We survey the current state of research on the knapsack problem in online and semi-online environments. In particular, we summarize what is known about models where different assumptions commonly made in online computation are relaxed: namely that online algorithms do not know the complete instances they are processing; have to make decisions that are irrevocable; and deal with an input chosen by a malicious adversary.

Graph coloring is a notoriously challenging problem, especially when considering the online setting where each arriving vertex must be colored immediately and irreversibly. Even on trees, which are trivially two-colorable, achieving anything better than a logarithmic competitive ratio becomes impossible if the order of arrival is adversarially determined. We investigate tree coloring in a slightly relaxed model where vertices arrive online but in random order, focusing specifically on algorithms with predictions of varying reliability. Furthermore, we extend our analysis to all two-colorable graphs and provide matching lower bounds for both cases.

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.

Partial vertex cover and partial dominating set are two well-investigated optimization problems. While they are W[1]-hard on general graphs, they have been shown to be fixed-parameter tractable on many sparse graph classes, including nowhere-dense classes. In this paper, we demonstrate that these problems are also fixed-parameter tractable with respect to the twin-width of a graph. Indeed, we establish a more general result: every graph property that can be expressed by a logical formula of the form ϕ≡∃ x₁⋯ ∃ x_k ∑_{α ∈ I} #y ψ_α(x₁,…,x_k,y) ≥ t, where ψ_α is a quantifier-free formula for each α ∈ I, t is an arbitrary number, and #y is a counting quantifier, can be evaluated in time f(d,k)n, where n is the number of vertices and d is the width of a contraction sequence that is part of the input. In addition to the aforementioned problems, this includes also connected partial dominating set and independent partial dominating set.

Abstract We analyze the competitive ratio and the advice complexity of the online unbounded knapsack problem. An instance is given as a sequence of n items with a size and a value each, and an algorithm has to decide whether or not and how often to pack each item into a knapsack of bounded capacity. The items are given online and the total size of the packed items must not exceed the knapsack’s capacity, while the objective is to maximize the total value of the packed items. While each item can only be packed once in the classical knapsack problem (also called the 0-1 knapsack problem), the unbounded version allows for items to be packed multiple times. We show that the simple unbounded knapsack problem, where the size of each item is equal to its value, allows for a competitive ratio of 2. We also analyze randomized algorithms and show that, in contrast to the 0-1 knapsack problem, one uniformly random bit cannot improve an algorithm’s performance. More randomness lowers the competitive ratio to less than 1 . 736 , but it can never be below 1 . 693 . In the advice complexity setting, we measure how many bits of information (so-called advice bits) the algorithm has to know to achieve some desired solution quality. For the simple unbounded knapsack problem, one advice bit lowers the competitive ratio to $$\varvec{3/2}$$ 3 / 2 . While this cannot be improved with fewer than $$\varvec{\log }_{\varvec{2}} \varvec{n} $$ log 2 n advice bits for instances of length n , a competitive ratio of $$\varvec{1}\varvec{+}\varvec{\varepsilon }$$ 1 + ε can be achieved with $$\varvec{O}\varvec{(}\varvec{\varepsilon }^{\varvec{-1}} \varvec{\cdot }\varvec{\log }\varvec{(}\varvec{n}\varvec{\varepsilon }^{\varvec{-1}}\varvec{))}$$ O ( ε - 1 · log ( n ε - 1 ) ) advice bits for any $$\varvec{\varepsilon }\varvec{>}\varvec{0}$$ ε > 0 . We further show that no amount of advice bounded by a function $$\varvec{f(n)}$$ f ( n ) allows an algorithm to be optimal. We also study the online general unbounded knapsack problem and show that it does not allow for any bounded competitive ratio for both deterministic and randomized algorithms, as well as for algorithms using fewer than $$\varvec{\log }_{\varvec{2}} \varvec{n}$$ log 2 n advice bits. We also provide a surprisingly simple algorithm that uses $$\varvec{O}\varvec{(}\varvec{\varepsilon }^{\varvec{-1}} \varvec{\cdot }\varvec{\log }\varvec{(}\varvec{n}\varvec{\varepsilon }^{\varvec{-1}}\varvec{))}$$

