$ \newcommand{\SVP}{\mathsf{SVP}} \newcommand{\NP}{\mathsf{NP}} \newcommand{\RTIME}{\mathsf{RTIME}} \newcommand{\RSUBEXP}{\mathsf{RSUBEXP}} \newcommand{\eps}ε \newcommand{\poly}{\mathop{\mathrm{poly}}} $We show that unless $\NP \subseteq \RTIME (2^{\poly(\log{n})})$, there is no polynomial-time algorithm approximating the Shortest Vector Problem ($\SVP$) on $n$-dimensional lattices in the $\ell_p$ norm ($1 \leq p< \infty$) to within a factor of $2^{(\log{n})^{1-\eps}}$ for any $\eps > 0$. This improves the previous best factor of $2^{(\log{n})^{1/2-\eps}}$ under the same complexity assumption due to Khot (J. ACM, 2005). Under the stronger assumption $\NP \nsubseteq \RSUBEXP$, we obtain a hardness factor of $n^{c/\log\log{n}}$ for some $c> 0$.
Our proof starts with Khot's $\SVP$ instances that are hard to approximate to within some constant. To boost the hardness factor we simply apply the standard tensor product of lattices. The main novelty is in the analysis, where we show that the lattices of Khot behave nicely under tensorization. At the heart of the analysis is a certain matrix inequality which was first used in the context of lattices by de Shalit and Parzanchevski (2006).
We give several improvements on the known hardness of the unique shortest vector problem. - We give a deterministic reduction from the shortest vector problem to the unique shortest vector problem. As a byproduct, we get deterministic NP-hardness for unique shortest vector problem in the $\ell_\infty$ norm. - We give a randomized reduction from SAT to uSVP_{1+1/poly(n)}. This shows that uSVP_{1+1/poly(n)} is NP-hard under randomized reductions. - We show that if GapSVP_γ\in coNP (or coAM) then uSVP_{\sqrtγ} \in coNP (coAM respectively). This simplifies previously known uSVP_{n^{1/4}} \in coAM …
We consider the Stackelberg shortest-path pricing problem, which is defined as follows. Given a graph G with fixed-cost and pricable edges and two distinct vertices s and t, we may assign prices to the pricable edges. Based on the predefined fixed costs and our prices, a customer purchases a cheapest s-t-path in G and we receive payment equal to the sum of prices of pricable edges belonging to the path. Our goal is to find prices maximizing the payment received from the customer. While Stackelberg shortest-path pricing was known to be APX-hard before, we provide the first explicit approximatio…
The shortest common superstring (SCS) problem has been studied at great length because of its connections to the de novo assembly problem in computational genomics. The base problem is APX-complete, but several generalizations of the problem have also been studied. In particular, previous results include that SCS with Negative strings (SCSN) is in Log-APX (though there is no known hardness result) and SCS with Wildcards (SCSW) is Poly-APX-hard. Here, we prove two new hardness results: (1) SCSN is Log-APX-hard (and therefore Log-APX-complete) by a reduction from Minimum Set Cover and (2) SCS wi…
$\newcommand{\NP}{\mathsf{NP}}\newcommand{\GapSVP}{\textrm{GapSVP}}$We give a simple proof that the (approximate, decisional) Shortest Vector Problem is $\NP$-hard under a randomized reduction. Specifically, we show that for any $p \geq 1$ and any constant $γ< 2^{1/p}$, the $γ$-approximate problem in the $\ell_p$ norm ($γ$-$\GapSVP_p$) is not in $\mathsf{RP}$ unless $\NP \subseteq \mathsf{RP}$. Our proof follows an approach pioneered by Ajtai (STOC 1998), and strengthened by Micciancio (FOCS 1998 and SICOMP 2000), for showing hardness of $γ$-$\GapSVP_p$ using locally dense lattices. We constru…
Given an $n$-vertex non-negatively real-weighted graph $G$, whose vertices are partitioned into a set of $k$ clusters, a \emph{clustered network design problem} on $G$ consists of solving a given network design optimization problem on $G$, subject to some additional constraint on its clusters. In particular, we focus on the classic problem of designing a \emph{single-source shortest-path tree}, and we analyze its computational hardness when in a feasible solution each cluster is required to form a subtree. We first study the \emph{unweighted} case, and prove that the problem is \np-hard. Howev…
