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We prove that in any $n$-vertex complete graph there is a collection $\mathcal{P}$ of $(1 + o(1))n$ paths that strongly separates any pair of distinct edges $e, f$, meaning that there is a path in $\mathcal{P}$ which contains $e$ but not $f$. Furthermore, for certain classes of $n$-vertex $\alpha n$-regular graphs we find a collection of $(\sqrt{3 \alpha + 1} - 1 + o(1))n$ paths that strongly separates any pair of edges. Both results are best-possible up to the $o(1)$ term.

The Steiner Multicycle problem consists of, given a complete graph, a weight function on its vertices, and a collection of pairwise disjoint non-unitary sets called terminal sets, finding a minimum weight collection of vertex-disjoint cycles in the graph such that, for every terminal set, all of its vertices are in a same cycle of the collection. This problem generalizes the Traveling Salesman problem and therefore is hard to approximate in general. On the practical side, it models a collaborative less-than-truckload problem with pickup and delivery locations. Using an algorithm for the Survivable Network Design problem and T -joins, we obtain a 3-approximation for the metric case, improving on the previous best 4-approximation. Furthermore, we present an (11/9)-approximation for the particular case of the Steiner Multicycle in which each edge weight is 1 or 2. This algorithm can be adapted to obtain a (7/6)-approximation when every terminal set contains at least 4 vertices. Finally, we devise an O(lg n)-approximation algorithm for the asymmetric version of the problem.

In 1996, Matheson and Tarjan proved that every near planar triangulation on $n$ vertices contains a dominating set of size at most $n/3$, and conjectured that this upper bound can be reduced to $n/4$ for planar triangulations when $n$ is sufficiently large. In this paper, we consider the analogous problem for independent dominating sets: What is the minimum $\epsilon$ for which every near planar triangulation on $n$ vertices contains an independent dominating set of size at most $\epsilon n$? We prove that $2/7 \leq \epsilon \leq 5/12$. Moreover, this upper bound can be improved to $3/8$ for planar triangulations, and to $1/3$ for planar triangulations with minimum degree 5., Comment: 9 pages

Trees with many leaves have applications on broadcasting, which is a method in networks for transferring a message to all recipients simultaneously. Internal nodes of a broadcasting tree require more expensive technology, because they have to forward the messages received. We address a problem that captures the main goal, which is to find spanning trees with few internal nodes in a given network. The Maximum Leaf Spanning Arborescence problem consists of, given a directed graph D, finding a spanning arborescence of D, if one exists, with the maximum number of leaves. This problem is known to be NP-hard in general and MaxSNP-hard on the class of rooted directed acyclic graphs. In this paper, we explore a relation between Maximum Leaf Spanning Arborescence in rooted directed acyclic graphs and maximum weight set packing. The latter problem is related to independent sets on particular classes of intersection graphs. Exploiting this relation, we derive a 7/5-approximation for Maximum Leaf Spanning Arborescence on rooted directed acyclic graphs, improving on the previous 3/2-approximation. The approach used might lead to improvements on the best approximation ratios for the weighted k-set packing problem.

A graph whose nodes have degree 1 or 3 is called a $\{1,3\}$-graph. Liu and Osserman associated a polytope to each $\{1,3\}$-graph and studied the Ehrhart quasi-polynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set $\{0,\frac14,\frac12,1\}$, which implies that the period of their Ehrhart quasi-polynomials is either 1, 2, or 4. We show that the period of the Ehrhart quasi-polynomial of these polytopes is at most 2 if the graph is a tree or a cubic graph, and it is equal to 4 otherwise. In the process of proving this theorem, several interesting combinatorial and geometric properties of these polytopes were uncovered, arising from the structure of their associated graphs. The tools developed here may find other applications in the study of Ehrhart quasi-polynomials and enumeration problems for other polytopes that arise from graphs. Additionally, we have identified some interesting connections with triangulations of 3-manifolds., Comment: 42 pages

Broadcasting in a computer network is a method of transferring a message to all recipients simultaneously. It is common in this situation to use a tree with many leaves to perform the broadcast, as internal nodes have to forward the messages received, while leaves are only receptors. We consider the subjacent problem of, given a directed graph~$D$, finding a spanning arborescence of D, if one exists, with the maximum number of leaves. In this paper, we concentrate on the class of rooted directed acyclic graphs, for which the problem is known to be MaxSNP-hard. A 2-approximation was previously known for this problem on this class of directed graphs. We improve on this result, presenting a (3/2)-approximation. We also adapt a result for the undirected case and derive an inapproximability result for the vertex-weighted version of Maximum Leaf Spanning Arborescence on rooted directed acyclic graphs.

