Foundations Of Computer Science Research Articles

A Discrepancy Lower Bound for Information Complexity

This paper provides the first general technique for proving information lower bounds on two-party unbounded-rounds communication problems. We show that the discrepancy lower bound, which applies to randomized communication complexity, also applies to information complexity. More precisely, if the discrepancy of a two-party function f with respect to a distribution $$\mu $$μ is $$Disc_\mu f$$Discμf, then any two party randomized protocol computing f must reveal at least $$\Omega (\log (1/Disc_\mu f))$$Ω(log(1/Discμf)) bits of information to the participants. As a corollary, we obtain that any two-party protocol for computing a random function on $$\{0,1\}^n\times \{0,1\}^n$${0,1}n?{0,1}n must reveal $$\Omega (n)$$Ω(n) bits of information to the participants. In addition, we prove that the discrepancy of the Greater-Than function is $$\Omega (1/\sqrt{n})$$Ω(1/n), which provides an alternative proof to the recent proof of Viola (Proceedings of the twenty-fourth annual ACM-SIAM symposium on discrete algorithms, SODA 2013, New Orleans, LA, USA, 6---8 Jan 2013, pp 632---651, 2013) of the $$\Omega (\log n)$$Ω(logn) lower bound on the communication complexity of this well-studied function and, combined with our main result, proves the tight $$\Omega (\log n)$$Ω(logn) lower bound on its information complexity. The proof of our main result develops a new simulation procedure that may be of an independent interest. In a followup breakthrough work of Kerenidis et al. (53rd annual IEEE symposium on foundations of computer science, FOCS 2012, New Brunswick, NJ, USA, 20---23 Oct 2012, pp 500---509, 2012), our simulation procedure served as a building block towards a proof that almost all known lower bound techniques for communication complexity (and not just discrepancy) apply to information complexity as well.

Oblivious routing in wireless mesh networks

As new network applications have arisen rapidly in recent years, it is becoming more difficult to predict the exact traffic pattern of a network. In consequence, a routing scheme based on a single traffic demand matrix often leads to a poor performance. Oblivious routing (Racke in Proceedings of the 43rd annual IEEE symposium on foundations of computer science 43---52, 2002) is a technique for tackling the traffic demand uncertainty problem. A routing scheme derived from this principle intends to achieve a predicable performance for a set of traffic matrixes. Oblivious routing can certainly be an effective tool to handle traffic demand uncertainty in a wireless mesh network (WMN). However, a WMN has an additional tool that a wireline network does not have: dynamic bandwidth allocation. A router in a WMN can dynamically assign bandwidth to its attached links. This capability has never been exploited previously in works on oblivious routing for a spatial time division multiple access (STDMA) based WMN. Another useful insight is that although it is impossible to know the exact traffic matrix, it is relatively easy to estimate the amount of the traffic routed through a link when the routing scheme is given. Based on these two insights, we propose a new oblivious routing framework for STDMA WMNs. Both analytical models and simulation results are presented in this paper to prove that the performance--in terms of throughput, queue lengths, and fairness--of the proposed scheme can achieve significant gains over conventional oblivious routing schemes for STDMA based WMNs.

Towards the price of leasing online

We consider online optimization problems in which certain goods have to be acquired in order to provide a service or infrastructure. Classically, decisions for such problems are considered as final: one buys the goods. However, in many real world applications, there is a shift away from the idea of buying goods. Instead, leasing is often a more flexible and lucrative business model. Research has realized this shift and recently initiated the theoretical study of leasing models (Anthony and Gupta in Proceedings of the integer programming and combinatorial optimization: 12th International IPCO Conference, Ithaca, NY, USA, June 25---27, 2007; Meyerson in Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), 23---25 Oct 2005, Pittsburgh, PA, USA, 2005; Nagarajan and Williamson in Discret Optim 10(4):361---370, 2013) We extend this line of work and suggest a more systematic study of leasing aspects for a class of online optimization problems. We provide two major technical results. We introduce the leasing variant of online set multicover and give an $$O\left( \log (mK)\log n\right) $$Olog(mK)logn-competitive algorithm (with n, m, and K being the number of elements, sets, and leases, respectively). Our results also imply improvements for the non-leasing variant of online set cover. Moreover, we extend results for the leasing variant of online facility location. Nagarajan and Williamson (Discret Optim 10(4):361---370, 2013) gave an $$O\left( K\log n\right) $$OKlogn-competitive algorithm for this problem (with n and K being the number of clients and leases, respectively). We remove the dependency on n (and, thereby, on time). In general, this leads to a bound of $$O\left( l_{\text {max}} \log l_{\text {max}} \right) $$Olmaxloglmax (with the maximal lease length $$l_{\text {max}} $$lmax). For many natural problem instances, the bound improves to $$O\left( K^2\right) $$OK2.

