We study graph search problems with imperfect detection. Both graph search and search with imperfect detection are well-studied subjects, but the natural combination is relatively new. In this setting, one is given an edge-weighted graph and a target is hidden at one of the vertices. One can walk through the graph to search for the hidden target and the goal is to find it as soon as possible. However, just visiting a vertex is not sufficient and a search needs to be done which takes a fixed amount of time. Each search attempt is only successful though with probability γ , given that the target is at the search location. For the hider, we consider the model where the target is hidden at random, and the adversarial model where the target is placed by an adversary. For the searcher, we consider both pathwise search and expanding search. For all cases, we obtain the first constant factor approximation guarantees.
We consider the buy-at-bulk facility location problem (BBFL), a problem combining the classic facility location problem with buy-at-bulk network design, which finds motivation in telecommunication networks. In it, we are given a graph with edge lengths, opening costs and demands for each vertex, and a monotone and subadditive capacity-cost function, and our task is to open facilities on a subset of the vertices and route the demand from each vertex to these facilities. The cost of a solution (which we want to minimize) is given by the opening costs of the chosen facilities, plus the cost on each edge, which is given by its length times the cost of providing enough capacity for the demands through the edge, given by the capacity-cost function. A common variant of the problem, the k-cable facility location problem (kCFL), considers the case where capacity is provided by buying copies of given cable types, each with a certain capacity and cost. We study BBFL on tree instances and show, for the unit-demand and splittable variants, that the problem admits a PTAS (a (1+ϵ ) -approximation for any ϵ > 0 ). We also consider kCFL in the new setting of cable-unsplittable demands, where the demand of a vertex cannot be split among multiple cables. We show that the problem is NP-hard to approximate to a factor better than 3/2 on stars, and then provide an algorithm for tree instances that outputs a solution with optimal cost, but which exceeds the capacity on each cable by a factor of 1+ϵ . As a consequence, we show that the problem has a 2-approximation algorithm on trees.
In this work, we study how to maintain a forest of arborescences of maximum arc cardinality under arc insertions while minimizing recourse—the total number of arcs changed in the maintained solution. This problem is the “arborescence version” of max cardinality matching. On the impossibility side, we observe that even in this insertion-only model, it is possible for m adversarial arc arrivals to necessarily incur (m · n) recourse, matching a trivial upper bound of O(m · n) . On the possibility side, we give an algorithm with expected recourse O(m ·log ^2 n) if all m arcs arrive uniformly at random.
We study online algorithms for maximum cardinality matchings with edge arrivals in graphs of low degree. Buchbinder, Segev, and Tkach showed that no online algorithm for maximum cardinality fractional matchings can achieve a competitive ratio larger than 4/(9-√(5))≈ 0.5914 even for graphs of maximum degree three. The negative result of Buchbinder et al. holds even when the graph is bipartite and edges are revealed according to vertex arrivals, i.e. once a vertex arrives, all edges are revealed that include the newly arrived vertex and one of the previously arrived vertices. In this work, we complement the negative result of Buchbinder et al. by providing an online algorithm for maximum cardinality fractional matchings with a competitive ratio at least 4/(9-√(5))≈ 0.5914 for graphs of maximum degree three. We also demonstrate that no online algorithm for maximum cardinality integral matchings can have the competitive guarantee 0.5807, establishing a gap between integral and fractional matchings for graphs of maximum degree three. Note that the work of Buchbinder et al. shows that for graphs of maximum degree two, there is no such gap between fractional and integral matchings, because for both of them the best achievable competitive ratio is 2/3. Also, our results demonstrate that for graphs of maximum degree three best possible competitive ratios for fractional matchings are the same in the vertex arrival and in the edge arrival models.
Given a set of geometric objects, A, in ℝ^d , a set of points P is a piercing set if every object in A includes a point in P. We consider the online version of the piercing set problem where recourse is allowed. In the traditional online setting, with irrevocable decisions, no algorithm can guarantee a competitive ratio better than (n) for piercing intervals, where n is the length of the input sequence. In this paper, we show a 2-competitive algorithm for the problem by allowing at most one recourse per input. We prove that to maintain an optimum piercing set for intervals, any online algorithm needs (n) amortised recourse. We also design an algorithm for piercing intervals which provides a trade-off between recourse and competitive ratio; the algorithm is (1+ϵ ) -competitive when allowed (1+2/ϵ ) recourse at each step. We extend our results to higher dimensions by obtaining a 2^d+1 -competitive algorithm with amortised two recourse for piercing axis-aligned hypercubes in ℝ^d .