Tuza (1981) conjectured that the size $\tau(G)$ of a minimum set of edges that intersects every triangle of a graph $G$ is at most twice the size $\nu(G)$ of a maximum set of edge-disjoint triangles of $G$. In this paper we present three results regarding Tuza's Conjecture. We verify it for graphs with treewidth at most $6$; we show that $\tau(G)\leq \frac{3}{2}\,\nu(G)$ for every planar triangulation $G$ different from $K_4$; and that $\tau(G)\leq\frac{9}{5}\,\nu(G) + \frac{1}{5}$ if $G$ is a maximal graph with treewidth 3. Our first result strengthens a result of Tuza, implying that $\tau(G) \leq 2\,\nu(G)$ for every $K_8$-free chordal graph $G$.

The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to graphs whose vertices have degree one or three. In this paper, we prove the scissors congruence conjecture, posed by Haase and McAllister, for this class of polytopes. The key ingredient in the proofs is the nearest neighbor interchange on graphs and a naturally arising piecewise unimodular transformation., Comment: 17 pages, with 10 figures, and a table

Let $\lpt(G)$ be the minimum cardinality of a set of vertices that intersects all longest paths in a graph $G$. Let $\omega(G)$ be the size of a maximum clique in $G$, and $\tw(G)$ be the treewidth of $G$. We prove that $ \lpt(G) \leq \max\{1,\omega(G)-2\}$ when $G$ is a connected chordal graph; that $\lpt(G) =1$ when $G$ is a connected bipartite permutation graph or a connected full substar graph; and that $\lpt(G) \leq \tw(G)$ for any connected graph $G$., Comment: 19 pages, 9 figures

Minimum $k$-Section denotes the NP-hard problem to partition the vertex set of a graph into $k$ sets of sizes as equal as possible while minimizing the cut width, which is the number of edges between these sets. When $k$ is an input parameter and $n$ denotes the number of vertices, it is NP-hard to approximate the width of a minimum $k$-section within a factor of $n^c$ for any $c<1$, even when restricted to trees with constant diameter. Here, we show that every tree $T$ allows a $k$-section of width at most $(k-1) (2 + 16n / diam(T) ) \Delta(T)$. This implies a polynomial-time constant-factor approximation for the Minimum $k$-Section Problem when restricted to trees with linear diameter and constant maximum degree. Moreover, we extend our results from trees to arbitrary graphs with a given tree decomposition., Comment: 15 pages, 5 figures

Minimum Bisection denotes the NP-hard problem to partition the vertex set of a graph into two sets of equal sizes while minimizing the width of the bisection, which is defined as the number of edges between these two sets. We first consider this problem for trees and prove that the minimum bisection width of every tree $T$ on $n$ vertices satisfies $MinBis(T) \leq 8 n \Delta(T) / diam(T)$. Second, we generalize this to arbitrary graphs with a given tree decomposition $(T,X)$ and give an upper bound on the minimum bisection width that depends on the structure of $(T,X)$. Moreover, we show that a bisection satisfying our general bound can be computed in time proportional to the encoding length of the tree decomposition when the latter is provided as input., Comment: 24 pages, 5 figures

Let $G=(V,E)$ be an $n$-vertex graph with $m$ edges. A function $f : V \cup E \rightarrow \{1, \ldots, n+m\}$ is an edge-magic labeling of $G$ if $f$ is bijective and, for some integer $k$, we have $f(u)+f(v)+f(uv) = k$ for every edge $uv \in E$. Furthermore, if $f(V) = \{1, \ldots, n\}$, then we say that $f$ is a super edge-magic labeling. A constellation, which is a collection of stars, is symmetric if the number of stars of each size is even except for at most one size. We prove that every symmetric constellation with an odd number of stars admits a super edge-magic labeling. We say that a caterpillar is of type $(r,s)$ if $r$ and $s$ are the sizes of its parts, where $r \leq s$. We also prove that every collection with an odd number of same-type caterpillars admits an edge-magic labeling.

We consider a game-theoretical problem called selfish 2-dimensional bin packing game, a generalization of the 1-dimensional case already treated in the literature. In this game, the items to be packed are rectangles, and the bins are unit squares. The game starts with a set of items arbitrarily packed in bins. The cost of an item is defined as the ratio between its area and the total occupied area of the respective bin. Each item is a selfish player that wants to minimize its cost. A migration of an item to another bin is allowed only when its cost is decreased. We show that this game always converges to a Nash equilibrium (a stable packing where no single item can decrease its cost by migrating to another bin). We show that the pure price of anarchy of this game is unbounded, so we address the particular case where all items are squares. We show that the pure price of anarchy of the selfish square packing game is at least 2.3634 and at most 2.6875. We also present analogous results for the strong Nash equilibrium (a stable packing where no nonempty set of items can simultaneously migrate to another common bin and decrease the cost of each item in the set). We show that the strong price of anarchy when all items are squares is at least 2.0747 and at most 2.3605.

