Publications - The Year Before Last

The existence of EFX allocations is a fundamental open problem in discrete
fair division. Given a set of agents and indivisible goods, the goal is to
determine the existence of an allocation where no agent envies another
following the removal of any single good from the other agent's bundle. Since
the general problem has been illusive, progress is made on two fronts: $(i)$
proving existence when the number of agents is small, $(ii)$ proving existence
of relaxations of EFX. In this paper, we improve results on both fronts (and
simplify in one of the cases).
We prove the existence of EFX allocations with three agents, restricting only
one agent to have an MMS-feasible valuation function (a strict generalization
of nice-cancelable valuation functions introduced by Berger et al. which
subsumes additive, budget-additive and unit demand valuation functions). The
other agents may have any monotone valuation functions. Our proof technique is
significantly simpler and shorter than the proof by Chaudhury et al. on
existence of EFX allocations when there are three agents with additive
valuation functions and therefore more accessible.
Secondly, we consider relaxations of EFX allocations, namely, approximate-EFX
allocations and EFX allocations with few unallocated goods (charity). Chaudhury
et al. showed the existence of $(1-\epsilon)$-EFX allocation with
$O((n/\epsilon)^{\frac{4}{5}})$ charity by establishing a connection to a
problem in extremal combinatorics. We improve their result and prove the
existence of $(1-\epsilon)$-EFX allocations with $\tilde{O}((n/
\epsilon)^{\frac{1}{2}})$ charity. In fact, some of our techniques can be used
to prove improved upper-bounds on a problem in zero-sum combinatorics
introduced by Alon and Krivelevich.

We consider the problem of maximizing the Nash social welfare when allocating
a set $G$ of indivisible goods to a set $N$ of agents. We study instances, in
which all agents have 2-value additive valuations: The value of a good $g \in
G$ for an agent $i \in N$ is either $1$ or $s$, where $s$ is an odd multiple of
$\frac{1}{2}$ larger than one. We show that the problem is solvable in
polynomial time. Akrami et at. showed that this problem is solvable in
polynomial time if $s$ is integral and is NP-hard whenever $s = \frac{p}{q}$,
$p \in \mathbb{N}$ and $q\in \mathbb{N}$ are co-prime and $p > q \ge 3$. For
the latter situation, an approximation algorithm was also given. It obtains an
approximation ratio of at most $1.0345$. Moreover, the problem is APX-hard,
with a lower bound of $1.000015$ achieved at $\frac{p}{q} = \frac{5}{4}$. The
case $q = 2$ and odd $p$ was left open.
In the case of integral $s$, the problem is separable in the sense that the
optimal allocation of the heavy goods (= value $s$ for some agent) is
independent of the number of light goods (= value $1$ for all agents). This
leads to an algorithm that first computes an optimal allocation of the heavy
goods and then adds the light goods greedily. This separation no longer holds
for $s = \frac{3}{2}$; a simple example is given in the introduction. Thus an
algorithm has to consider heavy and light goods together. This complicates
matters considerably. Our algorithm is based on a collection of improvement
rules that transfers any allocation into an optimal allocation and exploits a
connection to matchings with parity constraints.

We study the problem of fairly allocating a set of $m$ indivisible goods to a
set of $n$ agents. Envy-freeness up to any good (EFX) criteria -- which
requires that no agent prefers the bundle of another agent after removal of any
single good -- is known to be a remarkable analogous of envy-freeness when the
resource is a set of indivisible goods. In this paper, we investigate EFX
notion for the restricted additive valuations, that is, every good has some
non-negative value, and every agent is interested in only some of the goods.
We introduce a natural relaxation of EFX called EFkX which requires that no
agent envies another agent after removal of any $k$ goods. Our main
contribution is an algorithm that finds a complete (i.e., no good is discarded)
EF2X allocation for the restricted additive valuations. In our algorithm we
devise new concepts, namely "configuration" and "envy-elimination" that might
be of independent interest.
We also use our new tools to find an EFX allocation for restricted additive
valuations that discards at most $\lfloor n/2 \rfloor -1$ goods. This improves
the state of the art for the restricted additive valuations by a factor of $2$.

