In the sequential resource allocation problem there is a single divisible resource that is divided over a number of clients. Allocations are made in a predetermined order and only upon arrival at a client their demand for the resource is revealed; only the probability distribution of the demand of every client is known to the supplier. We consider this problem from a fairness perspective, where the aim is to balance allocations between individual clients. Several allocation policies have been proposed in the literature. In this work, we introduce a new, non-adaptive policy based on linear programming that can also incorporate group fairness. In addition, we provide an extensive computational study to compare allocation policies on several fairness measures. Using an optimized implementation of existing methods, we are able to evaluate significantly larger problem instances than those previously considered in the literature.

We generalize the polynomial-time solvability of $k$-\textsc{Diverse Minimum s-t Cuts} (De Berg et al., ISAAC'23) to a wider class of combinatorial problems whose solution sets have a distributive lattice structure. We identify three structural conditions that, when met by a problem, ensure that a $k$-sized multiset of maximally-diverse solutions -- measured by the sum of pairwise Hamming distances -- can be found in polynomial time. We apply this framework to obtain polynomial time algorithms for finding diverse minimum $s$-$t$ cuts and diverse stable matchings. Moreover, we show that the framework extends to two other natural measures of diversity. Lastly, we present a simpler algorithmic framework for finding a largest set of pairwise disjoint solutions in problems that meet these structural conditions.

There is arbitrariness in optimum solutions of graph-theoretic problems that can give rise to unfairness. Incorporating fairness in such problems, however, can be done in multiple ways. For instance, fairness can be defined on an individual level, for individual vertices or edges of a given graph, or on a group level. In this work, we analyze in detail two individual-fairness measures that are based on finding a probability distribution over the set of solutions. One measure guarantees uniform fairness, i.e., entities have equal chance of being part of the solution when sampling from this probability distribution. The other measure maximizes the minimum probability for every entity of being selected in a solution. In particular, we reveal that computing these individual-fairness measures is in fact equivalent to computing the fractional covering number and the fractional partitioning number of a hypergraph. In addition, we show that for a general class of problems that we classify as independence systems, these two measures coincide. We also analyze group fairness and how this can be combined with the individual-fairness measures. Finally, we establish the computational complexity of determining group-fair solutions for a variant of the matching problem.

Abstract. We study Dominating Set and Independent Set for dynamic graphs in the vertex-arrival model. We say that a dynamic algorithm for one of these problems is [Formula: see text]- stable when it makes at most [Formula: see text] changes to its output independent set or dominating set upon the arrival of each vertex. We study trade-offs between the stability parameter [Formula: see text] of the algorithm and the approximation ratio it achieves. We obtain the following results: (i) We show that there is a constant [Formula: see text] such that any dynamic [Formula: see text]-approximation algorithm for Dominating Set has stability parameter [Formula: see text], even for bipartite graphs of maximum degree 4. (ii) We present algorithms with very small stability parameters for Dominating Set in the setting where the arrival degree of each vertex is upper bounded by [Formula: see text]. In particular, we give a 1-stable [Formula: see text]-approximation algorithm, a 3-stable [Formula: see text]-approximation algorithm, and an [Formula: see text]-stable [Formula: see text]-approximation algorithm. (iii) We show that there is a constant [Formula: see text] such that any dynamic [Formula: see text]-approximation algorithm for Independent Set has stability parameter [Formula: see text], even for bipartite graphs of maximum degree 3. (iv) Finally, we present a 2-stable [Formula: see text]-approximation algorithm for Independent Set, in the setting where the average degree of the graph is upper bounded by some constant [Formula: see text] at all times. We extend this latter algorithm to the fully dynamic model where vertices can also be deleted, achieving a 6-stable [Formula: see text]-approximation algorithm.

Abstract. Let [Formula: see text] be a set of points in [Formula: see text], where each point [Formula: see text] has an associated transmission range, denoted [Formula: see text]. The range assignment [Formula: see text] induces a directed communication graph [Formula: see text] on [Formula: see text], which contains an edge [Formula: see text] iff [Formula: see text]. In the broadcast range-assignment problem, the goal is to assign the ranges such that [Formula: see text] contains an arborescence rooted at a designated root node and the cost [Formula: see text] of the assignment is minimized. We study the dynamic version of this problem. In particular, we study trade-offs between the stability of the solution—the number of ranges that are modified when a point is inserted into or deleted from [Formula: see text]—and its approximation ratio. To this end we study [Formula: see text]- stable algorithms, which are algorithms that modify the range of at most [Formula: see text] points when they update the solution. We also introduce the concept of a stable approximation scheme, or SAS for short. A SAS is an update algorithm [Formula: see text] that, for any given fixed parameter [Formula: see text], is [Formula: see text]-stable and that maintains a solution with approximation ratio [Formula: see text], where the stability parameter [Formula: see text] only depends on [Formula: see text] and not on the size of [Formula: see text]. We study such trade-offs in three settings. (1) For the problem in [Formula: see text], we present a SAS with [Formula: see text]. Furthermore, we prove that this is tight in the worst case: any SAS for the problem must have [Formula: see text]. We also present 1-, 2-, and 3-stable algorithms with constant approximation ratio. (2) For the problem in [Formula: see text] (that is, when the underlying space is a circle) we prove that no SAS exists. This is in spite of the fact that, for the static problem in [Formula: see text], we prove that an optimal solution can always be obtained by cutting the circle at an appropriate point and solving the resulting problem in [Formula: see text]. (3) For the problem in [Formula: see text], we also prove that no SAS exists, and we present a [Formula: see text]-stable [Formula: see text]-approximation algorithm. Most results generalize to the setting where, for any given constant [Formula: see text], the range-assignment cost is [Formula: see text].

