Subexponential Algorithm Research Articles

The dihedral hidden subgroup problem

Abstract The hidden subgroup problem (HSP) is a cornerstone problem in quantum computing, which captures many problems of interest and provides a standard framework algorithm for their study based on Fourier sampling, one class of techniques known to provide quantum advantage, and which succeeds for some groups but not others. The quantum hardness of the HSP problem for the dihedral group is a critical question for post-quantum cryptosystems based on learning with errors and also appears in subexponential algorithms for constructing isogenies between elliptic curves over a finite field. In this article, we give an updated overview of the dihedral hidden subgroup problem as approached by the “standard” quantum algorithm for HSP on finite groups, detailing the obstructions for strong Fourier sampling to succeed and summarizing other known approaches and results. In our treatment, we “contrast and compare” as much as possible the cyclic and dihedral cases, with a view to determining bounds for the success probability of a quantum algorithm that uses m m coset samples to solve the HSP on these groups. In the last sections, we prove a number of no-go results for the dihedral coset problem (DCP), motivated by a connection between DCP and cloning of quantum states. The proofs of these no-go results are then adapted to give nontrivial upper bounds on the success probability of a quantum algorithm that uses m m coset samples to solve DCP.

FPT approximation and subexponential algorithms for covering few or many edges

We study the α-Fixed Cardinality Graph Partitioning (α-FCGP) problem, the generic local graph partitioning problem introduced by Bonnet et al. [Algorithmica 2015]. In this problem, we are given a graph G, two numbers k,p and 0≤α≤1, the question is whether there is a set S⊆V of size k with a specified coverage function covα(S) at least p (or at most p for the minimization version). The coverage function covα(⋅) counts edges with exactly one endpoint in S with weight α and edges with both endpoints in S with weight 1−α. α-FCGP generalizes a number of fundamental graph problems such as Densestk-Subgraph, Maxk-Vertex Cover, and Max(k,n−k)-Cut.A natural question in the study of α-FCGP is whether the algorithmic results known for its special cases, like Maxk-Vertex Cover, could be extended to more general settings. One of the simple but powerful methods for obtaining parameterized approximation [Manurangsi, SOSA 2019] and subexponential algorithms [Fomin et al. IPL 2011] for Maxk-Vertex Cover is based on the greedy vertex degree orderings. The main insight of our work is that the idea of greedy vertex degree ordering could be used to design fixed-parameter approximation schemes (FPT-AS) for α>0 and subexponential-time algorithms for the problem on apex-minor free graphs for maximization with α>1/3 and minimization with α<1/3.4

Inapproximability of Matrix \(\boldsymbol{p \rightarrow q}\) Norms

We study the problem of computing the norm of a matrix , defined as . This problem generalizes the spectral norm of a matrix and the Grothendieck problem ( , ) and has been widely studied in various regimes. When , the problem exhibits a dichotomy: constant factor approximation algorithms are known if , and the problem is hard to approximate within almost polynomial factors when . The regime when , known as hypercontractive norms, is particularly significant for various applications but much less well understood. The case with and was studied by Barak et al. [Proceedings of the 44th Annual ACM Symposium on Theory of Computing, 2012, pp. 307–326], who gave subexponential algorithms for a promise version of the problem (which captures small-set expansion) and also proved hardness of approximation results based on the exponential time hypothesis. However, no NP-hardness of approximation is known for these problems for any . We prove the first NP-hardness result (under randomized reductions) for approximating hypercontractive norms. We show that for any with , is hard to approximate within assuming . En route to the above result, we also prove almost tight results for the case when with .

