We analyze the competitive ratio of the proportional online knapsack problem with removal and limited recourse. In contrast to the classical online knapsack problem, packed items can be removed and a limited number of removed items can be re-inserted to the knapsack. The variant with removal only was analyzed by Iwama and Taketomi (ICALP, 2002). We show that even a single use of recourse can improve the performance of an algorithm. We give lower bounds for a constant number of k ≥ 1 uses of recourse in total, matching upper bounds for 1 ≤ k ≤ 3 , and a general upper bound for any value of k . For a variant where a constant number of k ≥ 1 uses of recourse can be used per step, we give tight bounds for all k ≥ 1 . We further look at a scenario where an algorithm is informed when the instance ends and give improved upper bounds in both variants for this case.

We present an approximation algorithm for the Prize-collecting Ordered Traveling Salesman Problem (PCOTSP), which simultaneously generalizes the Prize-collecting TSP and the Ordered TSP. The Prize-collecting TSP is well-studied and has a long history, with the current best approximation factor slightly below 1.6, shown by Blauth, Klein and Nägele [IPCO 2024]. The best approximation ratio for Ordered TSP is 3/2+1/e, presented by Böhm, Friggstad, Mömke, Spoerhase [SODA 2025] and Armbruster, Mnich, Nägele [Approx 2024]. The former also present a factor 2.2131 approximation algorithm for Multi-Path-TSP. We present a 2.097-approximation algorithm for PCOTSP, which is, to the best of our knowledge, the first result for this problem. Key ideas in our approach are to sample a set of trees and then to probabilistically pick up some vertices, and to use the pruning ideas of Blauth, Klein, Nägele [IPCO 2024] on the sampled vertices. While the sampling probability of vertices for our problem is lower than for PCTSP, intuitively leaving less spare penalty to spend, we leverage the cycle structure induced by the sampled trees together with a simple combinatorial algorithm to bring the approximation factor below 2.1. Our techniques extend to Prize-collecting Multi-Path TSP, building on results from Böhm, Friggstad, Mömke, Spoerhase [SODA 2025], leading to a 2.41-approximation.

We give improved approximations for two metric Traveling Salesman Problem (TSP) variants. In Ordered TSP (OTSP) we are given a linear ordering on a subset of nodes o1,. .., ok. The TSP solution must have that o i+1 is visited at some point after Oi for each 1 ≤ i ≤ k. This is the special case of Precedence- Constrained TSP (PTSP) in which the precedence constraints are given by a single chain on a subset of nodes. In k-Person TSP Path (k-TSPP), we are given pairs of nodes (s1, t1), …, (sk, tk ). The goal is to find an si-ti path with minimum total cost such that every node is visited by at least one path.

We investigate semi-streaming algorithms for the Traveling Salesman Problem (TSP). Specifically, we focus on a variant known as the (1,2)-TSP, where the distances between any two vertices are either one or two. Our primary emphasis is on the closely related Maximum Path Cover Problem, which aims to find a collection of vertex-disjoint paths that covers the maximum number of edges in a graph. We propose an algorithm that, for any ε > 0, achieves a (2/3-ε)-approximation of the maximum path cover size for an n-vertex graph, using poly(1/ε) passes. This result improves upon the previous 1/2-approximation by Behnezhad et al. [Soheil Behnezhad et al., 2023] in the semi-streaming model. Building on this result, we design a semi-streaming algorithm that constructs a tour for an instance of (1,2)-TSP with an approximation factor of (4/3 + ε), improving upon the previous 3/2-approximation factor algorithm by Behnezhad et al. [Soheil Behnezhad et al., 2023]. Furthermore, we extend our approach to develop an approximation algorithm for the Maximum TSP (Max-TSP), where the goal is to find a Hamiltonian cycle with the maximum possible weight in a given weighted graph G. Our algorithm provides a (7/12 - ε)-approximation for Max-TSP in poly(1/(ε)) passes, improving on the previously known (1/2-ε)-approximation obtained via maximum weight matching in the semi-streaming model.

