Conditional Lower Bounds Research Articles

Graphs cannot be indexed in polynomial time for sub-quadratic time string matching, unless SETH fails

The string matching problem on a node-labeled graph G=(V,E) asks whether a given pattern string P equals the concatenation of node labels of some path in G. This is a basic primitive in various problems in bioinformatics, graph databases, or networks, and it was recently proven that solving it in time O(|E|1−ϵ|P|) or O(|E||P|1−ϵ) is hard under OVH, the Orthogonal Vectors Hypothesis, and thus under SETH, the Strong Exponential Time Hypothesis (Equi et al., 2019 [11]). We consider its indexed version, where the graph is indexed to support string queries. We show that, under OVH, no polynomial-time index of the graph performed in time O(|E|α) can support querying P in time O(|P|+|E|δ|P|β), with either δ<1 or β<1.We present our techniques as a general framework, introducing the notion of linear independent-components (lic) reduction, from which we derive our result. This allows us to also translate the quadratic conditional lower bound of Backurs and Indyk (2015) [48] for the problem of matching a query string inside a text, under edit distance, into an analogous tight quadratic lower bound for its indexed version. This improves the recent result of Cohen-Addad, Feuilloley and Starikovskaya (2019) [49], with a slightly different boundary condition. We also apply our technique to obtain the first quadratic indexing lower bounds for Fréchet distance and rooted unlabeled subtree-isomorphism queries.

Polyline simplification has cubic complexity

$\DeclareMathOperator{\poly}{poly}$In the classic polyline simplification problem, given a polygonal curve~$P$ consisting of $n$ vertices and an error threshold $\delta \geq 0$, we want to replace $P$ by a subsequence~$Q$ of minimal size such that the distance between the polygonal curves $P$ and $Q$ is at most $\delta$. The distance between curves is usually measured using the Hausdorff or continuous Frechet distance. These distance measures can be applied globally, i.e., to the whole curves $P$ and $Q$, or locally, i.e., to each simplified subcurve and the line segment that it was replaced with separately (and then taking the maximum). This gives rise to four problem variants: Global-Hausdorff simplification (known to be NP-hard), Local-Hausdorff simplification (can be solved in time $O(n^3)$), Global-Frechet simplification (can be solved in time $O(k n^5)$, where $k$ is the size of the optimum simplification), and Local-Fr\'{e}chet simplification (can be solved in time $O(n^3)$).Our contribution is as follows:Cubic time for all variants:\ For Global-Frechet simplification, we design an algorithm running in time $O(n^3)$. This shows that all three problems (Local-Hausdorff, Local-Frechet, and Global-Frechet) can be solved in cubic time. All these algorithms work over a general metric space such as $(\mathbb{R}^d,L_p)$, but the hidden constant depends on $p$ and (linearly) on~$d$.Cubic conditional lower bound: We provide evidence that in high dimensions, cubic time is essentially optimal for all three problems (Local-Hausdorff, Local-Frechet, and Global-Frechet). Specifically, improving the cubic time to $O(n^{3-\epsilon} \poly(d))$ for polyline simplification over $(\mathbb{R}^d,L_p)$ for $p = 1$ would violate plausible conjectures. We obtain similar results for all $p \in [1,\infty), p \ne 2$.In total, in high dimensions and over general $L_p$-norms we resolve the complexity of polyline simplification with respect to Local-Hausdorff, Local-Frechet, and Global-Frechet, by a providing new algorithm and conditional lower bounds.

Split Contraction

The edit operation that contracts edges, which is a fundamental operation in the theory of graph minors, has recently gained substantial scientific attention from the viewpoint of Parameterized Complexity. In this article, we examine an important family of graphs, namely, the family of split graphs, which in the context of edge contractions is proven to be significantly less obedient than one might expect. Formally, given a graph G and an integer k , S PLIT C ONTRACTION asks whether there exists X ⊆ E ( G ) such that G / X is a split graph and | X | ≤ k . Here, G / X is the graph obtained from G by contracting edges in X . Guo and Cai [Theoretical Computer Science, 2015] claimed that S PLIT C ONTRACTION is fixed-parameter tractable. However, our findings are different. We show that S PLIT C ONTRACTION , despite its deceptive simplicity, is W[1]-hard. Our main result establishes the following conditional lower bound: Under the Exponential Time Hypothesis, S PLIT C ONTRACTION cannot be solved in time 2 o (ℓ 2 )⋅ n O (1), where ℓ is the vertex cover number of the input graph. We also verify that this lower bound is essentially tight. To the best of our knowledge, this is the first tight lower bound of the form 2 o (ℓ 2 )⋅ n O (1) for problems parameterized by the vertex cover number of the input graph. In particular, our approach to obtain this lower bound borrows the notion of harmonious coloring from Graph Theory, and might be of independent interest.