In the proportional knapsack problem, we are given a knapsack of some capacity and a set of variably sized items. The goal is to pack a selection of these items that fills the knapsack as much as possible. The online version of this problem reveals the items and their sizes not all at once but one by one. For each item, the algorithm has to decide immediately whether to pack it or not. We consider a natural variant of this online knapsack problem, which has been coined removable knapsack. It differs from the classical variant by allowing the removal of any packed item from the knapsack. Repacking is impossible, however: Once an item is removed, it is gone for good. We analyze the advice complexity of this problem. It measures how many advice bits an omniscient oracle needs to provide for an online algorithm to reach any given competitive ratio, which is - understood in its strict sense - just the algorithm’s approximation factor. The online knapsack problem is known for its peculiar advice behavior involving three jumps in competitivity. We show that the advice complexity of the version with removability is quite different but just as interesting: The competitivity starts from the golden ratio when no advice is given. It then drops down to 1+ε for a constant amount of advice already, which requires logarithmic advice in the classical version. Removability comes as no relief to the perfectionist, however: Optimality still requires linear advice as before. These results are particularly noteworthy from a structural viewpoint for the exceptionally slow transition from near-optimality to optimality. Our most important and demanding result shows that the general knapsack problem, which allows an item’s value to differ from its size, exhibits a similar behavior for removability, but with an even more pronounced jump from an unbounded competitive ratio to near-optimality within just constantly many advice bits. This is a unique behavior among the problems considered in the literature so far. An advice analysis is interesting in its own right, as it allows us to measure the information content of a problem and leads to structural insights. But it also provides insurmountable lower bounds, applicable to any kind of additional information about the instances, including predictions provided by machine-learning algorithms and artificial intelligence. Unexpectedly, advice algorithms are useful in various real-life situations, too. For example, they provide smart strategies for cooperation in winner-take-all competitions, where several participants pool together to implement different strategies and share the obtained prize. Further illustrating the versatility of our advice-complexity bounds, our results automatically improve some of the best known lower bounds on the competitive ratio for removable knapsack with randomization. The presented advice algorithms also automatically yield deterministic algorithms for established deterministic models such as knapsack with a resource buffer and various problems with more than one knapsack. In their seminal paper introducing removability to the knapsack problem, Iwama and Taketomi have indeed proposed a multiple knapsack problem for which we can establish a one-to-one correspondence with the advice model; this paper therefore even provides a comprehensive analysis for this up until now neglected problem.

It is known that first-order logic with some counting extensions can be efficiently evaluated on graph classes with bounded expansion, where depth-$r$ minors have constant density. More precisely, the formulas are $\exists x_1 ... x_k \#y φ(x_1,...,x_k, y)>N$, where $φ$ is an FO-formula. If $φ$ is quantifier-free, we can extend this result to nowhere dense graph classes with an almost linear FPT run time. Lifting this result further to slightly more general graph classes, namely almost nowhere dense classes, where the size of depth-$r$ clique minors is subpolynomial, is impossible unless FPT=W[1]. On the other hand, in almost nowhere dense classes we can approximate such counting formulas with a small additive error. Note those counting formulas are contained in FOC({<}) but not FOC1(P). In particular, it follows that partial covering problems, such as partial dominating set, have fixed parameter algorithms on nowhere dense graph classes with almost linear running time.

Abstract Parameterized complexity allows us to analyze the time complexity of problems with respect to a natural parameter depending on the problem. Reoptimization looks for solutions or approximations for problem instances when given solutions to neighboring instances. We combine both techniques, in order to better classify the complexity of problems in the parameterized setting. Specifically, we see that some problems in the class of compositional problems, which do not have polynomial kernels under standard complexity-theoretic assumptions, do have polynomial kernels under the reoptimization model for some local modifications. We also observe that, for some other local modifications, these same problems do not have polynomial kernels unless $$\mathbf{NP}\subseteq \mathbf{coNP/poly}$$ NP ⊆ coNP / poly . We find examples of compositional problems, whose reoptimization versions do not have polynomial kernels under any of the considered local modifications. Finally, in another negative result, we prove that the reoptimization version of Connected Vertex Cover does not have a polynomial kernel unless Set Cover has a polynomial compression. In a different direction, looking at problems with polynomial kernels, we find that the reoptimization version of Vertex Cover has a polynomial kernel of size $$\varvec{2k+1}$$ 2 k + 1 using crown decompositions only, which improves the size of the kernel achievable with this technique in the classic problem.