In the known Interval Scheduling problem with Machine Availabilities (ISMA), each machine has a contiguous availability interval, and each job has a specific time interval which has to be scheduled. The objective is to schedule all jobs such that the machines’ availability intervals are respected or to decide that there exists no such schedule. We extend ISMA by introducing machine capacities and flexible machine end times. Using machine capacities we model parallel processing of multiple jobs per machine, which leads to the Multithread Interval Scheduling with Machine Availabilities (MISMA). Limited machine availabilities are usually due to maintenance. Time slots for maintenance at the end of a processing period are often predetermined by staff schedules before the slots are assigned to specific machines. This motivates a variant of MISMA in which the end times of the machines’ availability intervals can be permuted, the Flexible Multithread ISMA (FLEXMISMA). In this paper, we determine a tight classification of conditions that are required for obtaining a polynomial-time algorithm for both MISMA and FLEXMISMA. More specifically, we show that FLEXMISMA is at least as hard as MISMA. For FLEXMISMA, we present polynomial-time algorithms for instances (i) with at most two available machines at a time, and (ii) with constantly many parallel jobs at each point in time, which both also solve MISMA; (iii) with arbitrarily many machines of capacity one each, in which case MISMA is known to be NP-hard; and (iv) with jobs having length one or two, for which the complexity of MISMA remains open Furthermore, we complement result (i) by showing that both problems are NP-hard already for instances with three machines as a special case of the Vertex-Disjoint Paths problem. In contrast to (iii), we prove that increasing the capacity of machines from one to two renders FLEXMISMA NP-hard as well for arbitrarily many machines.

We study the Maximum Independent Set of Rectangles (MISR) problem, where we are given a set of axis-parallel rectangles in the plane and the goal is to select a subset of non-overlapping rectangles of maximum cardinality. In a recent breakthrough, Mitchell [46] obtained the first constant-factor approximation algorithm for MISR. His algorithm achieves an approximation ratio of 10 and it is based on a dynamic program that intuitively recursively partitions the input plane into special polygons called corner-clipped rectangles (CCRs), without intersecting certain special horizontal line segments called fences. In this paper, we present a 3-approximation algorithm for MISR which is also based on a recursive partitioning scheme. First, we use a partition into a class of axis-parallel polygons with constant complexity each that are more general than CCRs. This allows us to provide an arguably simpler analysis and at the same time already improves the approximation ratio to 6. Then, using a more elaborate charging scheme and a recursive partitioning into general axis-parallel polygons with constant complexity, we improve our approximation ratio to 3. In particular, we construct a recursive partitioning based on more general fences which can be sequences of up to O(1) line segments each. This partitioning routine and our other new ideas may be useful for future work towards a PTAS for MISR.

The unsplittable flow on a path (UFP) problem is a well-studied optimization problem, and it has applications in various settings like bandwidth allocation, caching, and scheduling. We are given a path with capacities on its edges and a set of n tasks, each of them defined via a demand, a subpath, and a profit. The goal is to select the most profitable set of tasks that together respect the edge capacities, i.e., for each edge e the total demand of the selected tasks whose subpath contains e is at most the capacity of e. The best known polynomial time approximation algorithm for UFP is a (5/3 + ∊)-approximation [Grandoni et al., STOC 2018]. It is an important open question whether the problem admits a PTAS. Informally, a task is large if its demand is at least an ∊-fraction of the capacity of some edge on its path, and small otherwise. If all tasks are large, a PTAS can be obtained via dynamic programming: intuitively each edge e is used by only O(1) relevant tasks in the optimal solution OPT. The same approach fails for small tasks since then this number can be up to Ω(n) which would yield an exponential number of states. In this paper we introduce a novel randomized sketching technique to address this issue. We model the computation of a solution as a solitary game where tasks are presented one by one to a player, who has to decide for each task i whether to select i (hence getting its profit) or not. When a small task i is selected, with some probability its demand is rounded up to some large value (and then i behaves like a large task), and otherwise down to zero (and then i can be “forgotten” afterwards), so that in expectation the demand of i does not change. The optimal strategy to play this game can be computed using similar ideas as used in the DP for large tasks. Furthermore, the expected profit of this strategy is at least as large as the profit of OPT. One complication is that the player's solution might be infeasible, e.g., when too many tasks are rounded down. Still, via probabilistic arguments, we can use it to construct a feasible UFP solution which is 1 + + ∊ < 1.269 approximate in expectation. It is potentially possible that a more sophisticated probabilistic analysis gives a PTAS for the problem. We believe that randomized sketching might turn out to be useful to address also other problems in which “large” and “small” objects interact, for example in packing, scheduling, or resource allocation settings, in particular when dynamic programming works if there are only large objects.