We present exponential and super factorial lower bounds on the number of Hamiltonian cycles passing through any edge of the basis graphs of a graphic, generalized Catalan and uniform matroids. All lower bounds were obtained by a common general strategy based on counting appropriated cycles of length four in the corresponding matroid basis graph., Comment: Reformulation of Propositions 7 and 8. Lemmas 9 and 10, and Theorem 11 are combined in the current Theorem 9

In the k-center problem, given a metric space V and a positive integer k, one wants to select k elements (centers) of V and an assignment from V to centers, minimizing the maximum distance between an element of V and its assigned center. One of the most general variants is the capacitated {\alpha}-fault-tolerant k-center, where centers have a limit on the number of assigned elements, and, if {\alpha} centers fail, there is a reassignment from V to non-faulty centers. In this paper, we present a new approach to tackle fault tolerance, by selecting and pre-opening a set of backup centers, then solving the obtained residual instance. For the {0,L}-capacitated case, we give approximations with factor 6 for the basic problem, and 7 for the so called conservative variant, when only clients whose centers failed may be reassigned. Our algorithms improve on the previously best known factors of 9 and 17, respectively. Moreover, we consider the case with general capacities. Assuming {\alpha} is constant, our method leads to the first approximations for this case. We also derive approximations for the capacitated fault- tolerant k-supplier problem.

The problem of clustering a set of points moving on the line consists of the following: given positive integers n and k, the initial position and the velocity of n points, find an optimal k-clustering of the points. We consider two classical quality measures for the clustering: minimizing the sum of the clusters diameters and minimizing the maximum diameter of a cluster. For the former, we present polynomial-time algorithms under some assumptions and, for the latter, a (2.71 + epsilon)-approximation.

We consider questions related to the existence of spanning trees in graphs with the property that after the removal of any path in the tree the graph remains connected. We show that, for planar graphs, the existence of trees with this property is closely related to the Hamiltonicity of the graph. For graphs with a 1- or 2-vertex cut, the Hamiltonicity also plays a central role. We also deal with spanning trees satisfying this property restricted to paths arising from fundamental cycles. The cycle space of a graph can be generated by the fundamental cycles of any spanning tree, and Tutte showed, that for a 3-connected graph, it can be generated by nonseparating cycles. We are also interested in the existence of a fundamental basis consisting of nonseparating cycles., Comment: 10 pages

A MILP model for an extended version of the Flexible Job Shop Scheduling problem is proposed. The extension allows the precedences between operations of a job to be given by an arbitrary directed acyclic graph rather than a linear order. The goal is the minimization of the makespan. Theoretical and practical advantages of the proposed model are discussed. Numerical experiments show the performance of a commercial exact solver when applied to the proposed model. The new model is also compared with a simple extension of the model described by \"Ozg\"uven, \"Ozbakir, and Yavuz (Mathematical models for job-shop scheduling problems with routing and process plan flexibility, Applied Mathematical Modelling, 34:1539--1548, 2010), using instances from the literature and instances inspired by real data from the printing industry., Comment: 15 pages, 2 figures, 4 tables. Optimization Letters, 2013

In 1966 Gallai asked whether all longest paths in a connected graph have nonempty intersection. This is not true in general and various counterexamples have been found. However, the answer to Gallai's question is positive for several well-known classes of graphs, as for instance connected outerplanar graphs, connected split graphs, and 2-trees. A graph is series-parallel if it does not contain $K_4$ as a minor. Series-parallel graphs are also known as partial 2-trees, which are arbitrary subgraphs of 2-trees. We present a proof that every connected series-parallel graph has a vertex that is common to all of its longest paths. Since 2-trees are maximal series-parallel graphs, and outerplanar graphs are also series-parallel, our result captures these two classes in one proof and strengthens them to a larger class of graphs. We also describe how this vertex can be found in linear time., Comment: 17 pages

We consider the unit-demand envy-free pricing problem, which is a unit-demand auction where each bidder receives an item that maximizes his utility, and the goal is to maximize the auctioneer's profit. This problem is NP-hard and unlikely to be in APX. We present four new MIP formulations for it and experimentally compare them to a previous one due to Shioda, Tun\c{c}el, and Myklebust. We describe three models to generate different random instances for general unit-demand auctions, that we designed for the computational experiments. Each model has a nice economic interpretation. Aiming approximation results, we consider the variant of the problem where the item prices are restricted to be chosen from a geometric series, and prove that an optimal solution for this variant has value that is a fraction (depending on the series used) of the optimal value of the original problem. So this variant is also unlikely to be in APX.