Computing unique maximum matchings in $$O(m)$$ O ( m ) time for König–Egerváry graphs and unicyclic graphs

Let $$\alpha \left( G\right) $$źG denote the maximum size of an independent set of vertices and $$\mu \left( G\right) $$μG be the cardinality of a maximum matching in a graph $$G$$G. A matching saturating all the vertices is a perfect matching. If $$\alpha \left( G\right) +\mu \left( G\right) =\left| V(G)\right| $$źG+μG=V(G), then $$G$$G is called a Konig---Egervary graph. A graph is unicyclic if it is connected and has a unique cycle. It is known that a maximum matching can be found in $$O(m\cdot \sqrt{n})$$O(m·n) time for a graph with $$n$$n vertices and $$m$$m edges. Bartha (Proceedings of the 8th joint conference on mathematics and computer science, Komarno, Slovakia, 2010) conjectured that a unique perfect matching, if it exists, can be found in $$O(m)$$O(m) time. In this paper we validate this conjecture for Konig---Egervary graphs and unicylic graphs. We propose a variation of Karp---Sipser leaf-removal algorithm (Karp and Sipser in Proceedings of the 22nd annual IEEE symposium on foundations of computer science, 364---375, 1981) , which ends with an empty graph if and only if the original graph is a Konig---Egervary graph with a unique perfect matching (obtained as an output as well). We also show that a unicyclic non-bipartite graph $$G$$G may have at most one perfect matching, and this is the case where $$G$$G is a Konig---Egervary graph.

Special Section on the Fifty-Third IEEE Annual Symposium on Foundations of Computer Science (FOCS 2012)

This special section comprises eight fully refereed papers whose extended abstracts were presented at the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2012) in New Brunswick, New Jersey, October 20--23, 2012. The unrefereed conference versions of these papers were published by IEEE in the FOCS 2012 proceedings. The regular conference program consisted of 79 papers chosen from among 248 submissions. These were selected by a program committee consisting of Susanne Albers, Nikhil Bansal, Mark Braverman, Harry Buhrman, Moses Charikar, Anupam Gupta, Adam Klivans, Swastik Kopparty, Yishay Mansour, Rasmus Pagh, Rafael Pass, Ramamohan Paturi, Sofya Raskhodnikova, Aaron Roth, Tim Roughgarden (chair), Alexander Russell, Amit Sahai, C. Seshadhri, Berthold Vöcking, Jan Vondrak, and David Woodruff. The papers invited to this special section were also selected with the input of the program committee. The eight papers in this section span a broad range of topics, including dynamical systems, cryptography, approximation algorithms, hardness of approximation, complexity theory, and combinatorics. Each paper underwent an extensive refereeing process; I thank the authors and the anonymous referees for their efforts. In addition, I would like to thank SICOMP Editor-in-Chief Leonard Schulman and SIAM Senior Publications Coordinator Heather Blythe for their help in preparing this special section.

Explicit Linear Kernels via Dynamic Programming

Several algorithmic meta-theorems on kernelization have appeared in the last years, starting with the result of Bodlaender et al. [(Meta) kernelization, in Proceedings of the 50th IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, 2009, pp. 629--638] on graphs of bounded genus, then generalized by Fomin et al. [Bidimensionality and kernels, in Proceedings of the 21st ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, Philadephia, 2010, pp. 503--510] to graphs excluding a fixed minor, and by Kim et al. [Linear kernels and single-exponential algorithms via protrusion decompositions, in Proceedings of the 40th International Colloquium on Automata, Languages and Programming (ICALP), Lecture Notes in Comput. Sci., 7965 (2013), pp. 613--624] to graphs excluding a fixed topological minor. Typically, these results guarantee the existence of linear or polynomial kernels on sparse graph classes for problems satisfying some generic conditions, but, mainly due to their generality, i...

