry over to problems involving edge capacities. Our main technical contribution is to give a "crossbar" of a polynomial size of the tree width of the graph. [ABSTRACT FROM AUTHOR]

In the (k, λ)-subgraph problem, we are given an undirected graph G = (V, E) with edge costs and two positive integers k and λ, and the goal is to find a minimum cost simple λ-edge-connected subgraph of G with at least k nodes. This generalizes several classical problems, such as the minimum cost k-spanning tree problem, or k-MST (which is a (k, 1)-subgraph), and the minimum cost λ-edge-connected spanning subgraph (which is a (|V(G)|, λ-subgraph). The only previously known results on this problem [L. C. Lau, J. S. Naor, M. R. Salavatipour, and M. Singh, SIAM J. Comput., 39 (2009), pp. 1062-1087], [C. Chekuri and N. Korula, in Proceedings of the IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS), Bangalore, India, LIPIcs 2, Schloss Dagstuhl--Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2008, pp. 119-130] show that the (k, 2)-subgraph problem has an O(log² n)-approximation (even for 2-node-connectivity) and that the (k, λ)-subgraph problem in general is almost as hard as the densest k-subgraph problem. In this paper we show that if the edge costs are metric (i.e., satisfy the triangle in equality), like in the k-MST problem, then there is an O(1)-approximation algorithm for the (k, λ)-subgraph problem. This essentially generalizes the k-MST constant factor approximability to higher connectivity. [ABSTRACT FROM AUTHOR]

Recently lexicographic breadth first search (LexBFS) has been shown to be a very powerful tool for the development of linear time, easily implementable recognition algorithms for various families of graphs. In this paper, we add to this work by producing a simple two LexBFS sweep algorithm to recognize the family of cographs. This algorithm extends to other related graph families such as P4-reducible, P4-sparse, and distance hereditary. It is an open question whether our cograph recognition algorithm can be extended to a similarly easy algorithm for modular decomposition. [ABSTRACT FROM AUTHOR]

We discuss two approximation paradigms that were used to construct many approximation algorithms during the last two decades, the primal-dual schema and the local ratio technique. Recently, primal-dual algorithms were devised by first constructing a local ratio algorithm and then transforming it into a primal-dual algorithm. This was done in the case of the $2$-approximation algorithms for the feedback vertex set problem and in the case of the first primal-dual algorithms for maximization problems. Subsequently, the nature of the connection between the two paradigms was posed as an open question by Williamson [Math. Program., 91 (2002), pp. 447--478]. In this paper we answer this question by showing that the two paradigms are equivalent. [ABSTRACT FROM AUTHOR]

The decycling number $\nabla(G)$ of a graph $G$ is the smallest number of vertices which can be removed from $G$ so that the resultant graph contains no cycles. In this paper, we study the decycling number for the family of graphs consisting of the Cartesian product of two cycles. We completely solve the problem of determining the decycling number of $C_m \square C_n$ for all $m$ and $n$. Moreover, we find a vertex set $T$ that yields a maximum induced tree in $C_m\square C_n$. [ABSTRACT FROM AUTHOR]

Given $G=(V,E)$ an undirected graph and $A=\{n_1,n_2,n_3\}\subseteq V$ a specified set of terminal nodes, a 3-terminal cut is a subset of edges whose removal disconnects each terminal from the rest. Given a nonnegative cost vector $c\in{\mathbb R}^{|E|}_{+}$, the optimal 3-terminal cut problem is to find a 3-terminal cut of minimum cost. In this paper we consider the polyhedron LP($G,A$), the linear relaxation of the 3-terminal cuts polyhedron P($G,A$). We give a characterization of the pairs $(G,A)$ for which LP($G,A$) is integer. This result was conjectured by Cunningham in [{\it Reliability of Compter and Communications Networks}, AMS, Providence, RI, 1991, pp. 105--120]. [ABSTRACT FROM AUTHOR]

It is known that every positive integer n can be represented as a finite sum of the form $n={\textstyle\sum} a_i 2^{i}$, where $a_i\in \{0,1,-1\}$ for all i, and no two consecutive ai's are nonzero. Such sums are called nonadjacent representations. Nonadjacent representations are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications. In this paper, we investigate if other digit sets of the form {0,1, x}, where x is an integer, provide each positive integer with a nonadjacent representation. If a digit set has this property, we call it a nonadjacent digit set (NADS). We present an algorithm to determine if {0,1, x} is an NADS and, if it is, we present an algorithm to efficiently determine the nonadjacent representation of any positive integer. We also present some necessary and sufficient conditions for {0,1, x} to be an NADS. These conditions are used to exhibit infinite families of integers x such that {0,1, x} is an NADS, as well as infinite families of x such that {0,1, x} is not an NADS. [ABSTRACT FROM AUTHOR]