Given the rapid rise in energy demand by data centers and computing systems
in general, it is fundamental to incorporate energy considerations when
designing (scheduling) algorithms. Machine learning can be a useful approach in
practice by predicting the future load of the system based on, for example,
historical data. However, the effectiveness of such an approach highly depends
on the quality of the predictions and can be quite far from optimal when
predictions are sub-par. On the other hand, while providing a worst-case
guarantee, classical online algorithms can be pessimistic for large classes of
inputs arising in practice.
This paper, in the spirit of the new area of machine learning augmented
algorithms, attempts to obtain the best of both worlds for the classical,
deadline based, online speed-scaling problem: Based on the introduction of a
novel prediction setup, we develop algorithms that (i) obtain provably low
energy-consumption in the presence of adequate predictions, and (ii) are robust
against inadequate predictions, and (iii) are smooth, i.e., their performance
gradually degrades as the prediction error increases.

We study sublinear time algorithms for estimating the size of maximum
matching. After a long line of research, the problem was finally settled by
Behnezhad [FOCS'22], in the regime where one is willing to pay an approximation
factor of $2$. Very recently, Behnezhad et al.[SODA'23] improved the
approximation factor to $(2-\frac{1}{2^{O(1/\gamma)}})$ using $n^{1+\gamma}$
time. This improvement over the factor $2$ is, however, minuscule and they
asked if even $1.99$-approximation is possible in $n^{2-\Omega(1)}$ time. We
give a strong affirmative answer to this open problem by showing
$(1.5+\epsilon)$-approximation algorithms that run in
$n^{2-\Theta(\epsilon^{2})}$ time. Our approach is conceptually simple and
diverges from all previous sublinear-time matching algorithms: we show a
sublinear time algorithm for computing a variant of the edge-degree constrained
subgraph (EDCS), a concept that has previously been exploited in dynamic
[Bernstein Stein ICALP'15, SODA'16], distributed [Assadi et al. SODA'19] and
streaming [Bernstein ICALP'20] settings, but never before in the sublinear
setting. Independent work: Behnezhad, Roghani and Rubinstein [BRR'23]
independently showed sublinear algorithms similar to our Theorem 1.2 in both
adjacency list and matrix models. Furthermore, in [BRR'23], they show
additional results on strictly better-than-1.5 approximate matching algorithms
in both upper and lower bound sides.

The circumference of a graph $G$ is the length of a longest cycle in $G$, or
$+\infty$ if $G$ has no cycle. Birmel\'e (2003) showed that the treewidth of a
graph $G$ is at most its circumference minus one. We strengthen this result for
$2$-connected graphs as follows: If $G$ is $2$-connected, then its treedepth is
at most its circumference. The bound is best possible and improves on an
earlier quadratic upper bound due to Marshall and Wood (2015).

We prove that for any triangle-free intersection graph of $n$ axis-parallel
segments in the plane, the independence number $\alpha$ of this graph is at
least $\alpha \ge n/4 + \Omega(\sqrt{n})$. We complement this with a
construction of a graph in this class satisfying $\alpha \le n/4 + c \sqrt{n}$
for an absolute constant $c$, which demonstrates the optimality of our result.