Finding Collisions in Interactive Protocols---Tight Lower Bounds on the Round and Communication Complexities of Statistically Hiding Commitments

We study the round and communication complexities of various cryptographic protocols. We give tight lower bounds on the round and communication complexities of any fully black-box reduction of a statistically hiding commitment scheme from one-way permutations and from trapdoor permutations. As a corollary, we derive similar tight lower bounds for several other cryptographic protocols, such as single-server private information retrieval, interactive hashing, and oblivious transfer that guarantees statistical security for one of the parties. Our techniques extend the collision-finding oracle due to Simon [Advances in Cryptology---EUROCRYPT'98, Lecture Notes in Comput. Sci. 1403, Springer, Berlin, 1998, pp. 334--345] to the setting of interactive protocols and the reconstruction paradigm of Gennaro and Trevisan [Proceedings of the 41st Annual Symposium on Foundations of Computer Science (FOCS), IEEE Press, Piscataway, NJ, 2000, pp. 305--313].

Making the Long Code Shorter

The long code is a central tool in hardness of approximation, especially in questions related to the Unique Games Conjecture. We construct a new code that is exponentially more efficient, but can still be used in many of these applications. Using the new code we obtain exponential improvements over several known results, including the following: (1) For any $\varepsilon>0$, we show the existence of an $n$-vertex graph $G$ where every set of $o(n)$ vertices has expansion $1-\varepsilon$, but $G$'s adjacency matrix has more than $\exp(\log^{\delta}n)$ eigenvalues larger than $1-\varepsilon$, where $\delta$ depends only on $\varepsilon$. This answers an open question of Arora, Barak, and Steurer [Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science, 2010, pp. 563--572], who asked whether one can improve over the noise graph on the Boolean hypercube that has ${\rm poly}(\log n)$ such eigenvalues. (2) A gadget that reduces Unique Games instances with linear constraints modulo $K$ into instances with alphabet $k$ with a blowup of $k^{{\rm polylog}(K)}$, improving over the previously known gadget with blowup of $k^{\Omega(K)}$. (3) An $n$-variable integrality gap for Unique Games that survives $\exp({\rm poly}(\log\log n))$ rounds of the semidefinite programming version of the Sherali--Adams hierarchy, improving on the previously known bound of ${\rm poly}(\log\log n)$. We show a connection between the local testability of linear codes and Small-Set Expansion in certain related Cayley graphs and use this connection to derandomize the noise graph on the Boolean hypercube.

Hitting-Sets for ROABP and Sum of Set-Multilinear Circuits

We give an $n^{O(\log n)}$-time ($n$ is the input size) blackbox polynomial identity testing algorithm for unknown-order read-once oblivious arithmetic branching programs (ROABPs). The best time complexity known for blackbox polynomial identity testing (PIT) for this class was $n^{O(\log^2 n)}$ due to Forbes, Saptharishi, and Shpilka [Proceedings of the 2014 ACM Symposium on Theory of Computing, 2014, pp. 867--875]. Moreover, their result holds only when the individual degree is small, while we do not need any such assumption. With this, we match the time complexity for the unknown-order ROABP with the known-order ROABP (due to Forbes and Shpilka [Proceedings of the 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, 2013, pp. 243--252]) and also with the depth-3 set-multilinear circuits (due to Agrawal, Saha, and Saxena [Proceedings of the 2013 ACM Symposium on Theory of Computing, 2013, pp. 321--330]). Our proof is simpler and involves a new technique called basis isolation. The depth-3 ...

Development of the Binary Number System and the Foundations of Computer Science

This paper discusses the formalization of the binary number system and the groundwork that was laid for the future of digital circuitry, computers, and the field of computer science. The goal of this paper is to show how Gottfried Leibniz formalized the binary number system and solidified his thoughts through an analysis of the Chinese I Ching. In addition, Leib- niz's work in logic and with computing machines is presented. This work laid the foundation for Boolean algebra and digital circuitry which was continued by George Boole, Augustus De Mor- gan, and Claude Shannon in the centuries following. Some have coined Leibniz the world's first