Finding minimum dominating set and maximum independent set for graphs in the classical online setup are notorious due to their disastrous $Ω(n)$ lower bound of the competitive ratio that even holds for interval graphs, where $n$ is the number of vertices. In this paper, inspired by Newton number, first, we introduce the independent kissing number $ζ$ of a graph. We prove that the well known online greedy algorithm for dominating set achieves optimal competitive ratio $ζ$ for any graph. We show that the same greedy algorithm achieves optimal competitive ratio $ζ$ for online maximum independent set of a class of graphs with independent kissing number $ζ$. For minimum connected dominating set problem, we prove that online greedy algorithm achieves an asymptotic competitive ratio of $2(ζ-1)$, whereas for a family of translated convex objects the lower bound is $\frac{2ζ-1}{3}$. Finally, we study the value of $ζ$ for some specific families of geometric objects: fixed and arbitrary oriented unit hyper-cubes in $I\!\!R^d$, congruent balls in $I\!\!R^3$, fixed oriented unit triangles, fixed and arbitrary oriented regular polygons in $I\!\!R^2$. For each of these families, we also present lower bounds of the minimum connected dominating set problem.

Given a connected undirected graph G, a spanning tree is a subgraph T of G such that V(T) = V(G) and T is a tree. A collection of 𝓁 spanning trees T₁,…,T_{𝓁} is {{pairwise k-diverse}} if for every i ≠ j, |E(T_i) △ E(T_j)| ≥ k. Given a connected undirected graph G and integers p, q, k, 𝓁, {Leaf&Internal-Constrained Diverse Spanning Trees} asks whether there are 𝓁 distinct spanning trees T₁,…,T_{𝓁} of G that are {{pairwise k-diverse}} such that each tree has at least p leaves and at least q internal vertices. Similarly, {Leaf&Non-terminal-Constrained Diverse Spanning Trees} takes a connected undirected graph G, V_NT ⊆ V(G), and three integers p, k, 𝓁, and asks if G has 𝓁 spanning trees that are {{pairwise k-diverse}}, and each has at least p leaves and contains the vertices of V_NT as internal. We consider these two problems from the kernelization perspective and provide polynomial kernels for {Leaf&Internal-Constrained Diverse Spanning Trees} and {Leaf&Non-terminal-Constrained Diverse Spanning Trees}, when parameterized by p + q + k + 𝓁 and p + |V_NT| + k + 𝓁, respectively.

For the problem of delivering a package from a source node to a destination node in a graph using a set of drones, we study the setting where the movements of each drone are restricted to a certain subgraph of the given graph. We consider the objectives of minimizing the delivery time (problem DDT) and of minimizing the total energy consumption (problem DDC). For general graphs, we show a strong inapproximability result and a matching approximation algorithm for DDT as well as NP-hardness and a 2-approximation algorithm for DDC. For the special case of a path, we show that DDT is NP-hard if the drones have different speeds. For trees, we give optimal algorithms under the assumption that all drones have the same speed or the same energy consumption rate. The results for trees extend to arbitrary graphs if the subgraph of each drone is isometric.

We introduce the problem of selecting patient-donor pairs in a kidney exchange program to undergo a crossmatch test, and we model this selection problem as a two-stage stochastic integer programming problem. The optimal solutions of this new formulation yield a larger expected number of realized transplants than previous approaches based on internal recourse or subset recourse. We settle the computational complexity of the selection problem by showing that it remains NP-hard even for maximum cycle length equal to two. Furthermore, we investigate to what extent different algorithmic approaches, including one based on Benders decomposition, are able to solve instances of the model. We empirically investigate the computational efficiency of this approach by solving randomly generated instances and study the corresponding running times as a function of maximum cycle length, and of the presence of nondirected donors. Summary of Contribution: This paper deals with an important and very complex issue linked to the optimization of transplant matchings in kidney exchange programs, namely, the inherent uncertainty in the assessment of compatibility between donors and recipients of transplants. Although this issue has previously received some attention in the optimization literature, most attempts to date have focused on applying recourse to solutions selected within restricted spaces. The present paper explicitly formulates the maximization of the expected number of transplants as a two-stage stochastic integer programming problem. The formulation turns out to be computationally difficulty, both from a theoretical and from a numerical perspective. Different algorithmic approaches are proposed and tested experimentally for its solution. The quality of the kidney exchanges produced by these algorithms compares favorably with that of earlier models.