We consider the approximate pattern matching problem under the edit distance.
Given a text $T$ of length $n$, a pattern $P$ of length $m$, and a threshold
$k$, the task is to find the starting positions of all substrings of $T$ that
can be transformed to $P$ with at most $k$ edits. More than 20 years ago, Cole
and Hariharan [SODA'98, J. Comput.'02] gave an $\mathcal{O}(n+k^4 \cdot n/
m)$-time algorithm for this classic problem, and this runtime has not been
improved since.
Here, we present an algorithm that runs in time $\mathcal{O}(n+k^{3.5}
\sqrt{\log m \log k} \cdot n/m)$, thus breaking through this long-standing
barrier. In the case where $n^{1/4+\varepsilon} \leq k \leq
n^{2/5-\varepsilon}$ for some arbitrarily small positive constant
$\varepsilon$, our algorithm improves over the state-of-the-art by polynomial
factors: it is polynomially faster than both the algorithm of Cole and
Hariharan and the classic $\mathcal{O}(kn)$-time algorithm of Landau and
Vishkin [STOC'86, J. Algorithms'89].
We observe that the bottleneck case of the alternative $\mathcal{O}(n+k^4
\cdot n/m)$-time algorithm of Charalampopoulos, Kociumaka, and Wellnitz
[FOCS'20] is when the text and the pattern are (almost) periodic. Our new
algorithm reduces this case to a new dynamic problem (Dynamic Puzzle Matching),
which we solve by building on tools developed by Tiskin [SODA'10,
Algorithmica'15] for the so-called seaweed monoid of permutation matrices. Our
algorithm relies only on a small set of primitive operations on strings and
thus also applies to the fully-compressed setting (where text and pattern are
given as straight-line programs) and to the dynamic setting (where we maintain
a collection of strings under creation, splitting, and concatenation),
improving over the state of the art.

We introduce Hyperbard, a dataset of diverse relational data representations
derived from Shakespeare's plays. Our representations range from simple graphs
capturing character co-occurrence in single scenes to hypergraphs encoding
complex communication settings and character contributions as hyperedges with
edge-specific node weights. By making multiple intuitive representations
readily available for experimentation, we facilitate rigorous representation
robustness checks in graph learning, graph mining, and network analysis,
highlighting the advantages and drawbacks of specific representations.
Leveraging the data released in Hyperbard, we demonstrate that many solutions
to popular graph mining problems are highly dependent on the representation
choice, thus calling current graph curation practices into question. As an
homage to our data source, and asserting that science can also be art, we
present all our points in the form of a play.

We introduce seven foundational principles for creating a culture of
constructive criticism in computational legal studies. Beginning by challenging
the current perception of papers as the primary scholarly output, we call for a
more comprehensive interpretation of publications. We then suggest to make
these publications computationally reproducible, releasing all of the data and
all of the code all of the time, on time, and in the most functioning form
possible. Subsequently, we invite constructive criticism in all phases of the
publication life cycle. We posit that our proposals will help form our field,
and float the idea of marking this maturity by the creation of a modern
flagship publication outlet for computational legal studies.

How does neural connectivity in autistic children differ from neural
connectivity in healthy children or autistic youths? What patterns in global
trade networks are shared across classes of goods, and how do these patterns
change over time? Answering questions like these requires us to differentially
describe groups of graphs: Given a set of graphs and a partition of these
graphs into groups, discover what graphs in one group have in common, how they
systematically differ from graphs in other groups, and how multiple groups of
graphs are related. We refer to this task as graph group analysis, which seeks
to describe similarities and differences between graph groups by means of
statistically significant subgraphs. To perform graph group analysis, we
introduce Gragra, which uses maximum entropy modeling to identify a
non-redundant set of subgraphs with statistically significant associations to
one or more graph groups. Through an extensive set of experiments on a wide
range of synthetic and real-world graph groups, we confirm that Gragra works
well in practice.

We generalize the monotone local search approach of Fomin, Gaspers,
Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection between
parameterized approximation and exponential-time approximation algorithms for
monotone subset minimization problems. In a monotone subset minimization
problem the input implicitly describes a non-empty set family over a universe
of size $n$ which is closed under taking supersets. The task is to find a
minimum cardinality set in this family. Broadly speaking, we use approximate
monotone local search to show that a parameterized $\alpha$-approximation
algorithm that runs in $c^k \cdot n^{O(1)}$ time, where $k$ is the solution
size, can be used to derive an $\alpha$-approximation randomized algorithm that
runs in $d^n \cdot n^{O(1)}$ time, where $d$ is the unique value in $d \in
(1,1+\frac{c-1}{\alpha})$ such that
$\mathcal{D}(\frac{1}{\alpha}\|\frac{d-1}{c-1})=\frac{\ln c}{\alpha}$ and
$\mathcal{D}(a \|b)$ is the Kullback-Leibler divergence. This running time
matches that of Fomin et al. for $\alpha=1$, and is strictly better when
$\alpha >1$, for any $c > 1$. Furthermore, we also show that this result can be
derandomized at the expense of a sub-exponential multiplicative factor in the
running time.
We demonstrate the potential of approximate monotone local search by deriving
new and faster exponential approximation algorithms for Vertex Cover,
$3$-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex
Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we
get a $1.1$-approximation algorithm for Vertex Cover with running time $1.114^n
\cdot n^{O(1)}$, improving upon the previously best known $1.1$-approximation
running in time $1.127^n \cdot n^{O(1)}$ by Bourgeois et al. [DAM 2011].