Kuske and Schweikardt introduced the very expressive first-order counting logic FOC(P) to model database queries with counting operations. They showed that there is an efficient model-checking algorithm on graphs with bounded degree, while Grohe and Schweikardt showed that probably no such algorithm exists for trees of bounded depth. We analyze the fragment FO({>0}) of this logic. While we remove for example subtraction and comparison between two nonatomic counting terms, this logic remains quite expressive: We allow nested counting and comparison between counting terms and arbitrarily large numbers. Our main result is an approximation scheme of the model-checking problem for FO({>0}) that runs in linear fpt time on structures with bounded expansion. This scheme either gives the correct answer or says “I do not know.” The latter answer may only be given if small perturbations in the number-symbols of the formula could make it both satisfied and unsatisfied. This is complemented by showing that exactly solving the model-checking problem for FO({>0}) is already hard on trees of bounded depth and just slightly increasing the expressiveness of FO({>0}) makes even approximation hard on trees.

In the Online Simple Knapsack Problem we are given a knapsack of unit size 1. Items of size smaller or equal to 1 are presented in an iterative fashion and an algorithm has to decide whether to permanently reject or include each item into the knapsack without any knowledge about the rest of the instance. The goal is then to pack the knapsack as full as possible. In this work, we introduce a third option additional to those of packing and rejecting an item, namely that of reserving an item for the cost of a fixed fraction α of its size. An algorithm may pay this fraction in order to postpone its decision on whether to include or reject the item until after the last item of the instance was presented. While the classical Online Simple Knapsack Problem does not admit any constantly bounded competitive ratio in the deterministic setting, we find that adding the possibility of reservation makes the problem constantly competitive, with varying competitive ratios depending on the value of α. We give upper and lower bounds for the whole range of reservation costs, with tight bounds for costs up to 1/6 - an area that is strictly 2-competitive - , for costs between √2-1 and 1 - an area that is strictly (2+α)-competitive up to ϕ -1, and strictly 1/(1-α)-competitive above ϕ-1, where ϕ is the golden ratio. With our analysis, we find a counterintuitive characteristic of the problem: Intuitively, one would expect that the possibility of rejecting items becomes more and more helpful for an online algorithm with growing reservation costs. However, for higher reservation costs above √2-1, an algorithm that is unable to reject any items tightly matches the lower bound and is thus the best possible. On the other hand, for any positive reservation cost smaller than 1/6, any algorithm that is unable to reject any items performs considerably worse than one that is able to reject.

Abstract In online edge- and node-deletion problems the input arrives node by node and an algorithm has to delete nodes or edges in order to keep the input graph in a given graph class $$\Pi $$ Π at all times. We consider only hereditary properties $$\Pi $$ Π , for which optimal online algorithms exist and which can be characterized by a set of forbidden subgraphs $${{\mathcal{F}}}$$ F and analyze the advice complexity of getting an optimal solution. We give almost tight bounds on the Delayed Connected $${{\mathcal{F}}}$$ F -Node-Deletion Problem, where all graphs of the family $${\mathcal{F}}$$ F have to be connected and almost tight lower and upper bounds for the Delayed $$H$$ H -Node-Deletion Problem, where there is one forbidden induced subgraph H that may be connected or not. For the Delayed $$H$$ H -Node-Deletion Problem the advice complexity is basically an easy function of the size of the biggest component in H. Additionally, we give tight bounds on the Delayed Connected $${\mathcal{F}}$$ F -Edge-Deletion Problem, where we have an arbitrary number of forbidden connected graphs. For the latter result we present an algorithm that computes the advice complexity directly from $${\mathcal{F}}$$ F . We give a separate analysis for the Delayed Connected $$H$$ H -Edge-Deletion Problem, which is less general but admits a bound that is easier to compute.