In the Unsplittable Flow on a Path problem (UFP) we are given a path with edge capacities, and a set of tasks where each task is characterized by a subpath, a demand, and a weight. The goal is to select a subset of tasks of maximum total weight such that the total demand of the selected tasks using each edge e is at most the capacity of e. The problem admits a QPTAS [Bansal, Chakrabarti, Epstein, Schieber, STOC'06; Batra, Garg, Kumar, Mömke, Wiese, SODA'15]. After a long sequence of improvements [Bansal, Friggstad, Khandekar, Salavatipour, SODA'09; Bonsma, Schulz, Wiese, FOCS'11; Anagnostopoulos, Grandoni, Leonardi, Wiese, SODA'14; Grandoni, Mömke, Wiese, Zhou, STOC'18], the best known polynomial time approximation algorithm for UFP has an approximation ratio of 1+1/(e+1) + epsilon < 1.269 [Grandoni, Mömke, Wiese, SODA'22]. It has been an open question whether this problem admits a PTAS. In this paper, we solve this open question and present a polynomial time (1 + epsilon)-approximation algorithm for UFP.

Abstract We study the relationship between the competitive ratio and the tail distribution of randomized online problems. To this end, we identify a broad class of online problems for which the existence of a randomized online algorithm with constant expected competitive ratio r implies the existence of a randomized online algorithm that has a competitive ratio of $$(1+\varepsilon )r$$ ( 1 + ε ) r with high probability, measured with respect to the optimal profit or cost, respectively. The class of problems includes some of the well-studied online problems such as paging, k-server, and metrical task systems on finite metric spaces.

Unsplittable flow on a path (UFP) is an important and well-studied problem. We are given a path with capacities on its edges, and a set of tasks where for each task we are given a demand, a subpath, and a weight. The goal is to select the set of tasks of maximum total weight whose total demands do not exceed the capacity on any edge. UFP admits an (1+ε)-approximation with a running time of n^{O_{ε}(poly(log n))}, i.e., a QPTAS {[}Bansal et al., STOC 2006; Batra et al., SODA 2015{]} and it is considered an important open problem to construct a PTAS. To this end, in a series of papers polynomial time approximation algorithms have been developed, which culminated in a (5/3+ε)-approximation {[}Grandoni et al., STOC 2018{]} and very recently an approximation ratio of (1+1/(e+1)+ε) < 1.269 {[}Grandoni et al., 2020{]}. In this paper, we address the search for a PTAS from a different angle: we present a faster (1+ε)-approximation with a running time of only n^{O_{ε}(log log n)}. We first give such a result in the relaxed setting of resource augmentation and then transform it to an algorithm without resource augmentation. For this, we present a framework which transforms algorithms for (a slight generalization of) UFP under resource augmentation in a black-box manner into algorithms for UFP without resource augmentation, with only negligible loss.

Packing problems are an important class of optimization problems. The probably most well-known problem if this type is knapsack and many generalizations of it have been studied in the literature like Two-dimensional Geometric Knapsack (2DKP) and Unsplittable Flow on a Path (UFP). For the latter two problems, recently the first polynomial time approximation algorithms with better approximation ratios than 2 were presented [Gálvez et al., FOCS 2017][Grandoni et al., STOC 2018]. In this paper we break the barrier of 2 for the Storage Allocation Problem (SAP) which is a natural intermediate problem between 2DKP and UFP. We are given a path with capacitated edges and a set of tasks where each task has a start vertex, an end vertex, a size, and a profit. We seek to select the most profitable set of tasks that we can draw as non-overlapping rectangles underneath the capacity profile of the edges where the height of each rectangle equals the size of the corresponding task. This problem is motivated by settings of allocation resources like memory, bandwidths, etc. where each request needs a contiguous portion of the resource. The best known polynomial time approximation algorithm for SAP has an approximation ratio of 2+epsilon$ [Mömke and Wiese, ICALP 2015] and no better quasi-polynomial time algorithm is known. We present a polynomial time (63/32) < 1.969-approximation algorithm for the case of uniform edge capacities and a quasi-polynomial time (1.997)-approximation algorithm for non-uniform quasi-polynomially bounded edge capacities. Finally, we show that under slight resource augmentation we can obtain approximation ratios of 3/2 + epsilon in polynomial time and 1 + epsilon in quasi-polynomial time, both for arbitrary edge capacities.