Given a complete graph with an even number of vertices, and with each edge colored with one of two colors (say red or blue), an equitable Hamiltonian cycle is a Hamiltonian cycle that can be decomposed into two perfect matchings such that both perfect matchings have the same number of red edges. We show that, for any coloring of the edges, in any complete graph on at least 6 vertices, an equitable Hamiltonian cycle exists.

We introduce and investigate the problem of scheduling a single lock with parallel chambers. Special cases of this problem are related to interval scheduling. We focus on the existence of no-wait schedules and characterize their feasibility for a lock consisting of two chambers using new graph-theoretical concepts. We obtain a linear time algorithm for this special case. We also provide an efficient algorithm for the case where all chambers of the lock are identical. Furthermore, we describe a dynamic programming algorithm for the general case with arbitrary chambers. Finally, we indicate how our methods for the no-wait case can be applied to practical settings where waiting time is unavoidable.

Consider a problem where 4k given vectors need to be partitioned into k clusters of four vectors each. A cluster of four vectors is called a quad, and the cost of a quad is the sum of the component-wise maxima of the four vectors in the quad. The problem is to partition the given 4k vectors into k quads with minimum total cost. We analyze a straightforward matching-based algorithm and prove that this algorithm is a 3/2-approximation algorithm for this problem. We further analyze the performance of this algorithm on a hierarchy of special cases of the problem and prove that, in one particular case, the algorithm is a 5/4-approximation algorithm. Our analysis is tight in all cases except one.

We investigate the scheduling of series of consecutive locks. This setting occurs naturally along canals and waterways. We describe a problem that generalizes different models that have been studied in literature. Our contribution is to (i) provide two distinct mathematical programming formulations, and compare them empirically, (ii) show how these models allow for minimizing emission by having the speed of a ship as a decision variable, (iii) to compare, on realistic instances, the optimum solution found by solving the models with the outcome of a decentralized heuristic.

Abstract We consider a rooted tree graph with costs associated with the edges and profits associated with the vertices. Every subtree containing the root incurs the sum of the costs of its edges, and collects the sum of the profits of its nodes; the goal is the simultaneous minimization of the total cost and maximization of the total profit. This problem is related to the TSP with profits on graphs with a tree metric. We analyze the problem from a biobjective point of view. We show that finding all extreme supported efficient points can be done in polynomial time. The problem of finding all efficient points, however, is harder; we propose a practical FPTAS for solving this problem. Some special cases are considered where the particular profit/cost structure or graph topology allows the efficient points to be found in polynomial time. Our results can be extended to more general graphs with distance matrices satisfying the Kalmanson conditions. © 2012 Wiley Periodicals, Inc. NETWORKS, 2013

We consider the problem of deciding whether a given directed graph can be vertex partitioned into two acyclic subgraphs. Applications of this problem include testing rationality of collective consumption behavior, a subject in microeconomics. We prove that the problem is NP-complete even for oriented graphs and argue that the existence of a constant-factor approximation algorithm is unlikely for an optimization version that maximizes the number of vertices that can be colored using two colors while avoiding monochromatic cycles. We present three exact algorithms—namely, an integer-programming algorithm based on cycle identification, a backtracking algorithm, and a branch-and-check algorithm. We compare these three algorithms both on real-life instances and on randomly generated graphs. We find that for the latter set of graphs, every algorithm solves instances of considerable size within a few seconds; however, the CPU time of the integer-programming algorithm increases with the number of vertices in the graph more clearly than the CPU time of the two other procedures. For real-life instances, the integer-programming algorithm solves the largest instance in about a half hour, whereas the branch-and-check algorithm takes approximately 10 minutes and the backtracking algorithm less than 5 minutes. Finally, for every algorithm, we also study empirically the transition from a high to a low probability of a YES answer as a function of the number of arcs divided by the number of vertices.

Inland waterways form a natural network that is an existing, congestion free infrastructure with capacity for more traffic. The European commission promotes the transportation of goods by ship as it is a reliable, efficient and environmental friendly way of transport. A bottleneck for transportation over water are the locks that manage the water level. The lockmaster's problem concerns the optimal strategy for operating such a lock. In the lockmaster's problem we are given a lock, a set of ships coming from downstream that want to go upstream, and another set of ships coming from upstream that want to go downstream. We are given the arrival times of the ships and a constant lockage time; the goal is to minimize total waiting time of the ships. In this paper a dynamic programming algorithm (DP) is proposed that solves the lockmaster's problem in polynomial time. We extend this DP to different generalizations that consider weights, water usage, capacity, and (a fixed number of) multiple chambers. Finally, we prove that the problem becomes strongly NP-hard when the number of chambers is part of the input.