The Edge Multicut problem is a classical cut problem where given an
undirected graph $G$, a set of pairs of vertices $\mathcal{P}$, and a budget
$k$, the goal is to determine if there is a set $S$ of at most $k$ edges such
that for each $(s,t) \in \mathcal{P}$, $G-S$ has no path from $s$ to $t$. Edge
Multicut has been relatively recently shown to be fixed-parameter tractable
(FPT), parameterized by $k$, by Marx and Razgon [SICOMP 2014], and
independently by Bousquet et al. [SICOMP 2018]. In the weighted version of the
problem, called Weighted Edge Multicut one is additionally given a weight
function $\mathtt{wt} : E(G) \to \mathbb{N}$ and a weight bound $w$, and the
goal is to determine if there is a solution of size at most $k$ and weight at
most $w$. Both the FPT algorithms for Edge Multicut by Marx et al. and Bousquet
et al. fail to generalize to the weighted setting. In fact, the weighted
problem is non-trivial even on trees and determining whether Weighted Edge
Multicut on trees is FPT was explicitly posed as an open problem by Bousquet et
al. [STACS 2009]. In this article, we answer this question positively by
designing an algorithm which uses a very recent result by Kim et al. [STOC
2022] about directed flow augmentation as subroutine.
We also study a variant of this problem where there is no bound on the size
of the solution, but the parameter is a structural property of the input, for
example, the number of leaves of the tree. We strengthen our results by stating
them for the more general vertex deletion version.

The leafage of a chordal graph G is the minimum integer l such that G can be
realized as an intersection graph of subtrees of a tree with l leaves. We
consider structural parameterization by the leafage of classical domination and
cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018,
Algorithmica 2020] proved, among other things, that Dominating Set on chordal
graphs admits an algorithm running in time $2^{O(l^2)} n^{O(1)}$. We present a
conceptually much simpler algorithm that runs in time $2^{O(l)} n^{O(1)}$. We
extend our approach to obtain similar results for Connected Dominating Set and
Steiner Tree. We then consider the two classical cut problems MultiCut with
Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove
that the former is W[1]-hard when parameterized by the leafage and complement
this result by presenting a simple $n^{O(l)}$-time algorithm. To our surprise,
we find that Multiway Cut with Undeletable Terminals on chordal graphs can be
solved, in contrast, in $n^{O(1)}$-time.

For a graph $G$, a subset $S \subseteq V(G)$ is called a \emph{resolving set}
if for any two vertices $u,v \in V(G)$, there exists a vertex $w \in S$ such
that $d(w,u) \neq d(w,v)$. The {\sc Metric Dimension} problem takes as input a
graph $G$ and a positive integer $k$, and asks whether there exists a resolving
set of size at most $k$. This problem was introduced in the 1970s and is known
to be NP-hard~[GT~61 in Garey and Johnson's book]. In the realm of
parameterized complexity, Hartung and Nichterlein~[CCC~2013] proved that the
problem is W[2]-hard when parameterized by the natural parameter $k$. They also
observed that it is FPT when parameterized by the vertex cover number and asked
about its complexity under \emph{smaller} parameters, in particular the
feedback vertex set number. We answer this question by proving that {\sc Metric
Dimension} is W[1]-hard when parameterized by the feedback vertex set number.
This also improves the result of Bonnet and Purohit~[IPEC 2019] which states
that the problem is W[1]-hard parameterized by the treewidth. Regarding the
parameterization by the vertex cover number, we prove that {\sc Metric
Dimension} does not admit a polynomial kernel under this parameterization
unless $NP\subseteq coNP/poly$. We observe that a similar result holds when the
parameter is the distance to clique. On the positive side, we show that {\sc
Metric Dimension} is FPT when parameterized by either the distance to cluster
or the distance to co-cluster, both of which are smaller parameters than the
vertex cover number.