Recently, D. Corneil found a simple 3-sweep lexicographic breadth first search (LexBFS) algorithm for the recognition of proper interval graphs. We point out how to modify Corneil's algorithm to make it a certifying algorithm, and then describe a similar certifying 3-sweep LexBFS algorithm for the recognition of proper interval bigraphs. It follows from an earlier paper that the class of proper interval bigraphs is equal to the better known class of bipartite permutation graphs, and so we have a certifying algorithm for that class as well. All our algorithms run in time O(m+n), including the certification phase. The certificates of representability (the intervals) can be authenticated in time O(m + n). The certificates of nonrepresentability (the forbidden subgraphs) can be authenticated in time O(n). [ABSTRACT FROM AUTHOR]

Splitting off two edges su, sv in a graph G means deleting su, sv and adding a new edge uv. Let G = (V + s, E) be k-edge-connected in V (k ≥ 2) and let d(s) be even. Lovász proved that the edges incident to s can be split off in pairs in a such a way that the resulting graph on vertex set V is k-edge-connected. In this paper we investigate the existence of such complete splitting sequences when the set of split edges has to meet additional requirements. We prove structural properties of the set of those pairs u, v of neighbors of s for which splitting off su, sv destroys k-edge-connectivity. This leads to a new method for solving problems of this type. By applying this method we obtain a short proof for a recent result of Nagamochi and Eades on planarity-preserving complete splitting sequences and prove the following new results: let G and H be two graphs on the same set V + s of vertices and suppose that their sets of edges incident to s coincide. Let G (H) be k-edge-connected (l-edge-connected, respectively) in V (k, l ≥ 2) and let d(s) be even. Then there exists a pair su, sv which can be split off in both graphs preserving k-edge-connectivity in G (l-edge-connectivity in H, respectively), provided d(s) ≥ 6. If k and l are both even, then such a pair always exists. By using these edge-splitting results and the polymatroid intersection theorem we give a polynomial algorithm for the problem of simultaneously augmenting the edge-connectivity of two graphs by adding a (common) set of new edges of (almost) minimum size. [ABSTRACT FROM AUTHOR]

This paper gives a new and faster algorithm to find a 1-factor in a bipartite Δ-regular graph. The time complexity of this algorithm is O(nΔ + n log n log Δ), where n is the number of nodes. This implies an O(n log n log Δ + m log Δ) algorithm to edge-color a bipartite graph with n nodes, in edges, and maximum degree Δ. [ABSTRACT FROM AUTHOR]

The combinatorial group testing problem is, assuming the existence of up to d defectives among n items, to identify the defectives by as few tests as possible. In this paper, we study the problem for what values of n, given d, individual testing is optimal in nonadaptive group testing. Let N(d) denote the largest n for fixed d for which individual testing is optimal. We will show that N(d) = (d + 1)² under a prevalent constraint in practical nonadaptive algorithms and prove that N(d) = (d + 1) ² for d = 1, 2, 3, 4 without any constraint. [ABSTRACT FROM AUTHOR]

Consider the 2n (or O(n²)) min-cut problems on a graph with n nodes and nonnegative edge weights. Gomory and Hu [J. Soc. Indust. Appl. Math., 9 (1961), pp. 551–570] showed (essentially) that there are at most n-1 different min-cuts. They also described a compact structure (the flow equivalent tree) of size O(n) with the following property: for any pair of nodes, the value of a min-cut can be obtained from this structure. Furthermore, they showed how this structure can be found by solving only n - 1 min-cut problems. This paper contains generalizations of these results. For example, consider a k-terminal cut problem on a graph: for a given set of k nodes, delete a minimum weight set of edges (called a k-cut) so that each of the k nodes is in a different component. There are kn (or O(nk)) such problems. Hassin [Math. Oper. Res., 13 (1988), pp. 535–542] showed (essentially) that there are at most k-1n-1 (or O(nk-1)) different min k-cuts. We describe a compact structure of size O(nk-1) with the following property: for any k nodes, the value of a min k-cut can be obtained from this structure. We also show how this structure can be found by solving only k-1n-1 k-terminal cut problems. This work builds upon the results of Hassin [Math. Oper. Res., 13 (1988), pp. 535–542], [Oper. Res. Lett., 9 (1990), pp. 315–318], and [Lecture Notes in Comput. Sci. 450, 1990, pp. 228–299]. [ABSTRACT FROM AUTHOR]