Blömer and Seifert showed that $\mathsf{SIVP}_2$ is NP-hard to approximate by giving a reduction from $\mathsf{CVP}_2$ to $\mathsf{SIVP}_2$ for constant approximation factors as long as the $\mathsf{CVP}$ instance has a certain property. In order to formally define this requirement on the $\mathsf{CVP}$ instance, we introduce a new computational problem called the Gap Closest Vector Problem with Bounded Minima. We adapt the proof of Blömer and Seifert to show a reduction from the Gap Closest Vector Problem with Bounded Minima to $\mathsf{SIVP}$ for any $\ell_p$ norm for some constant approxima…
This work studies the hardness of finding independent sets in hypergraphs which are either 2-colorable or are almost 2-colorable, i.e. can be 2-colored after removing a small fraction of vertices and the incident hyperedges. To be precise, say that a hypergraph is (1-eps)-almost 2-colorable if removing an eps fraction of its vertices and all hyperedges incident on them makes the remaining hypergraph 2-colorable. In particular we prove the following results. For an arbitrarily small constant gamma > 0, there is a constant xi > 0, such that, given a 4-uniform hypergraph on n vertices which is (1…
We study the NP-hard Shortest Path Most Vital Edges problem arising in the context of analyzing network robustness. For an undirected graph with positive integer edge lengths and two designated vertices $s$ and $t$, the goal is to delete as few edges as possible in order to increase the length of the (new) shortest $st$-path as much as possible. This scenario has been studied from the viewpoint of parameterized complexity and approximation algorithms. We contribute to this line of research by providing refined computational tractability as well as hardness results. We achieve this by a systema…
This paper considers the computational hardness of computing expected outcomes and deciding (universal) (positive) almost-sure termination of probabilistic programs. It is shown that computing lower and upper bounds of expected outcomes is $Σ_1^0$- and $Σ_2^0$-complete, respectively. Deciding (universal) almost-sure termination as well as deciding whether the expected outcome of a program equals a given rational value is shown to be $Π^0_2$-complete. Finally, it is shown that deciding (universal) positive almost-sure termination is $Σ_2^0$-complete ($Π_3^0$-complete).
We study variants of the classic $s$-$t$ cut problem and prove the following improved hardness results assuming the Unique Games Conjecture (UGC). - For any constant $k \geq 2$ and $ε> 0$, we show that Directed Multicut with $k$ source-sink pairs is hard to approximate within a factor $k - ε$. This matches the trivial $k$-approximation algorithm. By a simple reduction, our result for $k = 2$ implies that Directed Multiway Cut with two terminals (also known as $s$-$t$ Bicut) is hard to approximate within a factor $2 - ε$, matching the trivial $2$-approximation algorithm. Previously, the best ha…
The Super-SAT or SSAT problem was introduced by Dinur et al.(2002,2003) to prove the NP-hardness of approximation of two popular lattice problems - Shortest Vector Problem(SVP) and Closest Vector Problem(CVP). They conjectured that SSAT is NP-hard to approximate to within a factor of $n^c$ ($c>0$ is constant), where $n$ is the size of the SSAT instance. In this paper we prove this conjecture assuming the Projection Games Conjecture(PGC), given by Moshkovitz (2012). This implies hardness of approximation of SVP and CVP within polynomial factors, assuming PGC. We also reduce SSAT to the Nearest …
We study the approximation hardness of the Shortest Superstring, the Maximal Compression and the Maximum Asymmetric Traveling Salesperson (MAX-ATSP) problem. We introduce a new reduction method that produces strongly restricted instances of the Shortest Superstring problem, in which the maximal orbit size is eight (with no character appearing more than eight times) and all given strings having length four. Based on this reduction method, we are able to improve the best up to now known approximation lower bound for the Shortest Superstring problem and the Maximal Compression problem by an order…