In the Directed Steiner Network problem, the input is a directed graph G, a
subset T of k vertices of G called the terminals, and a demand graph D on T.
The task is to find a subgraph H of G with the minimum number of edges such
that for every edge (s,t) in D, the solution H contains a directed s to t path.
In this paper we investigate how the complexity of the problem depends on the
demand pattern when G is planar. Formally, if \mathcal{D} is a class of
directed graphs closed under identification of vertices, then the
\mathcal{D}-Steiner Network (\mathcal{D}-SN) problem is the special case where
the demand graph D is restricted to be from \mathcal{D}. For general graphs,
Feldmann and Marx [ICALP 2016] characterized those families of demand graphs
where the problem is fixed-parameter tractable (FPT) parameterized by the
number k of terminals. They showed that if \mathcal{D} is a superset of one of
the five hard families, then \mathcal{D}-SN is W[1]-hard parameterized by k,
otherwise it can be solved in time f(k)n^{O(1)}.
For planar graphs an interesting question is whether the W[1]-hard cases can
be solved by subexponential parameterized algorithms. Chitnis et al. [SICOMP
2020] showed that, assuming the ETH, there is no f(k)n^{o(k)} time algorithm
for the general \mathcal{D}-SN problem on planar graphs, but the special case
called Strongly Connected Steiner Subgraph can be solved in time f(k)
n^{O(\sqrt{k})} on planar graphs. We present a far-reaching generalization and
unification of these two results: we give a complete characterization of the
behavior of every $\mathcal{D}$-SN problem on planar graphs. We show that
assuming ETH, either the problem is (1) solvable in time 2^{O(k)}n^{O(1)}, and
not in time 2^{o(k)}n^{O(1)}, or (2) solvable in time f(k)n^{O(\sqrt{k})}, but
not in time f(k)n^{o(\sqrt{k})}, or (3) solvable in time f(k)n^{O(k)}, but not
in time f(k)n^{o({k})}.

Physarum Polycephalum is a Slime mold that can solve the shortest path
problem. A mathematical model based on the Physarum's behavior, known as the
Physarum Directed Dynamics, can solve positive linear programs. In this paper,
we will propose a Physarum based dynamic based on the previous work and
introduce a new way to solve positive Semi-Definite Programming (SDP) problems,
which are more general than positive linear programs. Empirical results suggest
that this extension of the dynamic can solve the positive SDP showing that the
nature-inspired algorithm can solve one of the hardest problems in the
polynomial domain. In this work, we will formulate an accurate algorithm to
solve positive and some non-negative SDPs and formally prove some key
characteristics of this solver thus inspiring future work to try and refine
this method.

We study the online Traveling Salesman Problem (TSP) on the line augmented
with machine-learned predictions. In the classical problem, there is a stream
of requests released over time along the real line. The goal is to minimize the
makespan of the algorithm. We distinguish between the open variant and the
closed one, in which we additionally require the algorithm to return to the
origin after serving all requests. The state of the art is a $1.64$-competitive
algorithm and a $2.04$-competitive algorithm for the closed and open variants,
respectively \cite{Bjelde:1.64}. In both cases, a tight lower bound is known
\cite{Ausiello:1.75, Bjelde:1.64}.
In both variants, our primary prediction model involves predicted positions
of the requests. We introduce algorithms that (i) obtain a tight 1.5
competitive ratio for the closed variant and a 1.66 competitive ratio for the
open variant in the case of perfect predictions, (ii) are robust against
unbounded prediction error, and (iii) are smooth, i.e., their performance
degrades gracefully as the prediction error increases.
Moreover, we further investigate the learning-augmented setting in the open
variant by additionally considering a prediction for the last request served by
the optimal offline algorithm. Our algorithm for this enhanced setting obtains
a 1.33 competitive ratio with perfect predictions while also being smooth and
robust, beating the lower bound of 1.44 we show for our original prediction
setting for the open variant. Also, we provide a lower bound of 1.25 for this
enhanced setting.