We consider the one-dimensional bin packing problem with unit-capacity bins and item sizes chosen according to the discrete uniform distribution U{j,k}, 1 < j ≤ k, where each item size in {1/k, 2/k,…,j/k} has probability 1/j of being chosen. Note that for fixed j,k as m → ∝ the discrete distributions U{mj,mk} approach the continuous distribution U(0,j/k], where the item sizes are chosen uniformly from the interval (0,j/k]. We show that average-case behavior can differ substantially between the two types of distributions. In particular, for all j,k with j < k - 1, there exist on-line algorithms that have constant expected wasted space under U{j,k}, whereas no on-line algorithm has even o(n½) expected waste under U(0,u] for any 0 < u ≤ 1. Our U{j,k} result is an application of a general theorem of Courcoubetis and Weber [C. Courcoubetis and R.R. Weber, Probab. Engrg. Inform. Sci., 4 (1990), pp. 447–460] that covers all discrete distributions. Under each such distribution, the optimal expected waste for a random list of n items must be either Θ(n), Θ(n½), or O(1), depending on whether certain ‘perfect’ packings exist. The perfect packing theorem needed for the U{j,k} distributions is an intriguing result of independent combinatorial interest, and its proof is a cornerstone of the paper. [ABSTRACT FROM AUTHOR]

A transshipment problem with demands that exceed network capacity can be solved by sending flow in several waves. How can this be done in the minimum number, T, of waves, and at minimum cost, if costs are piecewise linear convex functions of the flow? In this paper, we show that this problem can be solved using min{m, log T, log(mΓU) / 1+log(mΓU)-log(mU)} maximum flow computations and one minimum (convex) cost flow computation. Here m is the number of arcs, Γ is the maximum supply or demand, and U is the maximum capacity. When there is only one sink, this problem can be solved in the same asymptotic time as one minimum (convex) cost flow computation. This improves upon the previous best algorithm to solve the problem without costs by a factor of k. Our solutions start with a stationary fractional flow and use rounding to transform this into an integral flow. The rounding procedure takes O(n) time. [ABSTRACT FROM AUTHOR]

In this paper we approach the problem of computing the characteristic polynomial of a matrix from the combinatorial viewpoint. We present several combinatorial characterizations of the coefficients of the characteristic polynomial in terms of walks and closed walks of different kinds in the underlying graph. We develop algorithms based on these characterizations and show that they tally with well-known algorithms arrived at independently from considerations in linear algebra. [ABSTRACT FROM AUTHOR]

Bender, Richmond, and Wormald showed that in almost all planar 3-connected triangulations (or dually, 3-connected cubic maps) with n edges, the largest 4-connected triangulation (or dually, the largest cyclically 4-edge-connected cubic component) has about n/2 edges [Random Structures Algorithms, 7 (1995), pp. 273-285]. In this paper, we derive some general results about the size of the largest component and apply them to a variety of types of planar maps. [ABSTRACT FROM AUTHOR]

In the well-solved edge-connectivity augmentation problem we must find a minimum cardinality set F of edges to add to a given undirected graph to make it k-edge-connected. This paper solves the generalization where every edge of F must go between two different sets of a given partition of the vertex set. A special case of this partition-constrained problem, previously unsolved, is increasing the edge-connectivity of a bipartite graph to k while preserving bipartiteness. Based on this special case we present an application of our results in statics. Our solution to the general partition-constrained problem gives a min-max formula for F which includes as a special case the original min-max formula of Cai and Sun [Networks, 19 (1989), pp. 151-172] for the problem without partition constraints. When k is even the rain-max formula for the partition-constrained problem is a natural generalization of the unconstrained version. However, this generalization fails when k is odd. We show that at most one more edge is needed when k is odd and we characterize the graphs that require such an extra edge. We give a strongly polynomial algorithm that solves our problem in time O(n(m + n log n) log n). Here n and m denote the number of vertices and distinct edges of the given graph, respectively. This bound is identical to the best-known time bound for the problem without partition constraints. Our algorithm is based on the splitting off technique of Lovász, like several known efficient algorithms for the unconstrained problem. However, unlike previous splitting algorithms, when k is odd our algorithm must handle obstacles that prevent all edges from being split off. Our algorithm is of interest even when specialized to the unconstrained problem, because it produces an asymptotically optimum number of distinct splits. [ABSTRACT FROM AUTHOR]