Abstract In reoptimization, one is given an optimal solution to a problem instance and a (locally) modified instance. The goal is to obtain a solution for the modified instance. We aim to use information obtained from the given solution in order to obtain a better solution for the new instance than we are able to compute from scratch. In this paper, we consider Steiner tree reoptimization and address the optimality requirement of the provided solution. Instead of assuming that we are provided an optimal solution, we relax the assumption to the more realistic scenario where we are given an approximate solution with an upper bound on its performance guarantee. We show that for Steiner tree reoptimization there is a clear separation between local modifications where optimality is crucial for obtaining improved approximations and those instances where approximate solutions are acceptable starting points. For some of the local modifications that have been considered in previous research, we show that for every fixed $$\varepsilon > 0$$ ε > 0 , approximating the reoptimization problem with respect to a given $$(1+\varepsilon )$$ ( 1 + ε ) -approximation is as hard as approximating the Steiner tree problem itself. In contrast, with a given optimal solution to the original problem it is known that one can obtain considerably improved results. Furthermore, we provide a new algorithmic technique that, with some further insights, allows us to obtain improved performance guarantees for Steiner tree reoptimization with respect to all remaining local modifications that have been considered in the literature: a required node of degree more than one becomes a Steiner node; a Steiner node becomes a required node; the cost of one edge is increased.

Size and weight limitations of low-earth orbit (LEO) small satellites make their operation rest on a fine balance between solar power infeed and power demands of communication technologies on board, buffered by on-board battery storage. As a result, the problem of planning battery-powered payload utilization together with intersatellite communication is extremely intricate. Nevertheless, there is a growing trend toward constellations and megaconstellations that are to be managed using sophisticated software support. Earlier work has leveraged cost-optimal reachability in priced timed automata for deriving near-optimal finite-horizon schedules to operate a single LEO satellite in orbit. This article harvests that work and improves it in several dimensions, all needed for true in-orbit applicability: 1) the battery representation is no longer bound to be linear, but can be kinetic, which means that the optimization problem includes nonlinearities; 2) the management is perpetuated by a receding horizon scheduling strategy; 3) the model is continuously improved with the latest telemetry received from orbit; 4) a tandem of satellites equipped with state-of-the-art intersatellite link transponders is considered; 5) the core optimization problem is now solved using dynamic programming with antichain-based pruning, which is proven to be optimal and despite all the additional features outperforms the earlier approach by orders of magnitude; 6) the entire approach is grounded in the concrete requirements of the GOM X-4 LEO mission; 7) care is taken to make the approach usable by the space engineers, and robust against failures of parts of the toolchain; and 8) an extensive test campaign validates accuracy, efficiency, scalability, and robustness with respect to the operational requirements and constraints of LEO constellations.

We consider the problem Minimum Error Correction (MEC). A MEC instance is an n x m matrix M with entries from {0,1,-}. Feasible solutions are composed of two binary m-bit strings, together with an assignment of each row of M to one of the two strings. The objective is to minimize the number of mismatches (errors) where the row has a value that differs from the assigned solution string. The symbol "-" is a wildcard that matches both 0 and 1. A MEC instance is gapless, if in each row of M all binary entries are consecutive. Gapless-MEC is a relevant problem in computational biology, and it is closely related to segmentation problems that were introduced by [Kleinberg-Papadimitriou-Raghavan STOC'98] in the context of data mining. Without restrictions, it is known to be UG-hard to compute an O(1)-approximate solution to MEC. For both MEC and Gapless-MEC, the best polynomial time approximation algorithm has a logarithmic performance guarantee. We partially settle the approximation status of Gapless-MEC by providing a quasi-polynomial time approximation scheme (QPTAS). Additionally, for the relevant case where the binary part of a row is not contained in the binary part of another row, we provide a polynomial time approximation scheme (PTAS).