A mixed graph has a set of vertices, a set of undirected egdes, and a set of
directed arcs. A proper coloring of a mixed graph $G$ is a function $c$ that
assigns to each vertex in $G$ a positive integer such that, for each edge $uv$
in $G$, $c(u) \ne c(v)$ and, for each arc $uv$ in $G$, $c(u) < c(v)$. For a
mixed graph $G$, the chromatic number $\chi(G)$ is the smallest number of
colors in any proper coloring of $G$. A directional interval graph is a mixed
graph whose vertices correspond to intervals on the real line. Such a graph has
an edge between every two intervals where one is contained in the other and an
arc between every two overlapping intervals, directed towards the interval that
starts and ends to the right.
Coloring such graphs has applications in routing edges in layered orthogonal
graph drawing according to the Sugiyama framework; the colors correspond to the
tracks for routing the edges. We show how to recognize directional interval
graphs, and how to compute their chromatic number efficiently. On the other
hand, for mixed interval graphs, i.e., graphs where two intersecting intervals
can be connected by an edge or by an arc in either direction arbitrarily, we
prove that computing the chromatic number is NP-hard.

We show fixed-parameter tractability of the Directed Multicut problem with
three terminal pairs (with a randomized algorithm). This problem, given a
directed graph $G$, pairs of vertices (called terminals) $(s_1,t_1)$,
$(s_2,t_2)$, and $(s_3,t_3)$, and an integer $k$, asks to find a set of at most
$k$ non-terminal vertices in $G$ that intersect all $s_1t_1$-paths, all
$s_2t_2$-paths, and all $s_3t_3$-paths. The parameterized complexity of this
case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved
fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and
Pilipczuk and Wahlstr\"{o}m proved the W[1]-hardness of the 4-terminal-pairs
case at SODA 2016.
On the technical side, we use two recent developments in parameterized
algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch,
Pilipczuk, Wahlstr\"{o}m, STOC 2022] we cast the problem as a CSP problem with
few variables and constraints over a large ordered domain.We observe that this
problem can be in turn encoded as an FO model-checking task over a structure
consisting of a few 0-1 matrices. We look at this problem through the lenses of
twin-width, a recently introduced structural parameter [Bonnet, Kim,
Thomass\'{e}, Watrigant, FOCS 2020]: By a recent characterization [Bonnet,
Giocanti, Ossona de Mendes, Simon, Thomass\'{e}, Toru\'{n}czyk, STOC 2022] the
said FO model-checking task can be done in FPT time if the said matrices have
bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If
any of the matrices in the said encoding has a large grid minor, a vertex
corresponding to the ``middle'' box in the grid minor can be proclaimed
irrelevant -- not contained in the sought solution -- and thus reduced.

We show that the $b$-Coloring problem is complete for the class XNLP when
parameterized by the pathwidth of the input graph. Besides determining the
precise parameterized complexity of this problem, this implies that b-Coloring
parameterized by pathwidth is $W[t]$-hard for all $t$, and resolves the
parameterized complexity of $b$-Coloring parameterized by treewidth.

One of the first application of the recently introduced technique of
\emph{flow-augmentation} [Kim et al., STOC 2022] is a fixed-parameter algorithm
for the weighted version of \textsc{Directed Feedback Vertex Set}, a landmark
problem in parameterized complexity. In this note we explore applicability of
flow-augmentation to other weighted graph separation problems parameterized by
the size of the cutset. We show the following. -- In weighted undirected graphs
\textsc{Multicut} is FPT, both in the edge- and vertex-deletion version. -- The
weighted version of \textsc{Group Feedback Vertex Set} is FPT, even with an
oracle access to group operations. -- The weighted version of \textsc{Directed
Subset Feedback Vertex Set} is FPT. Our study reveals \textsc{Directed
Symmetric Multicut} as the next important graph separation problem whose
parameterized complexity remains unknown, even in the unweighted setting.