In the traveling repairman problem with profits, a repairman (also known as the server) visits a subset of nodes in order to collect time-dependent profits. The objective consists of maximizing the total collected revenue. We restrict our study to the case of a single server with nodes located in the Euclidean plane. We investigate properties of this problem, and we derive a mathematical model assuming that the number of visited nodes is known in advance. We describe a tabu search algorithm with multiple neighborhoods, and we test its performance by running it on instances based on TSPLIB. We conclude that the tabu search algorithm finds good-quality solutions fast, even for large instances.

We consider the problem of assigning sensors to track targets so as to minimize the expected error in the resulting estimation for target locations. The so-called Focus of Attention problem deals with the special case where every target is tracked by one pair of range sensors. We provide a complete complexity and approximability analysis of the Focus Of Attention problem: We establish its strong NP-hardness, and we construct a polynomial time approximation scheme for it.

A combinatorial auction is an auction where multiple items are for sale simultaneously to a set of buyers. Furthermore, buyers are allowed to place bids on subsets of the available items. This paper focuses on a combinatorial auction where a bidder can express his preferences by means of a so-called ordered matrix bid. Ordered matrix bids are a bidding language that allows a compact representation of a bidder's preferences and was developed by Day [Day, R. W. 2004. Expressing preferences with price-vector agents in combinatorial auctions. Ph.D. thesis, University of Maryland, College Park]. We give an overview of how a combinatorial auction with matrix bids works. We discuss the relevance of recognizing whether a given matrix bid has properties related to elements of economic theory such as free disposal, subadditivity, submodularity, and the gross substitutes property. We show that verifying whether a matrix bid has these properties can be done in polynomial time by solving one or more shortest-path problems. Finally, we investigate to what extent randomly generated matrix bids satisfy these properties.

In this paper we discuss a special pallet-loading problem, which we encountered at a manufacturing company. In graph-theoretical terms, the problem is equivalent to partitioning a permutation graph into bounded-size cliques. We formulate the problem as an integer program, and present two exact algorithms for solving it. The first algorithm is a branch-and-price algorithm based on the integer-programming formulation; the second one is an algorithm based on the concept of bounded clique width. The latter algorithm was motivated by the structure present in the real-world instances. Test results are given, both for real-world instances and randomly generated instances. As far as we are aware, this is the first implementation of an algorithm based on bounded clique width.

In the weighted rectangle stabbing problem we are given a grid in $\mathbb{R}^2$ consisting of columns and rows each having a positive integral weight, and a set of closed axis‐parallel rectangles each having a positive integral demand. The rectangles are placed arbitrarily in the grid with the only assumption being that each rectangle is intersected by at least one column or row. The objective is to find a minimum‐weight (multi)set of columns and rows of the grid so that for each rectangle the total multiplicity of selected columns and rows stabbing it is at least its demand. A special case of this problem, called the interval stabbing problem, arises when each rectangle is intersected by exactly one row. We describe an algorithm called STAB, which is shown to be a constant‐factor approximation algorithm for different variants of this stabbing problem.

Random redundant data storage strategies have proven to be a good choice for efficient data storage in multimedia servers. These strategies lead to a retrieval problem in which it is decided for each requested data block which disk to use for its retrieval. In this paper, we give a complexity classification of retrieval problems for random redundant storage.

Abstract The clique partitioning problem (CPP) can be formulated as follows: Given is a complete graph G = (V, E), with edge weights wij ∈ ℝ for all {i, j} ∈ E. A subset A ⊆ E is called a clique partition if there is a partition of V into nonempty, disjoint sets V1,…, Vk, such that each Vp (p = 1,…, k) induces a clique (i.e., a complete subgraph), and A = ∪ {{i, j}|i, j ∈ Vp, i ≠ j}. The weight of such a clique partition A is defined as Σ{i,j}∈A wij. The problem is now to find a clique partition of maximum weight. The clique partitioning polytope P is the convex hull of the incidence vectors of all clique partitions of G. In this paper, we introduce several new classes of facet‐defining inequalities of P. These suffice to characterize all facet‐defining inequalities with right‐hand side 1 or 2. Also, we present a procedure, called patching, which is able to construct new facets by making use of already‐known facet‐defining inequalities. A variant of this procedure is shown to run in polynomial time. Finally, we give limited empirical evidence that the facet‐defining inequalities presented here can be of use in a cutting‐plane approach for the clique partitioning problem. © 2001 John Wiley & Sons, Inc.