Fine-grained quantum supremacy is a study of proving (nearly) tight time lower bounds for classical simulations of quantum computing under "fine-grained complexity" assumptions. We show that under conjectures on Orthogonal Vectors (OV), 3-SUM, All-Pairs Shortest Paths (APSP) and their variants, strong and weak classical simulations of quantum computing are impossible in certain exponential time with respect to the number of qubits. Those conjectures are widely used in classical fine-grained complexity theory in which polynomial time hardness is conjectured. All previous results of fine-grained…
In this paper, we study the complexity of computing the determinant of a matrix over a non-commutative algebra. In particular, we ask the question, "over which algebras, is the determinant easier to compute than the permanent?" Towards resolving this question, we show the following hardness and easiness of noncommutative determinant computation. * [Hardness] Computing the determinant of an n \times n matrix whose entries are themselves 2 \times 2 matrices over a field is as hard as computing the permanent over the field. This extends the recent result of Arvind and Srinivasan, who proved a sim…
$ \newcommand{\problem}[1]{\ensuremath{\mathrm{#1}} } \newcommand{\SVP}{\problem{SVP}} \newcommand{\ensuremath}[1]{#1} $We prove the following quantitative hardness results for the Shortest Vector Problem in the $\ell_p$ norm ($\SVP_p$), where $n$ is the rank of the input lattice. $\bullet$ For "almost all" $p > p_0 \approx 2.1397$, there no $2^{n/C_p}$-time algorithm for $\SVP_p$ for some explicit constant $C_p > 0$ unless the (randomized) Strong Exponential Time Hypothesis (SETH) is false. $\bullet$ For any $p > 2$, there is no $2^{o(n)}$-time algorithm for $\SVP_p$ unless the (randomized) G…
We consider the fine-grained complexity of sparse graph problems that currently have $\tilde{O}(mn)$ 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), and Eccentricities, which is the problem of computing, for each vertex in the graph, the length of a longest shortest path starting at that vertex. We introduce the notion of a sparse reduction which preserves the sparsity of graphs, and we present near linear-ti…
In the Densest $k$-Subgraph problem, given an undirected graph $G$ and an integer $k$, the goal is to find a subgraph of $G$ on $k$ vertices that contains maximum number of edges. Even though the state-of-the-art algorithm for the problem achieves only $O(n^{1/4 + \varepsilon})$ approximation ratio (Bhaskara et al., 2010), previous attempts at proving hardness of approximation, including those under average case assumptions, fail to achieve a polynomial ratio; the best ratios ruled out under any worst case assumption and any average case assumption are only any constant (Raghavendra and Steure…
A map $f: K \to \mathbb{R}^d$ of a simplicial complex is an almost embedding if $f(σ) \cap f(τ) = \varnothing$ whenever $σ, τ$ are disjoint simplices of $K$. Fix integers $d,k \geqslant 2$ such that $k+2 \leqslant d \leqslant\frac{3k}2+1$. Assuming that the "preimage of a cycle is a cycle" we prove $\mathbf{NP}$-hardness of the algorithmic problem of recognition of almost embeddability of finite $k$-dimensional complexes in $\mathbb{R}^d$. Assuming that $\mathbf{P} \ne \mathbf{NP}$ (and that the "preimage of a cycle is a cycle") we prove that the embedding obstruction is incomplete for $k$-dim…
The NP-hard Metric Dimension problem is to decide for a given graph G and a positive integer k whether there is a vertex subset of size at most k that separates all vertex pairs in G. Herein, a vertex v separates a pair {u,w} if the distance (length of a shortest path) between v and u is different from the distance of v and w. We give a polynomial-time computable reduction from the Bipartite Dominating Set problem to Metric Dimension on maximum degree three graphs such that there is a one-to-one correspondence between the solution sets of both problems. There are two main consequences of this:…