Formal models of learning from teachers need to respect certain criteria to
avoid collusion. The most commonly accepted notion of collusion-freeness was
proposed by Goldman and Mathias (1996), and various teaching models obeying
their criterion have been studied. For each model $M$ and each concept class
$\mathcal{C}$, a parameter $M$-$\mathrm{TD}(\mathcal{C})$ refers to the
teaching dimension of concept class $\mathcal{C}$ in model $M$---defined to be
the number of examples required for teaching a concept, in the worst case over
all concepts in $\mathcal{C}$.
This paper introduces a new model of teaching, called no-clash teaching,
together with the corresponding parameter $\mathrm{NCTD}(\mathcal{C})$.
No-clash teaching is provably optimal in the strong sense that, given any
concept class $\mathcal{C}$ and any model $M$ obeying Goldman and Mathias's
collusion-freeness criterion, one obtains $\mathrm{NCTD}(\mathcal{C})\le
M$-$\mathrm{TD}(\mathcal{C})$. We also study a corresponding notion
$\mathrm{NCTD}^+$ for the case of learning from positive data only, establish
useful bounds on $\mathrm{NCTD}$ and $\mathrm{NCTD}^+$, and discuss relations
of these parameters to the VC-dimension and to sample compression.
In addition to formulating an optimal model of collusion-free teaching, our
main results are on the computational complexity of deciding whether
$\mathrm{NCTD}^+(\mathcal{C})=k$ (or $\mathrm{NCTD}(\mathcal{C})=k$) for given
$\mathcal{C}$ and $k$. We show some such decision problems to be equivalent to
the existence question for certain constrained matchings in bipartite graphs.
Our NP-hardness results for the latter are of independent interest in the study
of constrained graph matchings.

For a positive integer $\ell \geq 3$, the $C_\ell$-Contractibility problem
takes as input an undirected simple graph $G$ and determines whether $G$ can be
transformed into a graph isomorphic to $C_\ell$ (the induced cycle on $\ell$
vertices) using only edge contractions. Brouwer and Veldman [JGT 1987] showed
that $C_4$-Contractibility is NP-complete in general graphs. It is easy to
verify that $C_3$-Contractibility is polynomial-time solvable. Dabrowski and
Paulusma [IPL 2017] showed that $C_{\ell}$-Contractibility is \NP-complete\ on
bipartite graphs for $\ell = 6$ and posed as open problems the status of the
problem when $\ell$ is 4 or 5. In this paper, we show that both
$C_5$-Contractibility and $C_4$-Contractibility are NP-complete on bipartite
graphs.

We introduce the notion of {\em fair cuts} as an approach to leverage
approximate $(s,t)$-mincut (equivalently $(s,t)$-maxflow) algorithms in
undirected graphs to obtain near-linear time approximation algorithms for
several cut problems. Informally, for any $\alpha\geq 1$, an $\alpha$-fair
$(s,t)$-cut is an $(s,t)$-cut such that there exists an $(s,t)$-flow that uses
$1/\alpha$ fraction of the capacity of \emph{every} edge in the cut. (So, any
$\alpha$-fair cut is also an $\alpha$-approximate mincut, but not vice-versa.)
We give an algorithm for $(1+\epsilon)$-fair $(s,t)$-cut in
$\tilde{O}(m)$-time, thereby matching the best runtime for
$(1+\epsilon)$-approximate $(s,t)$-mincut [Peng, SODA '16]. We then demonstrate
the power of this approach by showing that this result almost immediately leads
to several applications:
- the first nearly-linear time $(1+\epsilon)$-approximation algorithm that
computes all-pairs maxflow values (by constructing an approximate Gomory-Hu
tree). Prior to our work, such a result was not known even for the special case
of Steiner mincut [Dinitz and Vainstein, STOC '94; Cole and Hariharan, STOC
'03];
- the first almost-linear-work subpolynomial-depth parallel algorithms for
computing $(1+\epsilon)$-approximations for all-pairs maxflow values (again via
an approximate Gomory-Hu tree) in unweighted graphs;
- the first near-linear time expander decomposition algorithm that works even
when the expansion parameter is polynomially small; this subsumes previous
incomparable algorithms [Nanongkai and Saranurak, FOCS '17; Wulff-Nilsen, FOCS
'17; Saranurak and Wang, SODA '19].