Clique-Based Separators for Geometric Intersection Graphs

Let F be a set of n objects in the plane and let mathcal {G}^{times }(F) be its intersection graph. A balanced clique-based separator of mathcal {G}^{times }(F) is a set mathcal {mathcal {S}} consisting of cliques whose removal partitions mathcal {G}^{times }(F) into components of size at most delta n, for some fixed constant delta <1. The weight of a clique-based separator is defined as sum _{Cin mathcal {mathcal {S}}}log (|C|+1). Recently De Berg et al. (SIAM J. Comput. 49: 1291-1331. 2020) proved that if S consists of convex fat objects, then mathcal {G}^{times }(F) admits a balanced clique-based separator of weight O(sqrt{n}). We extend this result in several directions, obtaining the following results. (i) Map graphs admit a balanced clique-based separator of weight O(sqrt{n}), which is tight in the worst case. (ii) Intersection graphs of pseudo-disks admit a balanced clique-based separator of weight O(n^{2/3}log n). If the pseudo-disks are polygonal and of total complexity O(n) then the weight of the separator improves to O(sqrt{n}log n). (iii) Intersection graphs of geodesic disks inside a simple polygon admit a balanced clique-based separator of weight O(n^{2/3}log n). (iv) Visibility-restricted unit-disk graphs in a polygonal domain with r reflex vertices admit a balanced clique-based separator of weight O(sqrt{n}+rlog (n/r)), which is tight in the worst case. These results immediately imply sub-exponential algorithms for Maximum Independent Set (and, hence, Vertex Cover), for Feedback Vertex Set, and for q-Coloring for constant q in these graph classes.

Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

We study algorithmic properties of the graph class {textsc {Chordal}}{-ke}, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. It appears that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from {textsc {Chordal}}{-ke}. More precisely, we identify a large class of optimization problems on {textsc {Chordal}}{-ke} solvable in time 2^{{mathcal{O}}(sqrt{k}log k)}cdot n^{{mathcal{O}}(1)}. Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on {textsc {Chordal}}{-ke} when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. Besides subexponential time algorithms, the class of {textsc {Chordal}}{-ke} graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on {textsc {Chordal}}{-ke} graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on {textsc {Chordal}}{-ke} graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of {textsc {Chordal}}{-ke}, namely, {textsc {Interval}}{-ke} and {textsc {Split}}{-ke} graphs.

Fine-Grained Time Complexity of Constraint Satisfaction Problems

We study the constraint satisfaction problem (CSP) parameterized by a constraint language Γ (CSPΓ) and how the choice of Γ affects its worst-case time complexity. Under the exponential-time hypothesis (ETH), we rule out the existence of subexponential algorithms for finite-domain NP-complete CSPΓ problems. This extends to certain infinite-domain CSPs and structurally restricted problems. For CSPs with finite domain D and where all unary relations are available, we identify a relation S D such that the time complexity of the NP-complete problem CSP({ S D }) is a lower bound for all NP-complete CSPs of this kind. We also prove that the time complexity of CSP({ S D }) strictly decreases when |D| increases (unless the ETH is false) and provide stronger complexity results in the special case when |D|=3.

EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for M AXIMUM C LIQUE on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2 Õ( n 2/3 ) for M AXIMUM C LIQUE on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for M AX C LIQUE on disk and unit ball graphs. M AX C LIQUE on unit ball graphs is equivalent to finding, given a collection of points in R 3 , a maximum subset of points with diameter at most some fixed value. In stark contrast, M AXIMUM C LIQUE on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2 n 1−ɛ , unless the Exponential Time Hypothesis fails.

Relation collection using Pollard special-q sieving to solve integer factorization and discrete logarithm problem

The strength of many security protocols lies on the computational intractability of the integer factorization and discrete logarithm problems. Currently, the best-known techniques employed are number field sieve (NFS) family of algorithms. They come under the class of sub-exponential time algorithms. This class of algorithms comprises of multiple steps. The relation collection (sieving step) is one of the computationally costly and highly memory-dependent phase of these algorithms. This paper discusses various ways to improve the efficiency of the relation collection phase by using parallelization techniques. Experiments have been carried out by using function field sieve, which is one of the NFS family algorithms, to show the computation efficiency of parallelization techniques along with the suitable sieving techniques and the key parameters. The result of our basic implementation is compared with the parallelized version of it. The result analysis depicts that the relation collection phase can be improved by using parallelization techniques up to fourfold.

