Tight Conditional Lower Bounds for Longest Common Increasing Subsequence

Longest common subsequence (LCS) is a classic and central problem in combinatorial optimization. While LCS admits a quadratic time solution, recent evidence suggests that solving the problem may be impossible in truly subquadratic time. A special case of LCS wherein each character appears at most once in every string is equivalent to the longest increasing subsequence problem (LIS) which can be solved in quasilinear time. In this work, we present novel algorithms for approximating LCS in truly subquadratic time and LIS in truly sublinear time. Our approximation factors depend on the ratio of the optimal solution size over the input size. We denote this ratio by λ and obtain the following results for LCS and LIS without any prior knowledge of λ. • A truly subquadratic time algorithm for LCS with approximation factor O(λ^3). • A truly sublinear time algorithm for LIS with approximation factor O(λ^3). Triangle inequality was recently used by Boroujeni et al. [1] and Chakraborty et al.[2] to present new approximation algorithms for edit distance. Our techniques for LCS extend the notion of triangle inequality to non-metric settings.

A central goal of algorithmic research is to determine how fast computational problems can be solved in the worst case. Theorems from complexity theory state that there are problems that, on inputs of size n, can be solved in t(n) time but not in O(t(n)^{1-epsilon}) time for epsilon>0. The main challenge is to determine where in this hierarchy various natural and important problems lie. Throughout the years, many ingenious algorithmic techniques have been developed and applied to obtain blazingly fast algorithms for many problems. Nevertheless, for many other central problems, the best known running times are essentially those of their classical algorithms from the 1950s and 1960s. Unconditional lower bounds seem very difficult to obtain, and so practically all known time lower bounds are conditional. For years, the main tool for proving hardness of computational problems have been NP-hardness reductions, basing hardness on P neq NP. However, when we care about the exact running time (as opposed to merely polynomial vs non-polynomial), NP-hardness is not applicable, especially if the problem is already solvable in polynomial time. In recent years, a new theory has been developed, based on "fine-grained reductions" that focus on exact running times. Mimicking NP-hardness, the approach is to (1) select a key problem X that is conjectured to require essentially t(n) time for some t, and (2) reduce X in a fine-grained way to many important problems. This approach has led to the discovery of many meaningful relationships between problems, and even sometimes to equivalence classes. The main key problems used to base hardness on have been: the 3SUM problem, the CNF-SAT problem (based on the Strong Exponential Time Hypothesis (SETH)) and the All Pairs Shortest Paths Problem. Research on SETH-based lower bounds has flourished in particular in recent years showing that the classical algorithms are optimal for problems such as Approximate Diameter, Edit Distance, Frechet Distance, Longest Common Subsequence etc. In this talk I will give an overview of the current progress in this area of study, and will highlight some exciting new developments and their relationship to database theory.

Why is it hard to beat O(n2) for Longest Common Weakly Increasing Subsequence?

We provide evidence that computing the maximum flow value between every pair of nodes in a directed graph on n nodes, m edges, and capacities in the range [1‥ n ], which we call the All-Pairs Max-Flow problem, cannot be solved in time that is significantly faster (i.e., by a polynomial factor) than O ( n 3 ) even for sparse graphs, namely m = O ( n ); thus for general m , it cannot be solved significantly faster than O ( n 2 m ). Since a single maximum st -flow can be solved in time Õ( m √ n ) [Lee and Sidford, FOCS 2014], we conclude that the all-pairs version might require time equivalent to Ω ˜ ( n 3/2 ) computations of maximum st -flow, which strongly separates the directed case from the undirected one. Moreover, if maximum st -flow can be solved in time Õ( m ), then the runtime of Ω ˜ ( n 2 ) computations is needed. This is in contrast to a conjecture of Lacki, Nussbaum, Sankowski, and Wulff-Nilsen [FOCS 2012] that All-Pairs Max-Flow in general graphs can be solved faster than the time of O ( n 2 ) computations of maximum st -flow. Specifically, we show that in sparse graphs G = ( V , E , w ), if one can compute the maximum st -flow from every s in an input set of sources S ⊆ V to every t in an input set of sinks T ⊆ V in time O ((| S || T | m ) 1−ε ), for some | S |, | T | and a constant ε > 0, then MAX-CNF-SAT (maximum satisfiability of conjunctive normal form formulas) with n ′ variables and m ′ clauses can be solved in time m ′ O (1) 2 (1−δ) n ′ for a constant δ(ε) > 0, a problem for which not even 2 n ′ / poly ( n ′) algorithms are known. Such running time for MAX-CNF-SAT would in particular refute the Strong Exponential Time Hypothesis (SETH). Hence, we improve the lower bound of Abboud, Vassilevska-Williams, and Yu [STOC 2015], who showed that for every fixed ε > 0 and | S | = | T | = O (√ n ), if the above problem can be solved in time O ( n 3/2−ε ), then some incomparable (and intuitively weaker) conjecture is false. Furthermore, a larger lower bound than ours implies strictly super-linear time for maximum st -flow problem, which would be an amazing breakthrough. In addition, we show that All-Pairs Max-Flow in uncapacitated networks with every edge-density m = m ( n ) cannot be computed in time significantly faster than O ( mn ), even for acyclic networks. The gap to the fastest known algorithm by Cheung, Lau, and Leung [FOCS 2011] is a factor of O ( m ω−1 / n ), and for acyclic networks it is O ( n ω−1 ), where ω is the matrix multiplication exponent. Finally, we extend our lower bounds to the version that asks only for the maximum-flow values below a given threshold (over all source-sink pairs).

Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no (k-epsilon)^pw(G)poly(n) time algorithm for deciding if an n-vertex graph G with pathwidth pw admits a proper vertex coloring with k colors unless the Strong Exponential Time Hypothesis (SETH) is false, for any constant epsilon>0. We show here that nevertheless, when k>lfloor Delta/2 rfloor + 1, where Delta is the maximum degree in the graph G, there is a better algorithm, at least when there are few colorings. We present a Monte Carlo algorithm that given a graph G along with a path decomposition of G with pathwidth pw(G) runs in (lfloor Delta/2 rfloor + 1)^pw(G)poly(n)s time, that distinguishes between k-colorable graphs having at most s proper k-colorings and non-k-colorable graphs. We also show how to obtain a k-coloring in the same asymptotic running time. Our algorithm avoids violating SETH for one since high degree vertices still cost too much and the mentioned hardness construction uses a lot of them. We exploit a new variation of the famous Alon--Tarsi theorem that has an algorithmic advantage over the original form. The original theorem shows a graph has an orientation with outdegree less than k at every vertex, with a different number of odd and even Eulerian subgraphs only if the graph is k-colorable, but there is no known way of efficiently finding such an orientation. Our new form shows that if we instead count another difference of even and odd subgraphs meeting modular degree constraints at every vertex picked uniformly at random, we have a fair chance of getting a non-zero value if the graph has few k-colorings. Yet every non-k-colorable graph gives a zero difference, so a random set of constraints stands a good chance of being useful for separating the two cases.

In this paper, we study the Maximum Happy Vertices and the Maximum Happy Edges problems (MHV and MHE for short). Very recently, the problems attracted a lot of attention and were studied in Agrawal ’17, Aravind et al. ’16, Choudhari and Reddy ’18, Misra and Reddy ’17. Main focus of our work is lower bounds on the computational complexity of these problems. Established lower bounds can be divided into the following groups: NP-hardness of the above guarantee parameterization, kernelization lower bounds (answering questions of Misra and Reddy ’17), exponential lower bounds under the Set Cover Conjecture and the Exponential Time Hypothesis, and inapproximability results. Moreover, we present an \({\mathcal {O}}^*(\ell ^k)\) randomized algorithm for MHV and an \({\mathcal {O}}^*(2^k)\) algorithm for MHE, where \(\ell \) is the number of colors used and k is the number of required happy vertices or edges. These algorithms cannot be improved to subexponential taking proved lower bounds into account.

The similarity between a pair of time series, i.e., sequences of indexed values in time order, is often estimated by the dynamic time warping (DTW) distance, instead of any in the well-studied family of measures including the longest common subsequence (LCS) length and the edit distance. Although it may seem as if the DTW and the LCS(-like) measures are essentially different, we reveal that the DTW distance can be represented by the longest increasing subsequence (LIS) length of a sequence of integers, which is the LCS length between the integer sequence and itself sorted. For a given pair of time series of length n such that the dissimilarity between any elements is an integer between zero and c, we propose an integer sequence that represents any substring-substring DTW distance as its band-substring LIS length. The length of the produced integer sequence is O(c n^2), which can be translated to O(n^2) for constant dissimilarity functions. To demonstrate that techniques developed under the LCS(-like) measures are directly applicable to analysis of time series via our reduction of DTW to LIS, we present time-efficient algorithms for DTW-related problems utilizing the semi-local sequence comparison technique developed for LCS-related problems.