The k -core of a graph is its maximal subgraph with minimum degree at least k, and the core value of a vertex u is the largest k for which u is contained in the k -core of the graph. Among cohesive subgraphs, k -core and its variants have received a lot of attention recently, particularly on dynamic graphs, as reported by Hanauer, Henzinger, and Schulz in their recent survey on dynamic graph algorithms. We answer questions on k-core stated in the survey, proving that there is no efficient dynamic algorithm for k -core or to find (2 - ϵ ) -approximations for the core values, unless we can improve decade-long state-of-the-art algorithms in many areas including matrix multiplication and satisfiability, based on the established OMv conjecture and SETH. Our results show that there is no dynamic algorithm for k-core asymptotically faster than the trivial ones, and we derive them via proving hardness results for a dynamic version of the circuit value problem. This explains why most recent research papers in this area focus not on a generic efficient dynamic algorithm, but on finding a bounded algorithm, which is fast when few core values change per update. However, we also prove that such bounded algorithms do not exist, based on the OMv conjecture. We also obtain lower bounds for a directed version of the problem, and for the edge variant of the problem, known as k-truss. On the positive side, we present a polylogarithmic dynamic algorithm for 2-core.
In the (1-dimensional) bin packing problem, we are asked to pack all the given items into bins, each of capacity one, so that the number of non-empty bins is minimized. Zhu [Chaos, Solitons & Fractals 2016] proposed an approximation algorithm MM that sorts the item sequence in a non-increasing order by size at the beginning, and then repeatedly packs, into the current single open bin, first as many of the largest items in the remaining sequence as possible and then as many of the smallest items in the remaining sequence as possible. In this paper we prove that the asymptotic approximation ratio of MM is at most 1.5. Next, focusing on the fact that MM is at the intersection of two algorithm classes, max-min algorithms and 1-bounded space algorithms, we comprehensively analyze the theoretical performance bounds of each subclass derived from the two classes. Our results include a lower bound of 1.25 for the intersection of the two classes. Furthermore, we extend the theoretical analysis over algorithm classes to the cardinality constrained bin packing problem.
We study the online sorting problem, where n real numbers arrive in an online fashion, and the algorithm must immediately place each number into an array of size (1+ε ) n before seeing the next number. After all n numbers are placed into the array, the cost is defined as the sum over the absolute differences of all n-1 pairs of adjacent numbers in the array, ignoring empty array cells. Aamand, Abrahamsen, Beretta, and Kleist introduced the problem and obtained a deterministic algorithm with cost 2^O( √(log n ·loglog n +logε ^-1)) , and a lower bound of (log n / loglog n) for deterministic algorithms. We obtain a deterministic4 algorithm with quasi-polylogarithmic cost ( ε ^-1log n) ^O( loglog n) . Concurrent and independent work by Azar, Panigrahi, and Vardi achieves polylogarithmic cost O(ε ^-1log ^2 n) .
We introduce a framework for incremental-decremental maximization that captures the gradual transformation or renewal of infrastructures. In our model, an initial solution is transformed one element at a time and the utility of an intermediate solution is given by the sum of the utilities of the transformed and untransformed parts. We propose a simple randomized and a deterministic algorithm that both find an order in which to transform the elements while maintaining a large utility during all stages of transformation, relative to an optimum solution for the current stage. More specifically, our algorithms yield competitive solutions for utility functions of bounded curvature and/or generic submodularity ratio, and, in particular, for submodular functions, and gross substitute functions. Our results exhibit that incremental-decremental maximization is substantially more difficult than incremental maximization.
We consider.Q||Cmax, the problem of scheduling.n jobs on.m uniform machines while minimizing the makespan, in an online setting with migration of jobs. In this online setting, the jobs are inserted or deleted over time, and at each step, the goal is to compute a near-optimal solution while reassigning some jobs, such that the overall processing time of reassigned jobs, called migration, is bounded by some factor.ss times the processing time of the job added or removed. We propose an Efficient Polynomial Time Approximation Schemes (EPTAS) with an additional load error of.O( epmax) and constant amortized migration factor.ss, where.pmax is the maximum processing time in the instance over all steps. As an intermediate step, we obtain an Efficient Parameterized Approximation Scheme (EPAS), a.(1+ e)-competitive algorithm parameterized by.pmax and the number of different processing times.d in an instance, with.ss bounded in a function of.p(max),.d and.e.
