Fine-grained reductions from approximate counting to decision

The minimum degree algorithm is one of the most widely-used heuristics for reducing the cost of solving large sparse systems of linear equations. It has been studied for nearly half a century and has a rich history of bridging techniques from data structures, graph algorithms, and scientific computing. In this paper, we present a simple but novel combinatorial algorithm for computing an exact minimum degree elimination ordering in $O(nm)$ time, which improves on the best known time complexity of $O(n^3)$ and offers practical improvements for sparse systems with small values of $m$. Our approach leverages a careful amortized analysis, which also allows us to derive output-sensitive bounds for the running time of $O(\min\{m\sqrt{m^+}, \Delta m^+\} \log n)$, where $m^+$ is the number of unique fill edges and original edges that the algorithm encounters and $\Delta$ is the maximum degree of the input graph. Furthermore, we show there cannot exist an exact minimum degree algorithm that runs in $O(nm^{1-\varepsilon})$ time, for any $\varepsilon > 0$, assuming the strong exponential time hypothesis. This fine-grained reduction goes through the orthogonal vectors problem and uses a new low-degree graph construction called $U$-fillers, which act as pathological inputs and cause any minimum degree algorithm to exhibit nearly worst-case performance. With these two results, we nearly characterize the time complexity of computing an exact minimum degree ordering.

Consider positive integral solutions to the equation a0x0 + … + anxn = t. In the so called unbounded subset sum problem, the objective is to decide whether such a solution exists, whereas in the Frobenius problem, the objective is to compute the largest t such that there is no such solution. In this paper we study the algorithmic complexity of the unbounded subset sum, the Frobenius problem and a generalization of both problems. More precisely, we study pseudo-polynomial time algorithms with a running time that depends on the smallest number a0 or respectively the largest number an. For the parameter a0, we show that all considered problems are subquadratically equivalent to (min, +)-convolution, a fundamental algorithmic problem from the area of fine-grained complexity. By this equivalence, we obtain hardness results for the considered problems (based on the assumption that an algorithm with a subquadratic running time for (min, +)-convolution does not exist) as well as algorithms with improved running time. The proof for the equivalence makes use of structural properties of solutions, a technique that was developed in the area of integer programming. In case of the complexity of the problems parameterized by an, we present improved algorithms. For example we give a quasi linear time algorithm for the Frobenius problem as well as a hardness result based on the strong exponential time hypothesis.

Hitting forbidden subgraphs in graphs of bounded treewidth

This paper initiates the study of algorithms (minimizing cache misses) from the perspective of fine-grained complexity (conditional polynomial lower bounds). Specifically, we aim to answer why sparse graph problems are so hard, and why the Longest Common Subsequence problem gets a savings of a factor of the size of cache times the length of a cache line, but no more. We take the reductions and techniques from complexity and fine-grained complexity and apply them to the model to generate new (conditional) lower bounds as well as new faster algorithms. We also prove the existence of a hierarchy for the model, which motivates the fine-grained reductions. - Using fine-grained reductions, we give an algorithm for distinguishing 2 vs. 3 diameter and radius that runs in O(|E|^2/(MB)) cache misses, which for sparse graphs improves over the previous O(|V|^2/B) running time. - We give new reductions from radius and diameter to Wiener index and median. These reductions are new in both the RAM and models. - We show meaningful reductions between problems that have linear-time solutions in the RAM model. The reductions use low complexity (typically O(n/B)), and thus help to finely capture between I/O linear time O(n/B) and RAM linear O(n). - We generate new assumptions based on the difficulty of improving sparse graph problem running times in the model. We create conjectures that the current best known algorithms for Single Source Shortest Paths (SSSP), diameter, and radius are optimal. - From these I/O-model assumptions, we show that many of the known reductions in the word-RAM model can naturally extend to hold in the model as well (e.g., a lower bound on the complexity of Longest Common Subsequence that matches the best known running time). - We prove an analog of the Time Hierarchy Theorem in the model, further motivating the study of fine-grained algorithmic differences.