In a classical scheduling problem, we are given a set of $n$ jobs of unit
length along with precedence constraints and the goal is to find a schedule of
these jobs on $m$ identical machines that minimizes the makespan. This problem
is well-known to be NP-hard for an unbounded number of machines. Using standard
3-field notation, it is known as $P|\text{prec}, p_j=1|C_{\max}$.
We present an algorithm for this problem that runs in $O(1.995^n)$ time.
Before our work, even for $m=3$ machines the best known algorithms ran in
$O^\ast(2^n)$ time. In contrast, our algorithm works when the number of
machines $m$ is unbounded. A crucial ingredient of our approach is an algorithm
with a runtime that is only single-exponential in the vertex cover of the
comparability graph of the precedence constraint graph. This heavily relies on
insights from a classical result by Dolev and Warmuth (Journal of Algorithms
1984) for precedence graphs without long chains.

We study distributed systems, with a particular focus on graph problems and fault
tolerance.
Fault-tolerance in a microprocessor or even System-on-Chip can be improved by using
a fault-tolerant pulse propagation design. The existing design TRIX achieves this goal
by being a distributed system consisting of very simple nodes. We show that even in
the typical mode of operation without faults, TRIX performs significantly better than a
regular wire or clock tree: Statistical evaluation of our simulated experiments show that
we achieve a skew with standard deviation of O(log log H), where H is the height of the
TRIX grid.
The distance-r generalization of classic graph problems can give us insights on how
distance affects hardness of a problem. For the distance-r dominating set problem, we
present both an algorithmic upper and unconditional lower bound for any graph class
with certain high-girth and sparseness criteria. In particular, our algorithm achieves a
O(r · f(r))-approximation in time O(r), where f is the expansion function, which correlates
with density. For constant r, this implies a constant approximation factor, in constant
time. We also show that no algorithm can achieve a (2r + 1 − δ)-approximation for any
δ > 0 in time O(r), not even on the class of cycles of girth at least 5r. Furthermore, we
extend the algorithm to related graph cover problems and even to a different execution
model.
Furthermore, we investigate the problem of packet forwarding, which addresses the
question of how and when best to forward packets in a distributed system. These packets
are injected by an adversary. We build on the existing algorithm OED to handle more
than a single destination. In particular, we show that buffers of size O(log n) are sufficient
for this algorithm, in contrast to O(n) for the naive approach.

We propose an algorithm for robust recovery of the spherical harmonic
expansion of functions defined on the d-dimensional unit sphere
$\mathbb{S}^{d-1}$ using a near-optimal number of function evaluations. We show
that for any $f \in L^2(\mathbb{S}^{d-1})$, the number of evaluations of $f$
needed to recover its degree-$q$ spherical harmonic expansion equals the
dimension of the space of spherical harmonics of degree at most $q$ up to a
logarithmic factor. Moreover, we develop a simple yet efficient algorithm to
recover degree-$q$ expansion of $f$ by only evaluating the function on
uniformly sampled points on $\mathbb{S}^{d-1}$. Our algorithm is based on the
connections between spherical harmonics and Gegenbauer polynomials and leverage
score sampling methods. Unlike the prior results on fast spherical harmonic
transform, our proposed algorithm works efficiently using a nearly optimal
number of samples in any dimension d. We further illustrate the empirical
performance of our algorithm on numerical examples.