Subexponential algorithm for d-cluster edge deletion: Exception or rule?

We study the question of finding a set of at most k edges, whose removal makes the input n-vertex graph a disjoint union of s-clubs (graphs of diameter s). Komusiewicz and Uhlmann [DAM 2012] showed that Cluster Edge Deletion (i.e., for the case of 1-clubs (cliques)), cannot be solved in time 2o(k)nO(1) unless the Exponential Time Hypothesis (ETH) fails. But, Fomin et al. [JCSS 2014] showed that if the number of cliques in the output graph is restricted to d, then the problem (d-Cluster Edge Deletion) can be solved in time O(2O(dk)+m+n). We show that assuming ETH, there is no algorithm solving 2-Club Cluster Edge Deletion in time 2o(k)nO(1). Further, we show that the same lower bound holds in the case of s-Clubd-Cluster Edge Deletion for any s≥2 and d≥2.

Lower Bounds for the Parameterized Complexity of Minimum Fill-in and Other Completion Problems

In this work, we focus on several completion problems for subclasses of chordal graphs: M INIMUM F ILL -I N , I NTERVAL C OMPLETION , P ROPER I NTERVAL C OMPLETION , T RIVIALLY P ERFECT C OMPLETION , and T HRESHOLD C OMPLETION . In these problems, the task is to add at most k edges to a given graph to obtain a chordal, interval, proper interval, threshold, or trivially perfect graph, respectively. We prove the following lower bounds for all these problems, as well as for the related C HAIN C OMPLETION problem: • Assuming the Exponential Time Hypothesis, none of these problems can be solved in time 2 O ( n 1/2 /log c n ) or 2 O ( k 1/4 /log c k )· n O (1) , for some integer c . • Assuming the non-existence of a subexponential-time approximation scheme for M IN B ISECTION on d -regular graphs, for some constant d , none of these problems can be solved in time 2 o ( n ) or 2 o √k) }· n O (1) . For all the aforementioned completion problems, apart from P ROPER I NTERVAL C OMPLETION , FPT algorithms with running time of the form 2 O (√ k log k ) · n O (1) are known. Thus, the second result proves that a significant improvement of any of these algorithms would lead to a surprising breakthrough in the design of approximation algorithms for M IN B ISECTION . To prove our results, we use a reduction methodology based on combining the classic approach of starting with a sparse instance of 3-S AT , prepared using the Sparsification Lemma, with the existence of almost linear-size Probabilistically Checkable Proofs. Apart from our main results, we also obtain lower bounds excluding the existence of subexponential algorithms for the O PTIMUM L INEAR A RRANGEMENT problem, as well as improved, yet still not tight, lower bounds for F EEDBACK A RC S ET IN T OURNAMENTS .

A Framework for Exponential-Time-Hypothesis--Tight Algorithms and Lower Bounds in Geometric Intersection Graphs

We give an algorithmic and lower bound framework that facilitates the construction of subexponential algorithms and matching conditional complexity bounds. It can be applied to intersection graphs ...

Subexponential algorithms for variants of the homomorphism problem in string graphs

We consider subexponential algorithms finding weighted homomorphisms from intersection graphs of curves (string graphs) with n vertices to a fixed graph H. We provide a complete dichotomy: if H has no two vertices sharing two common neighbors, then the problem can be solved in time 2O(n2/3log⁡n), otherwise there is no subexponential algorithm, assuming the ETH. Then we consider locally constrained homomorphisms. We show that for each target graph H, the locally injective and locally bijective homomorphism problems can be solved in time 2O(nlog⁡n) in string graphs. For locally surjective homomorphisms we show a dichotomy for H being a path or a cycle. If H is P3 or C4, then the problem can be solved in time 2O(n2/3log3/2⁡n) in string graphs; otherwise, assuming the ETH, there is no subexponential algorithm. As corollaries, we obtain new results concerning the complexity of homomorphism problems in Pt-free graphs.