The similarity between a pair of time series, i.e., sequences of indexed values in time order, is often estimated by the dynamic time warping (DTW) distance, instead of any in the well-studied family of measures including the longest common subsequence (LCS) length and the edit distance. Although it may seem as if the DTW and the LCS(-like) measures are essentially different, we reveal that the DTW distance can be represented by the longest increasing subsequence (LIS) length of a sequence of integers, which is the LCS length between the integer sequence and itself sorted. For a given pair of time series of n integers between zero and c, we propose an integer sequence that represents any substring-substring DTW distance as its band-substring LIS length. The length of the produced integer sequence is O(c⁴ n²) or O(c² n²) depending on the variant of the DTW distance used, both of which can be translated to O(n²) for constant cost functions. To demonstrate that techniques developed under the LCS(-like) measures are directly applicable to analysis of time series via our reduction of DTW to LIS, we present time-efficient algorithms for DTW-related problems utilizing the semi-local sequence comparison technique developed for LCS-related problems.

SUBSET SUM and k-SAT are two of the most extensively studied problems in computer science, and conjectures about their hardness are among the cornerstones of fine-grained complexity. An important open problem in this area is to base the hardness of one of these problems on the other.Our main result is a tight reduction from k-SAT to SUBSET SUM on dense instances, proving that Bellman's 1962 pseudo-polynomial O* (T)-time algorithm for SUBSET SUM on n numbers and target T cannot be improved to time T1-e . 2°(n) for any e > 0, unless the Strong Exponential Time Hypothesis (SETH) fails.As a corollary, we prove a Direct-OR theorem for SUBSET SUM under SETH, offering a new tool for proving conditional lower bounds: It is now possible to assume that deciding whether one out of N given instances of SUBSET SUM is a YES instance requires time (NT)1-°(1). As an application of this corollary, we prove a tight SETH-based lower bound for the classical BICRITERIAs, t-PATH problem, which is extensively studied in Operations Research. We separate its complexity from that of SUBSET SUM: On graphs with m edges and edge lengths bounded by L, we show that the O(Lm) pseudo-polynomial time algorithm by Joksch from 1966 cannot be improved to O(L + m), in contrast to a recent improvement for Subset Sum (Bringmann, SODA 2017).

Previous chapter Next chapter Full AccessProceedings Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA)Tight running times for minimum ℓq-norm load balancing: beyond exponential dependencies on 1/∊Lin Chen, Liangde Tao, and José VerschaeLin Chen, Liangde Tao, and José Verschaepp.275 - 315Chapter DOI:https://doi.org/10.1137/1.9781611977073.14PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstract We consider a classical scheduling problem on m identical machines. For an arbitrary constant q > 1, the aim is to assign jobs to machines such that is minimized, where Ci is the total processing time of jobs assigned to machine i. It is well known that this problem is strongly NP-hard. Under mild assumptions, the running time of an (1 + ∊)-approximation algorithm for a strongly NP-hard problem cannot be polynomial on 1/∊, unless P = NP. For most problems in the literature, this translates into algorithms with running time at least as large as 2Ω(1/∊) + nO(1). For the natural scheduling problem above, we establish the existence of an algorithm which violates this threshold. More precisely, we design a PTAS that runs in time. This result is in sharp contrast to the closely related minimum makespan variant, where an exponential lower bound is known under the exponential time hypothesis (ETH). We complement our result with an essentially matching lower bound on the running time, showing that our algorithm is best-possible under ETH. The lower bound proof exploits new number-theoretical constructions for variants of progression-free sets, which might be of independent interest. Furthermore, we provide a fine-grained characterization on the running time of a PTAS for this problem depending on the relation between ∊ and the number of machines m. More precisely, our lower bound only holds when . Better algorithms, that go beyond the lower bound, exist for other values of m. In particular, there even exists an algorithm with running time polynomial in 1/∊ if we restrict ourselves to instances with m = Ω(1/∊ log2 1/∊). Previous chapter Next chapter RelatedDetails Published:2022eISBN:978-1-61197-707-3 https://doi.org/10.1137/1.9781611977073Book Series Name:ProceedingsBook Code:PRDA22Book Pages:xvii + 3771

Knapsack and Partition are two important additive problems whose fine-grained complexities in the $(1-\varepsilon)$-approximation setting are not yet settled. In this work, we make progress on both problems by giving improved algorithms. - Knapsack can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {2.2} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {2.25} )$ by Jin (ICALP'19). There is a known conditional lower bound of $(n+\varepsilon)^{2-o(1)}$ based on $(\min,+)$-convolution hypothesis. - Partition can be $(1 - \varepsilon)$-approximated in $\tilde O(n + (1/\varepsilon) ^ {1.25} )$ time, improving the previous $\tilde O(n + (1/\varepsilon) ^ {1.5} )$ by Bringmann and Nakos (SODA'21). There is a known conditional lower bound of $(1/\varepsilon)^{1-o(1)}$ based on Strong Exponential Time Hypothesis. Both of our new algorithms apply the additive combinatorial results on dense subset sums by Galil and Margalit (SICOMP'91), Bringmann and Wellnitz (SODA'21). Such techniques have not been explored in the context of Knapsack prior to our work. In addition, we design several new methods to speed up the divide-and-conquer steps which naturally arise in solving additive problems.