Assuming the Unique Games Conjecture, we show strong inapproximability results for two natural vertex deletion problems on directed graphs: for any integer $k\geq 2$ and arbitrary small $ε> 0$, the Feedback Vertex Set problem and the DAG Vertex Deletion problem are inapproximable within a factor $k-ε$ even on graphs where the vertices can be almost partitioned into $k$ solutions. This gives a more structured and therefore stronger UGC-based hardness result for the Feedback Vertex Set problem that is also simpler (albeit using the "It Ain't Over Till It's Over" theorem) than the previous hardne…
Given $n$ weighted points (positive or negative) in $d$ dimensions, what is the axis-aligned box which maximizes the total weight of the points it contains? The best known algorithm for this problem is based on a reduction to a related problem, the Weighted Depth problem [T. M. Chan, FOCS'13], and runs in time $O(n^d)$. It was conjectured [Barbay et al., CCCG'13] that this runtime is tight up to subpolynomial factors. We answer this conjecture affirmatively by providing a matching conditional lower bound. We also provide conditional lower bounds for the special case when points are arranged in…
A path in an edge-colored graph is \textit{rainbow} if no two edges of it are colored the same. The graph is said to be \textit{rainbow connected} if there is a rainbow path between every pair of vertices. If there is a rainbow shortest path between every pair of vertices, the graph is \textit{strong rainbow connected}. We consider the complexity of the problem of deciding if a given edge-colored graph is rainbow or strong rainbow connected. These problems are called \textsc{Rainbow connectivity} and \textsc{Strong rainbow connectivity}, respectively. We prove both problems remain $\NP$\hyp{}c…
We study the 2-ary constraint satisfaction problems (2-CSPs), which can be stated as follows: given a constraint graph $G=(V,E)$, an alphabet set $Σ$ and, for each $\{u, v\}\in E$, a constraint $C_{uv} \subseteq Σ\timesΣ$, the goal is to find an assignment $σ: V \to Σ$ that satisfies as many constraints as possible, where a constraint $C_{uv}$ is satisfied if $(σ(u),σ(v))\in C_{uv}$. While the approximability of 2-CSPs is quite well understood when $|Σ|$ is constant, many problems are still open when $|Σ|$ becomes super constant. One such problem is whether it is hard to approximate 2-CSPs to …
Multidimensional packing problems generalize the classical packing problems such as Bin Packing, Multiprocessor Scheduling by allowing the jobs to be $d$-dimensional vectors. While the approximability of the scalar problems is well understood, there has been a significant gap between the approximation algorithms and the hardness results for the multidimensional variants. In this paper, we close this gap by giving almost tight hardness results for these problems. 1. We show that Vector Bin Packing has no polynomial time $Ω( \log d)$ factor asymptotic approximation algorithm when $d$ is a large …
We present a series of almost settled inapproximability results for three fundamental problems. The first in our series is the subexponential-time inapproximability of the maximum independent set problem, a question studied in the area of parameterized complexity. The second is the hardness of approximating the maximum induced matching problem on bounded-degree bipartite graphs. The last in our series is the tight hardness of approximating the k-hypergraph pricing problem, a fundamental problem arising from the area of algorithmic game theory. In particular, assuming the Exponential Time Hypot…
We use the incompressibility method based on Kolmogorov complexity to determine the total number of bits of routing information for almost all network topologies. In most models for routing, for almost all labeled graphs $Θ(n^2)$ bits are necessary and sufficient for shortest path routing. By `almost all graphs' we mean the Kolmogorov random graphs which constitute a fraction of $1-1/n^c$ of all graphs on $n$ nodes, where $c > 0$ is an arbitrary fixed constant. There is a model for which the average case lower bound rises to $Ω(n^2 \log n)$ and another model where the average case upper bound …