In this paper we consider the airport and railway problem (AR), which combines capacitated facility location with network design, both in the general metric and the two-dimensional Euclidean space. An instance of the airport and railway problem consists of a set of points in the corresponding metric, together with a non-negative weight for each point, and a parameter k. The points represent cities, the weights denote costs of opening an airport in the corresponding city, and the parameter k is a maximum capacity of an airport. The goal is to construct a minimum cost network of airports and railways connecting all the cities, where railways correspond to edges connecting pairs of points, and the cost of a railway is equal to the distance between the corresponding points. The network is partitioned into components, where each component contains an open airport, and spans at most k cities. For the Euclidean case, any points in the plane can be used as Steiner vertices of the network. We obtain the first bicriteria approximation algorithm for AR for the general metric case, which yields a 4-approximate solution with a resource augmentation of the airport capacity k by a factor of 2. More generally, for any parameter 0 < p <= 1 where pk is an integer we develop a (4/3)(2 + 1/p)-approximation algorithm for metric AR with a resource augmentation by a factor of 1 + p. Furthermore, we obtain the first constant factor approximation algorithm that does not resort to resource augmentation for AR in the Euclidean plane. Additionally, for the Euclidean setting we provide a quasi-polynomial time approximation scheme for the same problem with a resource augmentation by a factor of 1 + mu on the airport capacity, for any fixed mu > 0.

In the unsplittable flow on a path problem (UFP) we are given a path with edge capacities and a collection of tasks. Each task is characterized by a subpath, a profit, and a demand. Our goal is to compute a maximum profit subset of tasks such that, for each edge e, the total demand of selected tasks that use e does not exceed the capacity of e. The current best polynomial-time approximation factor for this problem is 2+є for any constant є>0 [Anagostopoulos et al.-SODA 2014]. This is the best known factor even in the case of uniform edge capacities [Călinescu et al.-IPCO 2002, TALG 2011]. These results, likewise most prior work, are based on a partition of tasks into large and small depending on their ratio of demand to capacity over their respective edges: these algorithms invoke (1+є)-approximations for large and small tasks separately.

In the Many-visits Path TSP, we are given a set of $n$ cities along with their pairwise distances (or costs) $c(uv)$, and moreover each city $v$ comes with an associated positive integer request $r(v)$. The goal is to find a minimum-cost path, starting at city $s$ and ending at city $t$, that visits each city $v$ exactly $r(v)$ times. We present a $3/2$-approximation algorithm for the metric Many-visits Path TSP that runs in time polynomial in $n$ and polylogarithmic in the requests $r(v)$. Our algorithm can be seen as a generalization of the $3/2$-approximation algorithm for Path TSP by Zenklusen [Proceedings of SODA, 2019, pp. 1539--1549], which answered a long-standing open problem by providing an efficient algorithm which matches the approximation guarantee of Christofides' algorithm from 1976 for metric TSP. One of the key components of our approach is a polynomial-time algorithm to compute a connected, degree-bounded multigraph of minimum cost in an undirected graph with edge costs. We tackle this problem by generalizing a fundamental result of Király, Lau, and Singh [Combinatorica, 32 (2012), pp. 705--720] on the Minimum Bounded Degree Matroid Basis problem, and devise such an algorithm for generalized polymatroids, even allowing element multiplicities. Our result directly yields a $3/2$-approximation to the metric Many-visits TSP, as well as a $3/2$-approximation for the problem of scheduling classes of jobs with sequence-dependent setup times on a single machine so as to minimize the makespan.