We consider the Max-Buying Problem with Limited Supply, in which there are $n$ items, with $C_i$ copies of each item $i$, and $m$ bidders such that every bidder $b$ has valuation $v_{ib}$ for item $i$. The goal is to find a pricing $p$ and an allocation of items to bidders that maximizes the profit, where every item is allocated to at most $C_i$ bidders, every bidder receives at most one item and if a bidder $b$ receives item $i$ then $p_i \leq v_{ib}$. Briest and Krysta presented a 2-approximation for this problem and Aggarwal et al. presented a 4-approximation for the Price Ladder variant where the pricing must be non-increasing (that is, $p_1 \geq p_2 \geq \cdots \geq p_n$). We present an $e/(e-1)$-approximation for the Max-Buying Problem with Limited Supply and, for every $\varepsilon > 0$, a $(2+\varepsilon)$-approximation for the Price Ladder variant.

A repetition free Longest Common Subsequence (LCS) of two sequences x and y is an LCS of x and y where each symbol may appear at most once. Let R denote the length of a repetition free LCS of two sequences of n symbols each one chosen randomly, uniformly, and independently over a k-ary alphabet. We study the asymptotic, in n and k, behavior of R and establish that there are three distinct regimes, depending on the relative speed of growth of n and k. For each regime we establish the limiting behavior of R. In fact, we do more, since we actually establish tail bounds for large deviations of R from its limiting behavior. Our study is motivated by the so called exemplar model proposed by Sankoff (1999) and the related similarity measure introduced by Adi et al. (2007). A natural question that arises in this context, which as we show is related to long standing open problems in the area of probabilistic combinatorics, is to understand the asymptotic, in n and k, behavior of parameter R., Comment: 15 pages, 1 figure

A systematic technique to bound factor-revealing linear programs is presented. We show how to derive a family of upper bound factor-revealing programs (UPFRP), and show that each such program can be solved by a computer to bound the approximation factor of an associated algorithm. Obtaining an UPFRP is straightforward, and can be used as an alternative to analytical proofs, that are usually very long and tedious. We apply this technique to the Metric Facility Location Problem (MFLP) and to a generalization where the distance function is a squared metric. We call this generalization the Squared Metric Facility Location Problem (SMFLP) and prove that there is no approximation factor better than 2.04, assuming P $\neq$ NP. Then, we analyze the best known algorithms for the MFLP based on primal-dual and LP-rounding techniques when they are applied to the SMFLP. We prove very tight bounds for these algorithms, and show that the LP-rounding algorithm achieves a ratio of 2.04, and therefore has the best factor for the SMFLP. We use UPFRPs in the dual-fitting analysis of the primal-dual algorithms for both the SMFLP and the MFLP, improving some of the previous analysis for the MFLP., Comment: Additional variants with powers of metrics

We give a simpler proof of Seymour's Theorem on edge-coloring series-parallel multigraphs and derive a linear-time algorithm to check whether a given series-parallel multigraph can be colored with a given number of colors.

The determination of the stability of the long-lived consensus problem is a fundamental open problem in distributed systems. We concentrate on the memoryless binary case with geodesic paths. We offer a conjecture on the stability in this case, exhibit two classes of colourings which attain this conjectured bound, and improve the known lower bounds for all colourings. We also introduce a related parameter, which measures the stability only for certain geodesics, and for which we also prove lower bounds.

The primal-dual scheme has been used to provide approximation algorithms for many problems. Goemans and Williamson gave a (2-1/(n-1))-approximation for the Prize-Collecting Steiner Tree Problem that runs in O(n^3 log n) time. it applies the primal-dual scheme once for each of the n vertices of the graph. Johnson, Minkoff and Phillips proposed a faster implementation of Goemans and Williamson's algorithm. We give a proof that the approximation ratio of this implementation is exactly 2.

Given a finite alphabet X and an ordering on the letters, the map \sigma sends each monomial on X to the word that is the ordered product of the letter powers in the monomial. Motivated by a question on Groebner bases, we characterize ideals I in the free commutative monoid (in terms of a generating set) such that the ideal <\sigma(I)> generated by \sigma(I) in the free monoid is finitely generated. Whether there exists an ordering such that <\sigma(I)> is finitely generated turns out to be NP-complete. The latter problem is closely related to the recognition problem for comparability graphs., Comment: 27 pages, 2 postscript figures, uses gastex.sty