Given n positive integers, the Modular Subset Sum problem asks if a subset adds up to a given target t modulo a given integer m. This is a natural generalization of the Subset Sum problem (where m = +∞) with ties to additive combinatorics and cryptography.Recently, in [Bri17, KX17], efficient algorithms have been developed for the non-modular case, running in near-linear pseudo-polynomial time. For the modular case, however, the best known algorithm by Koiliaris and Xu [KX17] runs in time Õ(m5/4).In this paper, we present an algorithm running in Õ(m) randomized time, which matches a recent conditional lower bound of [ABHS17] based on the Strong Exponential Time Hypothesis. Interestingly, in contrast to most previous results on Subset Sum, our algorithm does not use the Fast Fourier Transform. Instead, it is able to simulate the “textbook” Dynamic Programming algorithm much faster, using ideas from linear sketching. This is one of the first applications of sketching-based techniques to obtain fast algorithms for exact combinatorial problems in an offline setting.

In this paper, we give a conditional lower bound of nω(k) on running time for the classic k-median and k-means clustering objectives (where n is the size of the input), even in low-dimensional Euclidean space of dimension four, assuming the Exponential Time Hypothesis (ETH). We also consider k-median (and k-means) with penalties where each point need not be assigned to a center, in which case it must pay a penalty, and extend our lower bound to at least three-dimensional Euclidean space. This stands in stark contrast to many other geometric problems such as the traveling salesman problem, or computing an independent set of unit spheres. While these problems benefit from the so-called (limited) blessing of dimensionality, as they can be solved in time nO(k1--1/d) or 2n1--1/d in d dimensions, our work shows that widely-used clustering objectives have a lower bound of nω(k), even in dimension four. We complete the picture by considering the two-dimensional case: we show that there is no algorithm that solves the penalized version in time less than [Equation], and provide a matching upper bound of [Equation]. The main tool we use to establish these lower bounds is the placement of points on the moment curve, which takes its inspiration from constructions of point sets yielding Delaunay complexes of high complexity.

A directed graph D is semicomplete if for every pair x, y of vertices of D, there is at least one arc between x and y. Thus, a tournament is a semicomplete digraph. In the Directed Component Order Connectivity (DCOC) problem, given a digraph D=(V,A) and a pair of natural numbers k and ell , we are to decide whether there is a subset X of V of size k such that the largest strongly connected component in D-X has at most ell vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for ell =1. We study the parameterized complexity of DCOC for general and semicomplete digraphs with the following parameters: k, ell ,ell +k and n-ell . In particular, we prove that DCOC with parameter k on semicomplete digraphs can be solved in time O^*(2^{16k}) but not in time O^*(2^{o(k)}) unless the Exponential Time Hypothesis (ETH) fails. The upper bound O^*(2^{16k}) implies the upper bound O^*(2^{16(n-ell )}) for the parameter n-ell . We complement the latter by showing that there is no algorithm of time complexity O^*(2^{o({n-ell })}) unless ETH fails. Finally, we improve (in dependency on ell ) the upper bound of Göke, Marx and Mnich (2019) for the time complexity of DCOC with parameter ell +k on general digraphs from O^*(2^{O(kell log (kell ))}) to O^*(2^{O(klog (kell ))}). Note that Drange, Dregi and van ’t Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time O^*(2^{o(klog ell )}) unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity O^*(2^{o(klog k)}).

The Longest Common Increasing Subsequence problem (LCIS) is a natural variant of the celebrated Longest Common Subsequence (LCS) problem. For LCIS, as well as for LCS, there is an $O(n^2)$-time algorithm and a SETH-based conditional lower bound of $O(n^{2-\varepsilon})$. For LCS, there is also the Masek-Paterson $O(n^2 / \log{n})$-time algorithm, which does not seem to adapt to LCIS in any obvious way. Hence, a natural question arises: does any (slightly) sub-quadratic algorithm exist for the Longest Common Increasing Subsequence problem? We answer this question positively, presenting a $O(n^2 / \log^a{n})$-time algorithm for $a = \frac{1}{6}-o(1)$. The algorithm is not based on memorizing small chunks of data (often used for logarithmic speedups, including the Four Russians Trick in LCS), but rather utilizes a new technique, bounding the number of significant symbol matches between the two sequences.