Preferential attachment graphs are random graphs designed to mimic properties of real word networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. We prove various structural asymptotic properties of this graph model. In particular, we show that the size of the largest r-shallow clique minor in Gⁿ_m is at most log(n)^{O(r²)}m^{O(r)}. Furthermore, there exists a one-subdivided clique of size log(n)^{1/4}. Therefore, preferential attachment graphs are asymptotically almost surely somewhere dense and algorithmic techniques developed for structurally sparse graph classes are not directly applicable. However, they are just barely somewhere dense. The removal of just slightly more than a polylogarithmic number of vertices asymptotically almost surely yields a graph with locally bounded treewidth.

Complex networks are everywhere. They appear for example in the form of biological networks, social networks, or computer networks and have been studied extensively. Efficient algorithms to solve problems on complex networks play a central role in today's society. Algorithmic meta-theorems show that many problems can be solved efficiently. Since logic is a powerful tool to model problems, it has been used to obtain very general meta-theorems. In this work, we consider all problems definable in first-order logic and analyze which properties of complex networks allow them to be solved efficiently. The mathematical tool to describe complex networks are random graph models. We define a property of random graph models called $α$-power-law-boundedness. Roughly speaking, a random graph is $α$-power-law-bounded if it does not admit strong clustering and its degree sequence is bounded by a power-law distribution with exponent at least $α$ (i.e. the fraction of vertices with degree $k$ is roughly $O(k^{-α})$). We solve the first-order model-checking problem (parameterized by the length of the formula) in almost linear FPT time on random graph models satisfying this property with $α\ge 3$. This means in particular that one can solve every problem expressible in first-order logic in almost linear expected time on these random graph models. This includes for example preferential attachment graphs, Chung-Lu graphs, configuration graphs, and sparse Erdős-Rényi graphs. Our results match known hardness results and generalize previous tractability results on this topic.

Many graph properties are expressible in first order logic. Whether a graph contains a clique or a dominating set of size k are two examples. For the solution size as its parameter the first one is W[1]-complete and the second one W[2]-complete meaning that both of them are hard problems in the worst-case. If we look at both problem from the aspect of average-case complexity, the picture changes. Clique can be solved in expected FPT time on uniformly distributed graphs of size n, while this is not clear for Dominating Set. We show that it is indeed unlikely that Dominating Set can be solved efficiently on random graphs: If yes, then every first-order expressible graph property can be solved in expected FPT time, too. Furthermore, this remains true when we consider random graphs with an arbitrary constant edge probability. We identify a very simple problem on random matrices that is equally hard to solve on average: Given a square boolean matrix, are there k rows whose logical AND is the zero vector? The related Even Set problem on the other hand turns out to be efficiently solvable on random instances, while it is known to be hard in the worst-case.

The non-uniform version of Turing machines with an extra advice input tape that depends on the length of the input but not the input itself is a well-studied model in complexity theory. We investigate the same notion of non-uniformity in weaker models, namely one-way finite automata. In particular, we are interested in the power of two-sided bounded-error randomization, and how it compares to determinism and non-determinism. We show that for unlimited advice, randomization is strictly stronger than determinism, and strictly weaker than non-determinism. However, when the advice is restricted to polynomial length, the landscape changes: the expressive power of determinism and randomization does not change, but the power of non-determinism is reduced to the extent that it becomes incomparable with randomization.

Network motifs are small patterns that occur in a network significantly more often than expected. They have gathered a lot of interest, as they may describe functional dependencies of complex networks and yield insights into their basic structure [Milo et al., 2002]. Therefore, a large amount of work went into the development of methods for network motif detection in complex networks [Kashtan et al., 2004; Schreiber and Schwöbbermeyer, 2005; Chen et al., 2006; Wernicke, 2006; Grochow and Kellis, 2007; Alon et al., 2008; Omidi et al., 2009]. The underlying problem of motif detection is to count how often a copy of a pattern graph H occurs in a target graph G. This problem is #W[1]-hard when parameterized by the size of H [Flum and Grohe, 2004] and cannot be solved in time f(|H|)n^o(|H|) under #ETH [Chen et al., 2005]. Preferential attachment graphs [Barabási and Albert, 1999] are a very popular random graph model designed to mimic complex networks. They are constructed by a random process that iteratively adds vertices and attaches them preferentially to vertices that already have high degree. Preferential attachment has been empirically observed in real growing networks [Newman, 2001; Jeong et al., 2003]. We show that one can count subgraph copies of a graph H in the preferential attachment graph G^n_m (with n vertices and nm edges, where m is usually a small constant) in expected time f(|H|) m^O(|H|^6) log(n)^O(|H|^12) n. This means the motif counting problem can be solved in expected quasilinear FPT time on preferential attachment graphs with respect to the parameters |H| and m. In particular, for fixed H and m the expected run time is O(n^(1+epsilon)) for every epsilon>0. Our results are obtained using new concentration bounds for degrees in preferential attachment graphs. Assume the (total) degree of a set of vertices at a time t of the random process is d. We show that if d is sufficiently large then the degree of the same set at a later time n is likely to be in the interval (1 +/- epsilon)d sqrt(n/t) (for epsilon > 0) for all n >= t. More specifically, the probability that this interval is left is exponentially small in d.