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.

We consider the fine-grained complexity of sparse graph problems that currently have O(mn) time algorithms, where m is the number of edges and n is the number of vertices in the input graph. This class includes several important path problems on both directed and undirected graphs, including APSP, MWC (Minimum Weight Cycle), Radius, Eccentricities, BC (Betweenness Centrality), etc. We introduce the notion of a sparse reduction which preserves the sparsity of graphs, and we present near linear-time sparse reductions between various pairs of graph problems in the O(mn) class. There are many sub-cubic reductions between graph problems in the O(mn) class, but surprisingly few of these preserve sparsity. In the directed case, our results give a partial order on a large collection of problems in the O(mn) class (along with some equivalences), and many of our reductions are very nontrivial. In the undirected case we give two nontrivial sparse reductions: from MWC to APSP, and from unweighted ANSC (all nodes shortest cycles) to unweighted APSP. We develop a new ‘bit-sampling’ method for these sparse reductions on undirected graphs, which also gives rise to improved or simpler algorithms for cycle finding problems in undirected graphs. We formulate the the notion of MWC hardness, which is based on the assumption that a minimum weight cycle in a directed graph cannot be computed in time polynomially smaller than mn. Our sparse reductions for directed path problems in the O(mn) class establish that several problems in this class, including 2-SiSP (second simple shortest path), s-t Replacement Paths, Radius, Eccentricities and BC are MWC hard. Our sparse reductions give MWC hardness a status for the O(mn) class similar to 3SUM hardness for the quadratic class, since they show sub-mn hardness for a large collection of fundamental and well-studied graph problems that have maintained an O(mn) time bound for over half a century. We also identify Eccentricities and BC as key problems in the O(mn) class which are simultaneously MWC-hard, SETH-hard and k-DSH-hard, where SETH is the Strong Exponential Time Hypothesis, and k-DSH is the hypothesis that a dominating set of size k cannot be computed in time polynomially smaller than nk. Our framework using sparse reductions is very relevant to real-world graphs, which tend to be sparse and for which the O(mn) time algorithms are the ones typically used in practice, and not the O(n3) time algorithms.

We resolve the fine-grained parameterized complexity of detecting and counting small patterns in planar graphs, assuming the Exponential Time Hypothesis. Given an n-vertex planar graph G and a k-vertex pattern graph P, we compute the number of (induced) copies of P in G in time 2O(√k) n O(1), if P is a matching, independent set, or connected bounded maximum degree graph, 2 O(k/logk) n O(1), for any pattern P. Our results significantly advance the state of the art of the planar graph isomorphism problem: No 2 o(k) n O(1) time algorithms where previously known for counting patterns even in very restricted cases such as independent sets in subgraphs of grids, Even for detection, no 2 O(k/logk) n O(1) time algorithms for unrestricted patterns P were previously known, Our run times cannot be improved assuming the Exponential Time Hypothesis (ETH). Our results are a corollary of the following general result: We compute the number of (induced) copies of P in G in 2O(√k)(σ(P)n) O(1) time, where σ(P) denotes the number of non-isomorphic separations of P of order O(√k). To obtain the first sub-exponential time algorithms, we introduce a new general technique that we call efficient inclusion-exclusion. This technique allows us to efficiently use hierarchies of separators for counting problems. To resolve the optimal complexity of planar subgraph isomorphism we provide another technique that we call balanced cycle sparsification. This technique allows us to obtain from one balanced cycle separator in a planar graphs a family of balanced cycle separators that have limited mutual overlap, implying one separator has few vertices of the pattern copy.