We study the optimal solution to a general two slope ski rental problem with a tail risk, i.e., the chance of the competitive ratio exceeding a specific value γ is bounded by a constant δ . This extends the recent study of tail bounds for ski rental initiated by [Dinitz et al. SODA 2024] to the two-slope version defined by [Lotker et al. IPL 2008]. In this version, even after “buying” we must still pay a rental cost at each time step, but the rental cost is lower after buying. This models many real-world “rent-or-buy” scenarios where a one-time investment decreases (but does not eliminate) the per-time cost. Despite this being a simple extension of the classical problem, we find that adding tail risk bounds creates a fundamentally different solution structure. For example, in our setting there is a possibility that we never buy in an optimal solution (which can also occur without tail bounds), but more strangely (and unlike the case without tail bounds or the classical case with tail bounds) we also show that the optimal solution might need to have nontrivial probabilities of buying even at finite points beyond the time corresponding to the buying cost. Moreover, in many regimes there does not exist a unique optimal solution. As our first contribution, we develop a series of structure theorems to characterize some features of optimal solutions. The complex structure of optimal solutions makes it more difficult to develop an algorithm to compute such a solution. As our second contribution, we utilize our structure theorems to design two algorithms: one based on a greedy algorithm combined with binary search that is fast but yields arbitrarily close to optimal solutions, and a slower algorithm based on linear programming which computes exact optimal solutions.
Consider a graph with n nodes and m edges, independent edge weights and lengths, and arbitrary distance demands for node pairs. The spanner problem asks for a minimum-weight subgraph that satisfies these demands via sufficiently short paths w.r.t. the edge lengths. For multiplicative alpha-spanners (where demands equal alpha times the original distances) and assuming that each edge's weight equals its length, the simple Greedy heuristic by Althöfer et al. (1993) is known to yield strong solutions, both in theory and practice. To obtain guarantees in more general settings, recent approximations typically abandon this simplicity and practicality. Still, so far, there is no known non-trivial approximation algorithm for the spanner problem in its most general form. We provide two surprisingly simple approximations algorithms. In general, our Adapted Greedy achieves the first unconditional approximation ratio of m, which is non-trivial due to the independence of weights and lengths. Crucially, it maintains all size and weight guarantees Greedy is known for, i.e., in the aforementioned multiplicative alpha-spanner scenario and even for additive +beta-spanners. Further, it generalizes some of these size guarantees to derive new weight guarantees. Our second approach, Randomized Rounding, establishes a graph transformation that allows a simple rounding scheme over a standard multicommodity flow LP. It yields an O(n log n)-approximation, assuming integer lengths and polynomially bounded distance demands. The only other known approximation guarantee in this general setting requires several complex subalgorithms and analyses, yet we match it up to a factor of O(n^1/5-eps) using standard tools. Further, on bounded-degree graphs, we yield the first O(log n) approximation ratio for constant-bounded distance demands (beyond multiplicative 2-spanners in unit-length graphs).
In the online general knapsack problem, an algorithm is presented with an item x=(s,v) of size s and value v and must irrevocably choose to pack such an item into the knapsack or reject it before the next item appears. The goal is to maximize the total value of the packed items without overflowing the knapsack’s capacit. As this classical setting is way too harsh for many real-life applications, we will analyze the online general knapsack problem under the reservation model. Here, instead of accepting or rejecting an item immediately, an algorithm can delay the decision of whether to pack the item by paying a fraction 0≤α of the size or the value of the item. This models many practical applications, where, for example, decisions can be delayed for some costs e.g. cancellation fees. We present results for both variants: First, for costs depending on the value of the items and then for costs depending on the size of the items. If the reservation costs depend on the value of the items, we find that no algorithm is competitive for reservation costs larger than 1/2 of the item value, and we find upper and lower bounds for the rest of the reservation factor range 0≤α < 1/2 . On the other hand, if the reservation costs depend on the size of the items, we find a matching upper and lower bound of 2 for every reservation factor α .
In this paper, we analyze FirstFit on online unit-length intervals, which can either be open or closed, further investigating FirstFit’s true performance. We develop a sophisticated counting method by generalizing the classic neighborhood bound, which limits the color FirstFit can assign an interval by counting the potential intersections. In the generalization, we show that for any interval, there is a critical interval intersecting it that can help reduce overestimation of the number of intersections, and it further helps bound the color an interval can be assigned. The technical challenge lies in reliably finding these critical intervals. Using this new technique, we provide a tight analysis showing that FirstFit uses at most 2ω colors when all endpoints are integral, and an upper bound of ⌈7/3ω⌉ - 2 in the general case of open and closed unit intervals, where ω denotes the optimal number of colors needed.
Interval-constrained online bipartite matching problem frequently occurs in medical appointment scheduling: Unit-time jobs representing patients arrive online and are assigned to a time slot within their given feasible time interval. We consider a variant of this problem where reassignments are allowed and extend it by a notion of time that is decoupled from the job arrival events. As jobs appear, the current point in time gradually advances, and once the time of a slot is passed, the job assigned to it is fixed and cannot be reassigned anymore. We analyze two algorithms for the problem with respect to the resulting matching size and the number of reassignments they make. We show that FirstFit with reassignments according to the shortest augmenting path rule is 2/3 -competitive with respect to the matching cardinality, and that the bound is tight. The competitive ratio remains 2/3 if we restrict the algorithm to make only a constant number k ⩾ 1 of reassignments per job arrival. This fills the gap between the known optimal algorithm that makes no reassignments, which is 1/2 -competitive, on the one hand, and an earliest-deadline-first strategy (EDF), which we prove to obtain a maximum matching in this over-time framework, but which suffers Ω (n^2) reassignments in the worst case, on the other. In the full paper, we show that FirstFit remains 2/3 -competitive also if the sets of feasible slots per job that are not intervals, and that this is the best possible deterministic competitive ratio, while EDF loses its optimality.