We study the coloring problem: Given a graph G, decide whether $c(G) \leq q$ or $c(G) \ge Q$, where c(G) is the chromatic number of G. We derive conditional hardness for this problem for any constant $3 \le q < Q$. For $q\ge 4$, our result is based on Khot's 2-to-1 conjecture [Khot'02]. For $q=3$, we base our hardness result on a certain `fish shaped' variant of his conjecture. We also prove that the problem almost coloring is hard for any constant $\eps>0$, assuming Khot's Unique Games conjecture. This is the problem of deciding for a given graph, between the case where one can 3-color all bu…
$ \newcommand{\eps}{\varepsilon} \newcommand{\problem}[1]{\ensuremath{\mathrm{#1}} } \newcommand{\CVP}{\problem{CVP}} \newcommand{\SVP}{\problem{SVP}} \newcommand{\CVPP}{\problem{CVPP}} \newcommand{\ensuremath}[1]{#1} $For odd integers $p \geq 1$ (and $p = \infty$), we show that the Closest Vector Problem in the $\ell_p$ norm ($\CVP_p$) over rank $n$ lattices cannot be solved in $2^{(1-\eps) n}$ time for any constant $\eps > 0$ unless the Strong Exponential Time Hypothesis (SETH) fails. We then extend this result to "almost all" values of $p \geq 1$, not including the even integers. This comes…
This paper considers the following problem. Two mixed-state quantum circuits Q and R are given, and the goal is to determine which of two possibilities holds: (i) Q and R act nearly identically on all possible quantum state inputs, or (ii) there exists some input state that Q and R transform into almost perfectly distinguishable outputs. This problem may be viewed as an abstraction of the following problem: given two physical processes described by sequences of local interactions, are the processes effectively the same or are they different? We prove that this problem is a complete promise pro…
In this paper, we prove an almost-optimal hardness for Max $k$-CSP$_R$ based on Khot's Unique Games Conjecture (UGC). In Max $k$-CSP$_R$, we are given a set of predicates each of which depends on exactly $k$ variables. Each variable can take any value from $1, 2, \dots, R$. The goal is to find an assignment to variables that maximizes the number of satisfied predicates. Assuming the Unique Games Conjecture, we show that it is NP-hard to approximate Max $k$-CSP$_R$ to within factor $2^{O(k \log k)}(\log R)^{k/2}/R^{k - 1}$ for any $k, R$. To the best of our knowledge, this result improves on al…
In recent years, significant progress has been made in explaining the apparent hardness of improving upon the naive solutions for many fundamental polynomially solvable problems. This progress has come in the form of conditional lower bounds -- reductions from a problem assumed to be hard. The hard problems include 3SUM, All-Pairs Shortest Path, SAT, Orthogonal Vectors, and others. In the $(\min,+)$-convolution problem, the goal is to compute a sequence $(c[i])^{n-1}_{i=0}$, where $c[k] = $ $\min_{i=0,\ldots,k} $ $\{a[i] $ $+$ $b[k-i]\}$, given sequences $(a[i])^{n-1}_{i=0}$ and $(b[i])_{i=0}^…
Stackelberg Pricing Games is a two-level combinatorial pricing problem studied in the Economics, Operation Research, and Computer Science communities. In this paper, we consider the decade-old shortest path version of this problem which is the first and most studied problem in this family. The game is played on a graph (representing a network) consisting of {\em fixed cost} edges and {\em pricable} or {\em variable cost} edges. The fixed cost edges already have some fixed price (representing the competitor's prices). Our task is to choose prices for the variable cost edges. After that, a clien…
The Euclidean $k$-means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of $n$ points in Euclidean space $R^d$, and the goal is to choose $k$ centers in $R^d$ so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general $k$ and a $(1+ε)$-approximation which runs in time $poly(n) 2^{O(k/ε)}$. At the othe…
We study the computational complexity of multi-stage robust optimization problems. Such problems are formulated with alternating min/max quantifiers and therefore naturally fall into a higher stage of the polynomial hierarchy. Despite this, almost no hardness results with respect to the polynomial hierarchy are known. In this work, we examine the hardness of robust two-stage adjustable and robust recoverable optimization with budgeted uncertainty sets. Our main technical contribution is the introduction of a technique tailored to prove $Σ^p_3$-hardness of such problems. We highlight a differen…