Optimality Program in Segment and String Graphs

Planar graphs are known to allow subexponential algorithms running in time 2^{O(sqrt{n})} or 2^{O(sqrt{n} log n)} for most of the paradigmatic problems, while the brute-force time 2^{varTheta (n)} is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in 2^{O(n^{2/3}log n)} by Fox and Pach (SODA’11), we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the Exponential Time Hypothesis (ETH). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time 2^{O(n^{2/3}log ^{O(1)}n)} on string graphs while an algorithm running in time 2^{o(n)} for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker lower bound, excluding a 2^{o(n^{2/3})} algorithm (under the ETH). The construction exploits the celebrated Erdős–Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set, but not to Min Dominating Set and Min Independent Dominating Set.

Bivariate complexity analysis of Almost Forest Deletion

In this paper we study a generalization of classic Feedback Vertex Set problem in the realm of multivariate complexity analysis. We say that a graph F is an l-forest if we can delete at most l edges from F to get a forest. That is, F is at most l edges away from being a forest. In this paper we introduce the Almost Forest Deletion problem, where given a graph G and integers k and l, the question is whether there exists a subset of at most k vertices such that its deletion leaves us an l-forest. We show that this problem admits an algorithm with running time 2O(k+l)nO(1) and a kernel of size O(kl(k+l)). We also show that the problem admits a 2O(tw)nO(1) algorithm on bounded treewidth graphs, using which we design a subexponential algorithm for the problem on planar graphs.

Exact exponential algorithms to find tropical connected sets of minimum size

Tropical Connected Set is strongly related to the Graph Motif problem which deals with vertex-colored graphs. Graph Motif has various applications in biology and metabolic networks, and has widely been studied in the last twenty years.The input of the Tropical Connected Set problem is a vertex-colored graph (G,c), where G=(V,E) is a graph and c is a vertex coloring assigning to each vertex of G a color. The task is to find a connected subset S⊆V of minimum size such that each color of G appears in S. This problem is known to be NP-complete, even when restricted to trees of height at most three. We study exact exponential algorithms to solve Tropical Connected Set. We present an O⁎(1.5359n) time algorithm for general graphs and an O⁎(1.2721n) time algorithm for trees. We also show that Tropical Connected Set on trees has no sub-exponential algorithm unless the Exponential Time Hypothesis fails.

A subexponential-time algorithm for the Maximum Independent Set Problem in [formula omitted]-free graphs

The Maximum Independent Set Problem is known to be NP-hard in general. In the last decades lots of effort were spent to find polynomial-time algorithms for Pt-free graphs. A recent result presents such an algorithm for P5-free graphs. On the other hand, the complexity status for larger t is still unknown. In this paper we present a subexponential algorithm for the Maximum (Weighted) Independent Set Problem in Pt-free graphs.

On the vertex cover $$P_3$$ P 3 problem parameterized by treewidth

Consider a graph G. A subset of vertices, F, is called a vertex cover \(P_t\) (\(VCP_t\)) set if every path of order t contains at least one vertex in F. Finding a minimum \(VCP_t\) set in a graph is is NP-hard for any integer \(t\ge 2\) and is called the \(MVCP_3\) problem. In this paper, we study the parameterized algorithms for the \(MVCP_3\) problem when the underlying graph G is parameterized by the treewidth. Given an n-vertex graph together with its tree decomposition of width at most p, we present an algorithm running in time \(4^{p}\cdot n^{O(1)}\) for the \(MVCP_3\) problem. Moreover, we show that for the \(MVCP_3\) problem on planar graphs, there is a subexponential parameterized algorithm running in time \(2^{O(\sqrt{k})}\cdot n^{O(1)}\) where k is the size of the optimal solution.