We introduce and study the complexity of Path Packing. Given a graph $G$ and a list of paths, the task is to embed the paths edge-disjoint in $G$. This generalizes the well known Hamiltonian-Path problem. Since Hamiltonian Path is efficiently solvable for graphs of small treewidth, we study how this result translates to the much more general Path Packing. On the positive side, we give an FPT-algorithm on trees for the number of paths as parameter. Further, we give an XP-algorithm with the combined parameters maximal degree, number of connected components and number of nodes of degree at least three. Surprisingly the latter is an almost tight result by runtime and parameterization. We show an ETH lower bound almost matching our runtime. Moreover, if two of the three values are constant and one is unbounded the problem becomes NP-hard. Further, we study restrictions to the given list of paths. On the positive side, we present an FPT-algorithm parameterized by the sum of the lengths of the paths. Packing paths of length two is polynomial time solvable, while packing paths of length three is NP-hard. Finally, even the spacial case EPC where the paths have to cover every edge in $G$ exactly once is already NP-hard for two paths on 4-regular graphs.

It is known that FO model-checking is fixed-parameter tractable on Erdős - Rényi graphs G(n,p(n)) if the edge-probability p(n) is sufficiently small [Grohe, 2001] (p(n)=O(n^epsilon/n) for every epsilon>0). A natural question to ask is whether this result can be extended to bigger probabilities. We show that for Erdős - Rényi graphs with vertex colors the above stated upper bound by Grohe is the best possible. More specifically, we show that there is no FO model-checking algorithm with average FPT run time on vertex-colored Erdős - Rényi graphs G(n,n^delta/n) (0 < delta < 1) unless AW[*]subseteq FPT/poly. This might be the first result where parameterized average-case intractability of a natural problem with a natural probability distribution is linked to worst-case complexity assumptions. We further provide hardness results for FO model-checking on other random graph models, including G(n,1/2) and Chung-Lu graphs, where our intractability results tightly match known tractability results [E. D. Demaine et al., 2014]. We also provide lower bounds on the size of shallow clique minors in certain Erdős - Rényi and Chung - Lu graphs.

Given a graph G=(V, E), an s-plex S\subseteq V is a vertex subset such that for v\in S the degree of v in G[S] is at least |S|-s. An s-plex bipartition \mathcal{P}=(V_1, V_2) is a bipartition of G=(V, E), V=V_1\uplus V_2, satisfying that both V_1 and V_2 are s-plexes. Given an instance G=(V, E) and a parameter k, the s-Plex Bipartition problem asks whether there exists an s-plex bipartition of G such that min{|V_1|, |V_2|\}\leq k. The s-Plex Bipartition problem is NP-complete. However, it is still open whether this problem is fixed-parameter tractable. In this paper, we give a fixed-parameter algorithm for 2-Plex Bipartition running in time O*(2.4143^k). A graph G = (V, E) is called defective (p, d)-colorable if it admits a vertex coloring with p colors such that each color class in G induces a subgraph of maximum degree at most d. A graph G admits an s-plex bipartition if and only if the complement graph of G, \bar{G}, admits a defective (2, s-1)-coloring such that one of the two color classes is of size at most k. By applying our fixed-parameter algorithm as a subroutine, one can find a defective (2,1)-coloring with one of the two colors of minimum cardinality for a given graph in O*(1.5539^n) time where n is the number of vertices in the input graph.