Session details: Semantics

This issue marks the first "real column" in the Semantics series. I am pleased that the guest author, Achim Jung, agreed to write this column. Achim is renowned for his research in denotational semantics, and in particular in domain theory, and he also established himself as a leader in the UK computer science academic community during his two terms as Head of Department at the University of Birmingham. Achim has chosen to focus on pedagogy with a column entitled, "Teaching Denotational Semantics". Achim played a pivotal role in establishing the Midlands Graduate School in the Foundations of Computer Science, which runs an annual series of summer schools and other events. He was instrumental in obtaining initial funding for the series, assuring its success. Achim uses the course he teaches at the MGS summer school as a guide for what to include in a denotational semantics course, providing both a great template from which to work, as well as a clear rationale for why a short course on denotational semantics should be taught in the way he outlines. It's an excellent overview that offers guidance for those who are just starting to teach such a course, as well as a source of new ideas for those who have an established course already in hand. Above and beyond the outline Achim gives us, there is an equally important question that his column answers: "Why should we teach denotational semantics?" Denotational semantics courses are fairly well represented in Europe, where the area has its origins. But on my side of the Atlantic, semantics is relegated to a day, or at best a week, in programming language courses. Achim's outline provides a laundry list of reasons why denotational semantics deserves a full course, offering a number of lessons that are applicable across the whole range of computer science. For instance, such a course demonstrates: ---Abstraction: abstracting away from low-level details allows one to treat a problem on a more abstract level, where it can be understood better and more easily solved. This teaches students to avoid irrelevant details, and to focus on the issues at the heart of the problem. ---Modularity: breaking a problem down into subproblems that are easier to solve, and then using solutions to the subproblems to build a solution to the main problem. This is one of the great lessons of abstract mathematics that comes to the fore in denotational semantics. ---Proof techniques: the stock and trade of denotational semantics is proving properties of programs and it provides numerous proof tools for this purpose. Achim points out several techniques that are standard components of the approach: structural induction, term model construction, logical relations and domain theory. ---Mathematical thinking: using mathematics as both a language in which to phrase problems precisely, and as a tool kit for solving those problems. These are some of the benefits of a semantics course for computer science students. As for mathematics students, Achim argues that they are not interested in such a course. That may be true, but mathematics students also have a lot to learn from a course like Achim's: it demonstrates how mathematics can be used effectively to solve hard problems in a related discipline. Denotational semantics also offers a great middle ground between the abstract definition-theorem-proof world of abstract mathematics, and the reliance on combinatorics and calculations prevalent in complexity theory and algorithmic analysis. This latter point applies as well for computer science students, who should find in a semantics course a chance to experience abstract reasoning as a means for gaining a broader perspective about their discipline, as well as a new suite of tools to use in solving its problems.

Randomized broadcast in radio networks with collision detection

We present a randomized distributed algorithm that in radio networks with collision detection broadcasts a single message in $$O(D + \log ^6 n)$$ rounds, with high probability. This time complexity is most interesting because of its optimal additive dependence on the network diameter $$D$$ . It improves over the currently best known $$O(D\log \frac{n}{D}\,+\,\log ^2 n)$$ algorithms, due to Czumaj and Rytter (Broadcasting algorithms in radio networks with unknown topology. In: Proceedings of the symposium on foundations of computer science, pp 492–501, 2003), and Kowalski and Pelc (Broadcasting in undirected ad hoc radio networks. In: Proceedings of the ACM SIGACT-SIGOPS symposium on principles of distributed computing, pp 73–82, 2003). These algorithms where designed for the model without collision detection and are optimal in that model. However, as explicitly stated by Peleg in his 2007 survey on broadcast in radio networks, it had remained an open question whether the bound can be improved with collision detection. We also study distributed algorithms for broadcasting $$k$$ messages from a single source to all nodes. This problem is a natural and important generalization of the single-message broadcast problem, but is in fact considerably more challenging and less understood. We show the following results: If the network topology is known to all nodes, then a $$k$$ -message broadcast can be performed in $$O(D + k\log n + \log ^2 n)$$ rounds, with high probability. If the topology is not known, but collision detection is available, then a $$k$$ -message broadcast can be performed in $$O(D + k\log n + \log ^6 n)$$ rounds, with high probability. The first bound is optimal and the second is optimal modulo the additive $$O(\log ^6 n)$$ term.