Reducing number field defining polynomials: an application to class group computations

In this paper we describe how to compute smallest monic polynomials that define a given number field $\mathbb{K}$. We make use of the one-to-one correspondence between monic defining polynomials of $\mathbb{K}$ and algebraic integers that generate $\mathbb{K}$. Thus, a smallest polynomial corresponds to a vector in the lattice of integers of $\mathbb{K}$ and this vector is short in some sense. The main idea is to consider weighted coordinates for the vectors of the lattice of integers of $\mathbb{K}$. This allows us to find the desired polynomial by enumerating short vectors in these weighted lattices. In the context of the subexponential algorithm of Biasse and Fieker for computing class groups, this algorithm can be used as a precomputation step that speeds up the rest of the computation. It also widens the applicability of their faster conditional method, which requires a defining polynomial of small height, to a much larger set of number field descriptions.

An Almost Optimal Algorithm for Computing Nonnegative Rank

Here, we give an algorithm for deciding if the nonnegative rank of a matrix $M$ of dimension $m \times n$ is at most $r$ which runs in time $(nm)^{O(r^2)}$. This is the first exact algorithm that runs in time singly exponential in $r$. This algorithm (and earlier algorithms) are built on methods for finding a solution to a system of polynomial inequalities (if one exists). Notably, the best algorithms for this task run in time exponential in the number of variables but polynomial in all of the other parameters (the number of inequalities and the maximum degree). Hence, these algorithms motivate natural algebraic questions whose solution have immediate algorithmic implications: How many variables do we need to represent the decision problem, and does $M$ have nonnegative rank at most $r$? A naive formulation uses $nr + mr$ variables and yields an algorithm that is exponential in $n$ and $m$ even for constant $r$. Arora et al. [Proceedings of STOC, 2012, pp. 145--162] recently reduced the number of variables to $2r^2 2^r$, and here we exponentially reduce the number of variables to $2r^2$ and this yields our main algorithm. In fact, the algorithm that we obtain is nearly optimal (under the exponential time hypothesis) since an algorithm that runs in time $(nm)^{o(r)}$ would yield a subexponential algorithm for $3$-SAT [Proceedings of STOC, 2012, pp. 145--162]. Our main result is based on establishing a normal form for nonnegative matrix factorization---which in turn allows us to exploit algebraic dependence among a large collection of linear transformations with variable entries. Additionally, we also demonstrate that nonnegative rank cannot be certified by even a very large submatrix of $M$, and this property also follows from the intuition gained from viewing nonnegative rank through the lens of systems of polynomial inequalities.

Subexponential Algorithms for Unique Games and Related Problems

Subexponential time approximation algorithms are presented for the U nique G ames and S mall -S et E xpansion problems. Specifically, for some absolute constant c , the following two algorithms are presented. (1) An exp( kn ϵ )-time algorithm that, given as input a k -alphabet unique game on n variables that has an assignment satisfying 1-ϵ c fraction of its constraints, outputs an assignment satisfying 1-ϵ fraction of the constraints. (2) An exp( n ϵ /δ)-time algorithm that, given as input an n -vertex regular graph that has a set S of δ n vertices with edge expansion at most ϵ c , outputs a set S' of at most δ n vertices with edge expansion at most ϵ. subexponential algorithm is also presented with improved approximation to M ax C ut , S parsest C ut , and V ertex C over on some interesting subclasses of instances. These instances are graphs with low threshold rank , an interesting new graph parameter highlighted by this work. Khot's Unique Games Conjecture (UGC) states that it is NP -hard to achieve approximation guarantees such as ours for U nique G ames . While the results here stop short of refuting the UGC, they do suggest that U nique G ames are significantly easier than NP -hard problems such as M ax 3-S at , M ax 3- Lin , L abel C over , and more, which are believed not to have a subexponential algorithm achieving a nontrivial approximation ratio. Of special interest in these algorithms is a new notion of graph decomposition that may have other applications. Namely, it is shown for every ϵ &gt;0 and every regular n -vertex graph G , by changing at most δ fraction of G 's edges, one can break G into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most n ϵ eigenvalues larger than 1-η, where η depends polynomially on ϵ. The subexponential algorithm combines this decomposition with previous algorithms for U nique G ames on graphs with few large eigenvalues [Kolla and Tulsiani 2007; Kolla 2010].