We consider the problem of scheduling a set of n preemptive jobs with deadlines arriving online on m identical machines with the goal of maximizing weighted throughput. The jobs being scheduled are parallelizable and their parallelism is modelled with the standard speed-up curves model. Each job J_i arrives at time r_i with an associated deadline d_i and profit p_i which is acquired if the job is completed by its deadline. Jobs also have corresponding speed-up functions _i : ^+ → ^+ . The speed-up function _i(y) describes the rate at which the job is processed when scheduled on y machines and jobs are allowed to have distinct speed-up functions. We give the first result for the throughput scheduling problem for jobs with speed-up curves using resource augmentation, by showing a O(1+ϵ ) speed, O(1/ϵ ^2) competitive algorithm.
We consider an unsplittable version of the minimum max-load multicommodity flow problem, where each demand must be routed along a single path. The objective is to minimize the maximum load on any edge in the network. In a seminal work, Schrijver, Seymour, and Winkler showed how to efficiently solve this problem on a cycle to within an additive term of 3/2 W of the optimal value, where W is the largest demand between any two nodes. We extend their result to outerplanar graphs and provide an efficient algorithm for this problem that exceeds the optimal value by an additive term of no more than O(W log k), where k is the number of faces in the graph. This implies an O(log k) approximation ratio. We also extend this result to planar graphs with bounded treewidth and demands on the outer face, for which we also achieve an additive O(W log k) error term.
A classic result of Williamson, Goemans, Mihail, and Vazirani [STOC 1993: 708–717] states that the problem of covering an uncrossable set family by a min-cost edge set admits approximation ratio 2, by a primal-dual algorithm with a reverse delete phase. Recently, Bansal, Cheriyan, Grout, and Ibrahimpur [ICALP 2023: 15:1-15:19] showed that this algorithm achieves approximation ratio 16 for a larger class of so called γ -pliable set families, that have much weaker uncrossing properties. In this paper we will improve the approximation ratio to 10. Using this result and other techniques, we also improve approximation ratios for the following two problems related to the Capacitated k-Edge Connected Spanning Subgraph (Cap- k -ECSS) problem.
In today's data-centric world, fast and effective compression of data is paramount. To measure success towards the second goal, Kempa and Prezza [STOC2018] introduce the string attractor, a combinatorial object unifying dictionary-based compression. Given a string T is an element of Sigma(n), a string attractor (k-attractor) is a set of positions Gamma subset of [1, n], such that every distinct substring (of length at most k) has at least one occurrence that contains one of the selected positions. String attractors are shown to be approximated by and thus measure the quality of many important dictionary compression algorithms such as Lempel-Ziv 77, the Burrows-Wheeler transform, straight line programs, and macro schemes. In order to handle massive amounts of data, compression often has to be achieved in a streaming fashion. Thus, practically applied compression algorithms, such as Lempel-Ziv 77, have been extensively studied in an online setting. To the best of our knowledge, there has been no such work, and therefore are no theoretical underpinnings, for the string attractor problem. We introduce a natural online variant of both the k-attractor and the string attractor problem. First, we show that the Lempel-Ziv factorization corresponds to the best online algorithm for this problem, resulting in an upper bound of O(log(n)) on the competitive ratio. On the other hand, we consider prefixes of the Fibonacci word, and show that any online algorithm has a cost growing with the length of the prefix, for a matching lower bound of Omega(log(n)). For the online k-attractor problem, we show tight (strict) k-competitiveness.
We consider the classic Vehicle Routing Problem with the additional property that all requests are ordered and the subtour of each server (or vehicle) must obey the fixed order. A scheduling version of this problem was introduced by Bosman et al. (2019). We study several metric spaces and objective functions and our results show that in some settings such a fixed order simplifies the problem, while in others it makes an easy problem become NP-hard. For general metrics, we show that c-capacitated VRP remains APX-hard in the fixed order setting for c = 3 and show that the well-known iterated tour partitioning algorithm yields a (2 - 1/c)-approximation. When all points are on the line, we show that the fixed order restriction makes VRP NP-hard to solve for minimizing total completion time or maximum completion time, in contrast to standard VRP. We also sketch how to obtain a PTAS in these settings for general metrics.