Krivine schemes are optimal

Grothendieck's inequality asserts that there exists a constant K>0 such that for every m,n and every m×n matrix A=(aij) with real entries and every choice of unit vectors x1,…,xm,y1,…,yn∈Sm+n−1, there are signs e1,…,em,δ1,…,δn∈{−1,1} satisfying ∑mi=1∑nj=1aij⟨xi,yj⟩≤K∑mi=1∑nj=1aijeiδj. The infimum over those K for which this inequality holds is called the (real) Grothendieck constant and is denoted KG. In order to investigate the exact value of KG, the authors introduce the following definition. A Borel probability measure μ on {−1,1}Rk×{−1,1}Rk is a k-dimensional Krivine scheme of quality K>0 if for every m,n and every choice of unit vectors x1,…,xm,y1,…,yn∈Sm+n−1, there exist new unit vectors x′1,…,x′m,y′1,…,y′n∈Sm+n−1 such that if G:Rm+n→Rk is a random k×(m+n) matrix whose entries are i.i.d. standard Gaussian random variables, then for all (i,j)∈{1,…,m}×{1,…,n} we have EG[∫{−1,1}Rk×{−1,1}Rkf(Gx′i)g(Gy′j)dμ(f,g)]=1K⟨xi,yj⟩. The authors note that the existence of a k-dimensional Krivine scheme of quality K>0 implies that KG≤K. A special one-dimensional Krivine scheme was introduced by J.-L. Krivine [C. R. Acad. Sci. Paris Ser. A-B 284 (1977), no. 8, A445–A446; MR0428414 (55 #1435)] in order to obtain a new upper bound on KG. Krivine conjectured that his bound is exact. This conjecture was refuted [M. Braverman et al., in 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science—FOCS 2011, 453–462, IEEE Computer Soc., Los Alamitos, CA, 2011; MR2932721] yielding a better upper bound on KG via a two-dimensional Krivine scheme. In the paper under review the authors prove the optimality of the Krivine scheme: For every k∈N there exists a k-dimensional Krivine scheme of quality (1+O(1/k))KG.

On the historical semantics of the notion of software architecture

This study outlines in some detail the semantic variety of the notion of ‘software architecture’ in the field of software engineering since the early 1970s. This paper shows that there are two schools of thought in software architecture, namely a material-substantialist and a formal-structuralist one. In the former school of thought, software architecture is basically regarded as a thing (device) on its own, whereas in the latter one, software architecture is basically considered a property, not a thing (device). From an ontological point of view, these two opinions are mutually exclusive. In their mutual exclusivity, however, they coincide with non-formalist versus formalist philosophies and interpretations of informatics or computer science in general, wherein software engineering –and, by implication, software architecture– is embedded. In this way, the field of software architecture mirrors an ongoing science-philosophical dispute about the characteristics and foundations of computer science or informatics as a scholarly and practical discipline. In summary it seems fair to say that the metaphor of ‘architecture’, with its distinct fine arts connotations, has been particularly attractive to software engineers because this metaphor has helped software engineers to circumvent the notorious scientific immaturity and shortage of classical engineering methods in the field of software engineering.

On the Correlation of Parity and Small-Depth Circuits

We prove that the correlation of a depth-$d$ unbounded fanin circuit of size $S$ with parity of $n$ variables is at most $2^{-\Omega(n/(\log S)^{d-1})}$.

Analyzing Walksat on Random Formulas

Let $\mathbf{\Phi}$ be a uniformly distributed random $k$-SAT formula with $n$ variables and $m$ clauses. We prove that the \tt Walksat algorithm from Papadimitriou [On selecting a satisfying truth assignment, in Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Soc., Los Alamitos, CA, 1991, pp. 163--169] and Schöning [A probabilistic algorithm for $k$-SAT and constraint satisfaction problems, in Proceedings of the 40th Annual IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Soc., Los Alamitos, CA, 1999, pp. 410--414] finds a satisfying assignment of $\mathbf{\Phi}$ in polynomial time with high probability if $m/n\leq\rho\cdot2^k/k$ for a certain constant $\rho>0$. This is an improvement by a factor of $\Theta(k)$ over the best previous analysis of \tt Walksat from Coja-Oghlan et al. [On smoothed $k$-CNF formulas and the Walksat algorithm, in Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), ACM, New York, SIAM, Philadelphia, 2009, pp. 451--460].

Coin Flipping with Constant Bias Implies One-Way Functions

It is well known (cf. Impagliazzo and Luby [in Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, 1989, pp. 230--235]) that the existence of almost all “interesting" cryptographic applications, i.e., ones that cannot hold information theoretically, implies one-way functions. An important exception where the above implication is not known, however, is the case of coin-flipping protocols. Such protocols allow honest parties to mutually flip an unbiased coin, while guaranteeing that even a cheating (efficient) party cannot bias the output of the protocol by much. Impagliazzo and Luby proved that coin-flipping protocols that are safe against negligible bias do imply one-way functions, and, very recently, Maji, Prabhakaran, and Sahai [in Proceedings of the 2001 51st Annual IEEE Symposium on Foundations of Computer Science, 2010, pp. 613--622] proved the same for constant-round protocols (with any nontrivial bias). For the general case, however, no such implication was known. We make progress towards answering the above fundamental question, showing that (strong) coin-flipping protocols safe against a constant bias (concretely, $\frac{\sqrt2 -1}2 - o(1)$) imply one-way functions.