A construction of Cayley digraphs associated to arc-colored regular digraphs is presented. The resulting Cayley digraphs, which we call Cayley regular covers, can be seen as a symmetrization of the original digraph. This construction is applied to the de Bruijn digraphs. By using the fact that they are iterated line digraphs of complete symmetric digraphs, valuable information about their Cayley regular covers regarding routings, diameter, hamiltonicity, faulttolerance properties and degree of symmetry is obtained. In particular, a shortest-path, self-routing algorithm is given for a family of Cayley digraphs which includes the well known butterfly network. These results can be applied to the design of permutation networks. The Cayley regular covers represent sets of permutations in the original digraph which can be performed without conflict. In particular, a sharply 2-transitive group of permutations on the de Bruijn network is presented which admits a simple shortest-path self-routing algorithm. By using the same construction, a Cayley digraph on the symmetric group on the nodes of the de Bruijn digraph of degree two is obtained. The techniques introduced in this paper can also be extended to other families of iterated line digraphs. [ABSTRACT FROM AUTHOR]

In this paper we modify Buchberger's S-pair reduction algorithm for computing a Gröbner basis of a toric ideal so as to apply it to an integer program (IP) in inequality form with fixed right-hand sides and fixed upper bounds on the variables. We formulate the algorithm in the original space and interpret the reduction steps geometrically. In fact, three variants of this algorithm are presented, and we give elementary proofs for their correctness. A relationship among these (exact) algorithms, iterative improvement heuristics, and the Kernighan-Lin procedure is established. Computational results are also presented. [ABSTRACT FROM AUTHOR]

The worst case length of a tour for the Euclidean traveling salesman problem produced by the nearest neighbor (NN) heuristic is studied in this paper. Nearest neighbor tours for a set of arbitrarily located points in the d-dimensional unit cube are considered. A technique is developed for bounding the worst case length of a tour. It is based on identifying sequences of coverings of [0,1]d. Each covering Pk consists of sets Ci, with diameter bounded by the diameter D(Pk) of the covering. For every sequence of coverings a bound is obtained that depends on the cardinality of the coverings and their diameters. The task of bounding the worst case length of an NN tour is reduced to finding appropriate sequences of coverings. Using coverings produced by the rectangular lattice with appropriately shrinking diameter, it is shown that the worst case length of an NN tour through Ar points in [0,1]d is bounded by [d√d/(d-1)]N(d-1)/d + o(N(d-1)/d). For the unit square the tighter bound 2.482→N + o(√N) is obtained using regular hexagonal lattice coverings. [ABSTRACT FROM AUTHOR]

In this paper we describe new constructions for de Bruijn sequences and Perfect Factors. These constructions are all based upon the idea of constructing one sequence (or set of sequences) from another. As a result of this fact, the sequences obtained from these construction methods possess simple decoding algorithms based on decoding the sequences used to construct them. Such decoding algorithms are of importance in position-location applications. [ABSTRACT FROM AUTHOR]

Generalizing the two-commodity flow theorem of Rothschild and Whinston [Oper. Res., 14 (1966), pp. 377-387] and the multiflow theorem of Lovász [Acta Mat. Akad. Sci. Hungaricae, 28 (1976), pp. 129-138] and Cherkasky [Ekonom.-Mat. Metody, 13 (1977), pp. 143-151], Karzanov and Lomonosov [Mathematical Programming, O. I. Larichev, ed., Institute for System Studies, 1978, pp. 59-66] in 1978 proved a min-max theorem on maximum multiflows. Their original proof is quite long and technical and relies on earlier investigations into metrics. The main purpose of the present paper is to provide a relatively simple proof of this theorem. Our proof relies on the locking theorem, which is another result of Karzanov and Lomonosov, and the polymatroid intersection theorem of Edmonds [Combinatorial Structures and Their Applications, R. Guy, H. Hanani, N. Sauer, and J. Schönheim, eds., Gordon and Breach, 1970, pp. 69-87]. For completeness, we also provide a simplified proof of the locking theorem. Finally, we introduce the notion of a node demand problem and, as another application of the locking theorem, we derive a feasibility theorem concerning it. The presented approach gives rise to (combinatorial) polynomial-time algorithms. [ABSTRACT FROM AUTHOR]