In the Unsplittable Flow on a Path problem (UFP) we are given a path with non-negative edge capacities and a set of tasks, each one characterized by a subpath, a demand, and a profit. Our goal is to select a subset of tasks of maximum total profit so that the total demand of the selected tasks on each edge does not exceed the respective edge capacity. UFP naturally captures several applications in bandwidth allocation, job scheduling, and caching. Following a sequence of improvements, the current best (polynomial time) approximation factor for UFP is 2 + ∊ [Anagnostopoulos et al. SODA'14]. UFP also admits a QPTAS [Bansal et al. STOC'06, Batra et al. SODA'15], and finding a PTAS is considered a challenging open problem. In this paper we make progress in the direction of the mentioned open problem. Informally, we introduce a technique to obtain real PTASs from PTASs with resource augmentation where edge capacities can be violated by a 1 + ∊ factor. While unfortunately we do not have a resource-augmentation PTAS for the general case of UFP, for many relevant special cases we have such an algorithm or we provide one in this paper. For example, our approach leads to a PTAS for the rooted case of UFP, where all tasks share a common edge. This is one of the simplest natural restrictions of UFP where the best-known approximation was 2 + ∊ (like for the general case). At a high level, our technique is to sacrifice a few tasks in the optimal solution (with a small loss of profit) in order to create a sufficient amount of slack capacity on each edge. This slack turns out to be large enough to substitute the additional capacity we would gain from resource augmentation. Crucial for our approach is that we obtain slack from tasks with relatively small and relatively large demand simultaneously. In all prior polynomial time approximation algorithms the sacrificed tasks came from only one of these two groups.

We introduce a new framework of Airport and Railway Problems, which combines capacitated facility location with network design. In this framework we are given a graph with weights on the vertices and on the edges, together with a parameter k. The vertices of the graph represent cities, and weights denote respectively the costs of opening airports in the cities and building railways that connect pairs of cities. The parameter $k$ can be thought of as the capacity of an airport. The goal is to construct a minimum cost network of airports and railways connecting the cities, where each connected component in the network spans at most k vertices, contains an open airport, and the network satisfies some additional requirements specific to the problem in the framework. We consider two problems in this framework. In the AR_F problem there are no additional requirements for the network. This problem is related to capacitated facility location. In the AR_P problem, we require each component to be a path with airports at both endpoints. AR_P is a relaxation of the capacitated vehicle routing problem (CVRP). We consider the problems in the two-dimensional Euclidean setting. We show that both AR_F and AR_P are NP-hard, even for uniform vertex weights (i.e., when the cost of building an airport is the same for all cities). On the positive side, we provide polynomial time approximation schemes for AR_F and AR_P when vertex weights are uniform. We also investigate AR_F and AR_P for k = infinity. In this setting we present an exact polynomial time algorithm for AR_F with general vertex costs, which also works for general edge costs. In contrast to AR_F, AR_P remains NP-hard when k = infinity, and we present a polynomial time approximation scheme for general vertex weights. We believe that our PTAS for AR_P with uniform vertex weights and arbitrary k brings us closer towards a PTAS for Euclidean CVRP, for which the main difficulty is to deal with paths of length at most k.

We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graph-TSP), we show that the approach gives a 1.461-approximation algorithm with respect to the Held-Karp lower bound. For graph-TSP restricted either to half-integral solutions to the Held-Karp relaxation or to a class of graphs that contains subcubic and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio 4/3. The framework also allows for generalizations in a natural way and leads to analogous results for the s , t -path traveling salesman problem on graphic metrics where the start and end vertices are prespecified.

In reoptimization problems, one is given an optimal solution to a problem instance and a local modification of the instance. The goal is to obtain a solution for the modified instance. The additional information about the instance provided by the given solution plays a central role: we aim to use that information in order to obtain better solutions than we are able to compute from scratch. In this paper, we consider Steiner tree reoptimization and address the optimality requirement of the provided solution. Instead of assuming that we are provided an optimal solution, we relax the assumption to the more realistic scenario where we are given an approximate solution with an upper bound on its performance guarantee. We show that for Steiner tree reoptimization there is a clear separation between local modifications where optimality is crucial for obtaining improved approximations and those instances where approximate solutions are acceptable starting points. For some of the local modifications that have been considered in previous research, we show that for every fixed epsilon > 0, approximating the reoptimization problem with respect to a given (1+epsilon)-approximation is as hard as approximating the Steiner tree problem itself (whereas with a given optimal solution to the original problem it is known that one can obtain considerably improved results). Furthermore, we provide a new algorithmic technique that, with some further insights, allows us to obtain improved performance guarantees for Steiner tree reoptimization with respect to all remaining local modifications that have been considered in the literature: a required node of degree more than one becomes a Steiner node; a Steiner node becomes a required node; the cost of one edge is increased.

