Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal

Lokshtanov, Daniel; Marx, Dániel; Saurabh, Saket

doi:10.1137/1.9781611973082.61

Cited by 56 publications

(28 citation statements)

References 33 publications

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Section: Acm Subject Classification Theory Of Computation → Problemsmentioning

confidence: 99%

“…SETH is used in the study of exact and fixed parameter tractable algorithms, see e.g [23,46] or the book by Cygan et al [24]. In this area, it implies, among other things, tight lower bounds for problems on graphs that have small treewidth or pathwidth [41,26,25].Closely related to SETH, the orthogonal vectors problem (OV) is, given two sets A and B of N vectors from {0, 1} D , to decide whether there are vectors a ∈ A and b ∈ B such that a and b are orthogonal in Z D . If D ≤ O(N 0.3 ) holds, the problem can be solved in timeÕ(N 2 ) using an algorithm based on fast rectangular matrix multiplication (see e.g.…”

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confidence: 99%

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More consequences of falsifying SETH and the orthogonal vectors conjecture

Abboud¹,

Bringmann

Dell

et al. 2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

106

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The Strong Exponential Time Hypothesis and the OV-conjecture are two popular hardness assumptions used to prove a plethora of lower bounds, especially in the realm of polynomialtime algorithms. The OV-conjecture in moderate dimension states there is no ε > 0 for which an O(N 2−ε ) poly(D) time algorithm can decide whether there is a pair of orthogonal vectors in a given set of size N that contains D-dimensional binary vectors.We strengthen the evidence for these hardness assumptions. In particular, we show that if the OV-conjecture fails, then two problems for which we are far from obtaining even tiny improvements over exhaustive search would have surprisingly fast algorithms. If the OV conjecture is false, then there is a fixed ε > 0 such that: ACM Subject Classification Theory of computation → Problems, reductions and completenessMore Consequences of Falsifying SETH and the Orthogonal Vectors Conjecture decide the satisfiability of bounded-width CNF formulas. SETH is used in the study of exact and fixed parameter tractable algorithms, see e.g [23,46] or the book by Cygan et al. [24]. In this area, it implies, among other things, tight lower bounds for problems on graphs that have small treewidth or pathwidth [41,26,25].Closely related to SETH, the orthogonal vectors problem (OV) is, given two sets A and B of N vectors from {0, 1} D , to decide whether there are vectors a ∈ A and b ∈ B such that a and b are orthogonal in Z D . If D ≤ O(N 0.3 ) holds, the problem can be solved in timeÕ(N 2 ) using an algorithm based on fast rectangular matrix multiplication (see e.g. [31]). SETH implies [54] that this algorithm is essentially as fast as possible; in particular, SETH implies the following hardness conjecture, which was given its name by Gao et al. [32].Conjecture 1.1 (Moderate-dimension OV Conjecture). There are no reals ε, δ > 0 such that OV for D = N δ can be solved 1 in time O(N 2−ε ).The moderate-dimension OV conjecture is used to study the fine-grained complexity of problems in P, for which it has remarkably strong and diverse implications. If the conjecture is true, then dozens of important problems from all across computer science exhibit running time lower bounds that match existing upper bounds up to subpolynomial factors. These include pattern matching and other problems in bioinformatics [7, 10, 40, 1], graph algorithms [47,6,32], computational geometry [16], formal languages [11,18], time-series analysis [2,19], and even economics [42] (see [58] for a more comprehensive list).Gao et al.[32] also named the low-dimension OV conjecture, which asserts that OV does not have subquadratic algorithms whenever D = ω(log N ) holds. The low-dimension implies the moderate-dimension variant of the OV conjecture, and both are implied by SETH [54]. Recent results on the hardness of approximation problems, such as Maximum Inner Product [5], rely on the stronger conjecture (perhaps also [12,14]). However, for the vast majority of OV-based hardness results, reducing the dimension only affects lower-order terms in the lo...

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Section: Acm Subject Classification Theory Of Computation → Problemsmentioning

confidence: 99%

mentioning

confidence: 99%

More consequences of falsifying SETH and the orthogonal vectors conjecture

Abboud¹,

Bringmann

Dell

et al. 2018

Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing

106

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“…Protrusion replacement is often applied in the context of restricted graph classes, where the protrusions to be replaced are known to have bounded treewidth and may even belong to a family of embeddable graphs such as planar graphs. With these application areas in mind, we develop our lower bounds to apply even when we wish only to have a representative for each equivalence class that contains a planar graph whose treewidth is t + O(1), for boundary size t. To find a large set of nonequivalent graphs we adapt a construction of Lokshtanov et al [16], which they used to prove that Independent Set on graphs of treewidth w cannot be solved in time O * ((2 − ε) w ) for any ε > 0 unless the Strong Exponential Time Hypothesis fails. We show that the graphs they construct can be made planar while increasing the treewidth (and in fact the pathwidth) by only a small additive term.…”

Section: Our Resultsmentioning

confidence: 99%

“…The following gadget, of which several variations were used in earlier work (cf. [14,Theorem 5.3] and [10,16]), will be useful in our construction. To ensure our construction yields a planar graph, we use a crossover gadget for Independent Set due to Garey et al [12].…”

Section: Defining Planar Graphs With Given Boundary Characteristicsmentioning

confidence: 99%

Lower bounds for protrusion replacement by counting equivalence classes

Jansen

Wulms

2020

Discrete Applied Mathematics

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Garnero et al. [SIAM J. Discrete Math. 2015, 29(4): recently introduced a framework based on dynamic programming to make applications of the protrusion replacement technique constructive and to obtain explicit upper bounds on the involved constants. They show that for several graph problems, for every boundary size t one can find an explicit set R t of representatives. Any subgraph H with a boundary of size t can be replaced with a representative H ∈ R t such that the effect of this replacement on the optimum can be deduced from H and H alone. Their upper bounds on the size of the graphs in R t grow triple-exponentially with t. In this paper we complement their results by lower bounds on the sizes of representatives, in terms of the boundary size t. For example, we show that each set of planar representatives R t for Independent Set or Dominating Set contains a graph with Ω(2 t / √ 4t) vertices. This lower bound even holds for sets that only represent the planar subgraphs of bounded pathwidth. To obtain our results we provide a lower bound on the number of equivalence classes of the canonical equivalence relation for Independent Set on t-boundaried graphs. We also find an elegant characterization of the number of equivalence classes in general graphs, in terms of the number of monotone functions of a certain kind. Our results show that the number of equivalence classes is at most 2 2 t , improving on earlier bounds of the form (t + 1) 2 t .

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“…Prior to our work there are others which solely discuss lower bounds on the running time; e. g., for graph problems (and under the very strong ETH assumption [22]) [25]. Another kind of tradeoff (between size of the separator and the sharpness) for graph problems was discussed in [17].…”

Section: Related Workmentioning

confidence: 99%