Special Section on the Fifty-Second IEEE Annual Symposium on Foundations of Computer Science (FOCS 2011)

This special section comprises 11 fully refereed papers whose extended abstracts were presented at the 52nd Annual IEEE Symposium on Foundations of Computer Science (FOCS 2011) in Palm Springs, California, October 23--25, 2011. The unrefereed conference versions of these papers were published by IEEE in the FOCS 2011 proceedings. The regular conference program consisted of 85 papers chosen from among 292 submissions. These were selected by a program committee consisting of Nir Ailon, Eli Ben-Sasson, Serge Fehr, Juan A. Garay, Russell Impagliazzo, Sandy Irani, Kamal Jain, Stefano Leonardi, Richard Lipton, Claire Mathieu, Dieter van Melkebeek, Ashwin Nayak, Rafail Ostrovsky (chair), Boaz Patt-Shamir, Tim Roughgarden, Rocco Servedio, Adam Smith, Kunal Talwar, Chris Umans, Chee Keng Yap, and Lisa Zhang. The papers invited to this special section were also selected with the input of the program committee. The 11 papers in this section span a broad range of topics, including cryptography, approximation algorithms, SAT solving algorithms, combinatorics, mechanism design, and derandomization. Each paper underwent an extensive refereeing process; we thank the authors and the anonymous referees for their efforts. In addition, we would like to thank Madhu Sudan, who was SICOMP’s editor-in-chief as this project began; Leonard Schulman, the current editor-in-chief; and SIAM Senior Publications Coordinator Heather Blythe for their help in preparing this special section.

Special Section on the Fifty-First Annual IEEE Symposium on Foundations of Computer Science (FOCS 2010)

This special section contains five selected papers from the 50th Annual Symposium on Foundations of Computer Science (FOCS 2010) sponsored by the IEEE Technical Committee on Mathematical Foundations of Computing. The conference was held in Las Vegas, Nevada, October 23-26, 2010. The conference program consisted of 81 papers, which the program committee selected from 270 submissions. The program committee was composed of Scott Aaronson, Dorit Aharonov, Eli Ben-Sasson, Julia Chuzhoy, Ryan O'Donnell, Roberto Grossi, Nick Harvey, Adam Kalai, Nicole Immorlica, Yuval Ishai, Lap Chi Lau, James Lee, Tal Malkin, Joe Mitchell, Dana Moshkovitz, S. Muthukrishnan, Christos Papadimitriou, Sofya Raskhodnikova, Steve Skiena, Mikkel Thorup, Luca Trevisan, and Eric Vigoda. Each of the five papers appearing in this issue was subject to the standard refereeing process of the SIAM Journal on Computing. In “The Monotone Complexity of $k$-Clique on Random Graphs, Ben Rossman proves that monotone circuits that solve the $k$-cli...

Constant Factor Lasserre Integrality Gaps for Graph Partitioning Problems

Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called balanced separator) that arises in many settings. For this problem, and variants such as the uniform sparsest cut problem where the goal is to minimize the fraction of pairs on opposite sides of the cut that are connected by an edge, there are large gaps between the known approximation algorithms and nonapproximability results. While no constant factor approximation algorithms are known, even APX-hardness is not known either. In this work we prove that for balanced separator and uniform sparsest cut, semidefinite programs from the Lasserre hierarchy (which are the most powerful relaxations studied in the literature) have an integrality gap bounded away from 1, even for $\Omega(n)$ levels of the hierarchy. This complements recent algorithmic results in Guruswami and Sinop [Proceedings of the 52nd Annual Symposium on Foundations of Computer Science (FOCS), 2011, pp. 482--491] which used the Lasserre hierarchy to give an approximation scheme for these problems (with runtime depending on the spectrum of the graph). Along the way, we make an observation that simplifies the task of lifting “polynomial constraints” (such as the global balance constraint in balanced separator) to higher levels of the Lasserre hierarchy.