We present a linear-time algorithm to compute a decomposition scheme for graphs G that have a set X ⊆ V ( G ), called a treewidth-modulator , such that the treewidth of G − X is bounded by a constant. Our decomposition, called a protrusion decomposition , is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k ) that has a finite integer index and such that Y es -instances have a treewidth-modulator of size O ( k ) admits a linear kernel on the class of H -topological-minor-free graphs, for any fixed graph H . This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus and H -minor-free graphs. Let F be a fixed finite family of graphs containing at least one planar graph. Given an n -vertex graph G and a non-negative integer k , P lanar - F -D eletion asks whether G has a set X ⊆ V ( G ) such that | X | ⩽ k and G − X is H -minor-free for every H ϵ F . As our second application, we present the first single-exponential algorithm to solve P lanar - F -D eletion . Namely, our algorithm runs in time 2 O ( k ) · n 2 , which is asymptotically optimal with respect to k . So far, single-exponential algorithms were only known for special cases of the family F .

A fundamental theorem of Courcelle states that every problem definable in Monadic Second-Order Logic (MSO) is solvable in linear time on graphs of bounded treewidth.In this paper, we report on our ongoing effort to develop a general purpose software tool designed to solve MSO-definable optimization and decision problems on graphs of small treewidth.We discuss the theoretical underpinnings of our tool and present experimental results, which indicate that for some natural optimization problems MSO based approaches might be a suitable alternative to ILP solvers.

One of the most important algorithmic meta-theorems is a famous result by Courcelle, which states that any graph problem definable in monadic second-order logic with edge-set quantifications (MSO2) is decidable in linear time on any class of graphs of bounded tree-width. In the parlance of parameterized complexity, this means that MSO2 model-checking is fixed-parameter tractable with respect to the tree-width as parameter. Recently, Kreutzer and Tazari proved a corresponding complexity lower-bound---that MSO2 model-checking is not even in XP wrt the formula size as parameter for graph classes that are subgraph-closed and whose tree-width is poly-logarithmically unbounded. Of course, this is not an unconditional result but holds modulo a certain complexity-theoretic assumption, namely, the Exponential Time Hypothesis (ETH). In this paper we present a closely related result. We show that even MSO1 model-checking with a fixed set of vertex labels, but without edge-set quantifications, is not in XP wrt the formula size as parameter for graph classes which are subgraph-closed and whose tree-width is poly-logarithmically unbounded unless the non-uniform ETH fails. In comparison to Kreutzer and Tazari, (1) we use a stronger prerequisite, namely non-uniform instead of uniform ETH, to avoid the effectiveness assumption and the construction of certain obstructions used in their proofs; and (2) we assume a different set of problems to be efficiently decidable, namely MSO1-definable properties on vertex labeled graphs instead of MSO2-definable properties on unlabeled graphs. Our result has an interesting consequence in the realm of digraph width measures: Strengthening a recent result, we show that no subdigraph-monotone measure can be algorithmically useful, unless it is within a poly-logarithmic factor of (undirected) tree-width.

We present a simple exact algorithm for the \is\ problem with a runtime bounded by $O(\rt^n \poly(n))$. This bound is obtained by, firstly, applying a new branching rule and, secondly, by a distinct, computer-aided case analysis. The new branching rule uses the concept of satellites and has previously only been used in an algorithm for sparse graphs. The computer-aided case analysis allows us to capture the behavior of our algorithm in more detail than in a traditional analysis. The main purpose of this paper is to demonstrate how a very simple algorithm can outperform more complicated ones if the right analysis of its running time is performed.

We present a bound of $m/5.769+O(\log n)$ on the pathwidth of graphs with m edges. Respective path decompositions can be computed in polynomial time. Using a well-known framework for algorithms that rely on tree decompositions, this directly leads to runtime bounds of $O^*(2^{m/5.769})$ for Max-2SAT and Max-Cut. Both algorithms require exponential space due to dynamic programming. If we agree to accept a slightly larger bound of $m/5.217+3$, we even obtain path decompositions with a rather simple structure: all bags share a large set of common nodes. Using branching based algorithms, this allows us to solve the same problems in polynomial space and time $O^*(2^{m/5.217})$.