Conditional Lower Bounds for All-Pairs Max-Flow

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 &gt; 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) &gt; 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 &gt; 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).

Tight Conditional Lower Bounds for Longest Common Increasing Subsequence

We consider the canonical generalization of the well-studied Longest Increasing Subsequence problem to multiple sequences, called k-LCIS: Given k integer sequences X_1,dots ,X_k of length at most n, the task is to determine the length of the longest common subsequence of X_1,dots ,X_k that is also strictly increasing. Especially for the case of k=2 (called LCIS for short), several algorithms have been proposed that require quadratic time in the worst case. Assuming the Strong Exponential Time Hypothesis (SETH), we prove a tight lower bound, specifically, that no algorithm solves LCIS in (strongly) subquadratic time. Interestingly, the proof makes no use of normalization tricks common to hardness proofs for similar problems such as Longest Common Subsequence. We further strengthen this lower bound (1) to rule out {mathcal {O}}left( (nL)^{1-varepsilon }right) time algorithms for LCIS, where L denotes the solution size, (2) to rule out {mathcal {O}}left( n^{k-varepsilon }right) time algorithms for k-LCIS, and (3) to follow already from weaker variants of SETH. We obtain the same conditional lower bounds for the related Longest Common Weakly Increasing Subsequence problem.

Algorithms and Conditional Lower Bounds for Planning Problems

We consider planning problems for graphs, Markov decision processes (MDPs), and games on graphs. While graphs represent the most basic planning model, MDPs represent interaction with nature and games on graphs represent interaction with an adversarial environment.We consider two planning problems where there are k different target sets, and the problems are as follows: (a) the coverage problem asks whether there is a plan for each individual target set, and (b) the sequential target reachability problem asks whether the targets can be reached in sequence. For the coverage problem, we present a linear-time algorithm for graphs, and quadratic conditional lower bound for MDPs and games on graphs.For the sequential target problem, we present a linear-time algorithm for graphs, a sub-quadratic algorithm for MDPs, and a quadratic conditional lower bound for games on graphs.Our results with conditional lower bounds establish (i) model-separation results showing that for the coverage problem MDPs and games on graphs are harder than graphs and for the sequential reachability problem games on graphs are harder than MDPs and graphs;and (ii) objective-separation results showing that for MDPs the coverage problem is harder than the sequential target problem.

Lower bounds on the mim-width of some graph classes

mim-width is a recent graph width measure that has seen applications in graph algorithms and problems related to propositional satisfiability. In this paper, we show linear lower bounds for the mim-width of strongly chordal split graphs, co-comparability graphs and circle graphs. This improves and refines lower bounds that were known before, some of them only conditionally. In the case of strongly chordal graphs not even a conditional lower bound was known before. All of the bounds given are optimal up to constants.

Improved Approximation for Fréchet Distance on c-Packed Curves Matching Conditional Lower Bounds

The Fréchet distance is a well studied and very popular measure of similarity of two curves. The best known algorithms have quadratic time complexity, which has recently been shown to be optimal assuming the Strong Exponential Time Hypothesis (SETH) [Bringmann, FOCS'14]. To overcome the worst-case quadratic time barrier, restricted classes of curves have been studied that attempt to capture realistic input curves. The most popular such class are [Formula: see text]-packed curves, for which the Fréchet distance has a [Formula: see text]-approximation in time [Formula: see text] [Driemel et al., DCG'12]. In dimension [Formula: see text] this cannot be improved to [Formula: see text] for any [Formula: see text] unless SETH fails [Bringmann, FOCS'14]. In this paper, exploiting properties that prevent stronger lower bounds, we present an improved algorithm with time complexity [Formula: see text]. This improves upon the algorithm by Driemel et al. for any [Formula: see text]. Moreover, our algorithm's dependence on [Formula: see text], [Formula: see text] and [Formula: see text] is optimal in high dimensions apart from lower order factors, unless SETH fails. Our main new ingredients are as follows: For filling the classical free-space diagram we project short subcurves onto a line, which yields one-dimensional separated curves with roughly the same pairwise distances between vertices. Then we tackle this special case in near-linear time by carefully extending a greedy algorithm for the Fréchet distance of one-dimensional separated curves.