In the S ubset F eedback V ertex S et (S ubset FVS) problem, the input is a graph G on n vertices and m edges, a subset of vertices T , referred to as terminals, and an integer k . The objective is to determine whether there exists a set of at most k vertices intersecting every cycle that contains a terminal. The study of parameterized algorithms for this generalization of the F eedback V ertex S et problem has received significant attention over the past few years. In fact, the parameterized complexity of this problem was open until 2011, when two groups independently showed that the problem is fixed parameter tractable. Using tools from graph minors,, Kawarabayashi and Kobayashi obtained an algorithm for S ubset FVS running in time O ( f ( k )ċ n 2 m ) [SODA 2012, JCTB 2012]. Independently, Cygan et al. [ICALP 2011, SIDMA 2013] designed an algorithm for S ubset FVS running in time 2 O ( k log k ) ċ n O (1) . More recently, Wahlström obtained the first single exponential time algorithm for S ubset FVS, running in time 4 k ċ n O (1) [SODA 2014]. While the 2 O ( k ) dependence on the parameter k is optimal under the Exponential Time Hypothesis, the dependence of this algorithm as well as those preceding it, on the input size is at least quadratic. In this article, we design the first linear time parameterized algorithms for S ubset FVS. More precisely, we obtain the following new algorithms for S ubset FVS. — A randomized algorithm for S ubset FVS running in time O (25.6 k ċ ( n + m )). — A deterministic algorithm for S ubset FVS running in time 2 O ( k log k ) ċ ( n + m ). Since it is known that assuming the Exponential Time Hypothesis, S ubset FVS cannot have an algorithm running in time 2 o ( k ) n O (1) , our first algorithm obtains the best possible asymptotic dependence on both the parameter as well as the input size. Both of our algorithms are based on “cut centrality,” in the sense that solution vertices are likely to show up in minimum size cuts between vertices sampled from carefully chosen distributions.

The Orthogonal Vectors problem ($\textsf{OV}$) asks: given $n$ vectors in $\{0,1\}^{O(\log n)}$, are two of them orthogonal? $\textsf{OV}$ is easily solved in $O(n^2 \log n)$ time, and it is a central problem in fine-grained complexity: dozens of conditional lower bounds are based on the popular hypothesis that $\textsf{OV}$ cannot be solved in (say) $n^{1.99}$ time. However, unlike the APSP problem, few other problems are known to be non-trivially equivalent to $\textsf{OV}$. We show $\textsf{OV}$ is truly-subquadratic equivalent to several fundamental problems, all of which (a priori) look harder than $\textsf{OV}$. A partial list is given below: ($\textsf{Min-IP}/\textsf{Max-IP}$) Find a red-blue pair of vectors with minimum (respectively, maximum) inner product, among $n$ vectors in $\{0,1\}^{O(\log n)}$. ($\textsf{Exact-IP}$) Find a red-blue pair of vectors with inner product equal to a given target integer, among $n$ vectors in $\{0,1\}^{O(\log n)}$. ($\textsf{Apx-Min-IP}/\textsf{Apx-Max-IP}$) Find a red-blue pair of vectors that is a 100-approximation to the minimum (resp. maximum) inner product, among $n$ vectors in $\{0,1\}^{O(\log n)}$. (Approx. $\textsf{Bichrom.-$\ell_p$-Closest-Pair}$) Compute a $(1 + \Omega(1))$-approximation to the $\ell_p$-closest red-blue pair (for a constant $p \in [1,2]$), among $n$ points in $\mathbb{R}^d$, $d \le n^{o(1)}$. (Approx. $\textsf{$\ell_p$-Furthest-Pair}$) Compute a $(1 + \Omega(1))$-approximation to the $\ell_p$-furthest pair (for a constant $p \in [1,2]$), among $n$ points in $\mathbb{R}^d$, $d \le n^{o(1)}$. We also show that there is a $\text{poly}(n)$ space, $n^{1-\epsilon}$ query time data structure for Partial Match with vectors from $\{0,1\}^{O(\log n)}$ if and only if such a data structure exists for $1+\Omega(1)$ Approximate Nearest Neighbor Search in Euclidean space.