We study the optimization version of constraint satisfaction problems (Max-CSPs) in the framework of parameterized complexity; the goal is to compute the maximum fraction of constraints that can be satisfied simultaneously. In standard CSPs, we want to decide whether this fraction equals one. The parameters we investigate are structural measures, such as the treewidth or the clique-width of the variable–constraint incidence graph of the CSP instance. We consider Max-CSPs with the constraint types AND, OR, PARITY, and MAJORITY, and with various parameters k. We attempt to fully classify them into the following three cases: 1. The exact optimum can be computed in FPT-time. 2. It is W[1]-hard to compute the exact optimum, but there is a randomized FPT approximation scheme (FPT-AS), which computes a (1-epsilon)-approximation in time f(k,epsilon) * poly(n). 3. There is no FPT-AS unless FPT=W[1]. For the corresponding standard CSPs, we establish FPT vs. W[1]-hardness results.

We study the unsplittable flow on a path problem which has received a lot of attention in the research community recently. Given is a path with capacities on its edges and a set of tasks where each task is characterized by a source and a sink vertex, a demand, and a profit. The goal is to find a subset of the tasks of maximum total profit such that all task demands from this subset can be routed simultaneously without violating the capacity constraints. The best known approximation results are a quasi-polynomial time-approximation scheme if the task demands are in a quasi-polynomial range [Bansal et al., STOC 2006] and a polynomial time (2 + ∊)-approximation algorithm [Anagnostopoulos et al., SODA 2014]. Finding a PTAS for it has remained an important open question. In this paper we make progress towards this goal. When the task densities—defined as the ratio of a task's profit and demand—lie in a constant range, we obtain a PTAS. We also improve the QPTAS of Bansal et al. by removing the assumption that the demands need to lie in a quasi-polynomial range. Our third result is a PTAS for the case where we are allowed to shorten the paths of the tasks by at most an ∊-fraction. This is particularly motivated by bandwidth allocation and scheduling applications of our problem if we are allowed to slightly increase the speed of the underlying transmission link/machine. Each of these results critically uses a sparsification lemma which we believe could be of independent interest. The lemma shows that in any (optimal) solution there exists an O(∊)-fraction (measured by weight) of its tasks whose removal creates, on each edge, a slack which is at least as large as the (1/∊)th largest demand using that edge. This slack can then be used to allow slight errors when estimating or rounding quantities arising in the computation.

We study the relationship between the competitive ratio and the tail distribution of randomized online problems. To this end, we define a broad class of online problems that includes some of the well-studied problems like paging, k-server and metrical task systems on finite metrics, and show that for these problems it is possible to obtain, given an algorithm with constant expected competitive ratio, another algorithm that achieves the same solution quality up to an arbitrarily small constant error with high probability; the "high probability" statement is in terms of the optimal cost. Furthermore, we show that our assumptions are tight in the sense that removing any of them allows for a counterexample to the theorem.

We present a framework for approximating the metric TSP based on a novel use of matchings. Traditionally, matchings have been used to add edges in order to make a given graph Eulerian, whereas our approach also allows for the removal of certain edges leading to a decreased cost. For the TSP on graphic metrics (graph-TSP), the approach yields a 1.461-approximation algorithm with respect to the Held-Karp lower bound. For graph-TSP restricted to a class of graphs that contains degree three bounded and claw-free graphs, we show that the integrality gap of the Held-Karp relaxation matches the conjectured ratio 4/3. The framework allows for generalizations in a natural way and also leads to a 1.586-approximation algorithm for the traveling salesman path problem on graphic metrics where the start and end vertices are prespecified.