Article Free Access Share on Exact sampling and approximate counting techniques Author: Mark Huber School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NY School of Operations Research and Industrial Engineering, Cornell University, Ithaca, NYView Profile Authors Info & Claims STOC '98: Proceedings of the thirtieth annual ACM symposium on Theory of computingMay 1998 Pages 31–40https://doi.org/10.1145/276698.276709Published:23 May 1998Publication History 25citation666DownloadsMetricsTotal Citations25Total Downloads666Last 12 Months66Last 6 weeks15 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF

Makespan scheduling on identical machines is one of the most basic and fundamental packing problem studied in the discrete optimization literature. It asks for an assignment of n jobs to a set of m identical machines that minimizes the makespan. The problem is strongly NPhard, and thus we do not expect a (1 + epsilon)-approximation algorithm with a running time that depends polynomially on 1/epsilon. Furthermore, Chen et al. [Chen/JansenZhang, SODA'13] recently showed that a running time of 2^{1/epsilon}^{1-delta} + poly(n) for any delta > 0 would imply that the Exponential Time Hypothesis (ETH) fails. A long sequence of algorithms have been developed that try to obtain low dependencies on 1/epsilon, the better of which achieves a running time of 2^{~O(1/epsilon^{2})} + O(n*log(n)) [Jansen, SIAM J. Disc. Math. 2010]. In this paper we obtain an algorithm with a running time of 2^{~O(1/epsilon)} + O(n*log(n)), which is tight under ETH up to logarithmic factors on the exponent. Our main technical contribution is a new structural result on the configuration-IP. More precisely, we show the existence of a highly symmetric and sparse optimal solution, in which all but a constant number of machines are assigned a configuration with small support. This structure can then be exploited by integer programming techniques and enumeration. We believe that our structural result is of independent interest and should find applications to other settings. In particular, we show how the structure can be applied to the minimum makespan problem on related machines and to a larger class of objective functions on parallel machines. For all these cases we obtain an efficient PTAS with running time 2^{~O(1/epsilon)} + poly(n).

Let F be a family of graphs. In the F -C ompletion problem, we are given an n -vertex graph G and an integer k as input, and asked whether at most k edges can be added to G so that the resulting graph does not contain a graph from F as an induced subgraph. It was shown recently that two special cases of F -C ompletion , namely, (i) the problem of completing into a chordal graph known as M inimum F ill-in (SIAM J. Comput. 2013), which corresponds to the case of F ={ C 4 , C 5 , C 6 , …}, and (ii) the problem of completing into a split graph (Algorithmica 2015), that is, the case of F ={ C 4 , 2 K 2 , C 5 }, are solvable in parameterized subexponential time 2 O (√ k log k ) n O (1) . The exploration of this phenomenon is the main motivation for our research on F -C ompletion . In this article, we prove that completions into several well-studied classes of graphs without long induced cycles and paths also admit parameterized subexponential time algorithms by showing that: —The problem T rivially P erfect C ompletion , which is F - C ompletion for F ={ C 4 , P 4 }, a cycle and a path on four vertices, is solvable in parameterized subexponential time 2 O (√ k log k ) n O (1) . —The problems known in the literature as P seudosplit C ompletion , the case in which F{2 K 2 , C 4 }, and T hreshold C ompletion , in which F =2 K 2 , P 4 , C 4 }, are also solvable in time 2 O (√ k log k ) n O }(1) . We complement our algorithms for F -C ompletion with the following lower bounds: —For F ={2 K 2 }, F = { C 4 }, F ={ P o 4 }, and F ={2 K 2 , P 4 }, F -C ompletion cannot be solved in time 2 o(k) n O (1) unless the Exponential Time Hypothesis (ETH) fails. Our upper and lower bounds provide a complete picture of the subexponential parameterized complexity of F -C ompletion problems for any F ⊆ {2 K 2 , C 4 , P 4 }.

Tight complexity bounds for FPT subgraph problems parameterized by the clique-width

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

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).