271 research outputs found
Precedence-constrained scheduling problems parameterized by partial order width
Negatively answering a question posed by Mnich and Wiese (Math. Program.
154(1-2):533-562), we show that P2|prec,|, the
problem of finding a non-preemptive minimum-makespan schedule for
precedence-constrained jobs of lengths 1 and 2 on two parallel identical
machines, is W[2]-hard parameterized by the width of the partial order giving
the precedence constraints. To this end, we show that Shuffle Product, the
problem of deciding whether a given word can be obtained by interleaving the
letters of other given words, is W[2]-hard parameterized by , thus
additionally answering a question posed by Rizzi and Vialette (CSR 2013).
Finally, refining a geometric algorithm due to Servakh (Diskretn. Anal. Issled.
Oper. 7(1):75-82), we show that the more general Resource-Constrained Project
Scheduling problem is fixed-parameter tractable parameterized by the partial
order width combined with the maximum allowed difference between the earliest
possible and factual starting time of a job.Comment: 14 pages plus appendi
Decision Problems on Copying and Shuffling
We study decision problems of the form: given a regular or linear
context-free language , is there a word of a given fixed form in , where
given fixed forms are based on word operations copy, marked copy, shuffle and
their combinations
Topological Sorting with Regular Constraints
We introduce the constrained topological sorting problem (CTS): given a regular language K and a directed acyclic graph G with labeled vertices, determine if G has a topological sort that forms a word in K. This natural problem applies to several settings, e.g., scheduling with costs or verifying concurrent programs. We consider the problem CTS[K] where the target language K is fixed, and study its complexity depending on K. We show that CTS[K] is tractable when K falls in several language families, e.g., unions of monomials, which can be used for pattern matching. However, we show that CTS[K] is NP-hard for K = (ab)^* and introduce a shuffle reduction technique to show hardness for more languages. We also study the special case of the constrained shuffle problem (CSh), where the input graph is a disjoint union of strings, and show that CSh[K] is additionally tractable when K is a group language or a union of district group monomials. We conjecture that a dichotomy should hold on the complexity of CTS[K] or CSh[K] depending on K, and substantiate this by proving a coarser dichotomy under a different problem phrasing which ensures that tractable languages are closed under common operators
GENROUTE: A genetic algorithm printed wire board (printed wire board (PWB) Router)
The major effort of this thesis was to develop an electronic circuit routing system that utilizes genetic algorithms to perform Printed Wire Board (PWB) routing rather than brute force exhaustive searching methods. This problem can be classified as an NP-Hard optimization problem searching a large solution space. Some desirable characteristics of an electronic routing system are that it: Minimize the number of potential solutions Minimize the number of board layers and tap holes Minimize trace lengths and the number of jogs Minimize trace cross-talk and the board capacitance
Algorithms, Bounds, and Strategies for Entangled XOR Games
We study the complexity of computing the commuting-operator value
of entangled XOR games with any number of players. We introduce necessary and
sufficient criteria for an XOR game to have , and use these
criteria to derive the following results:
1. An algorithm for symmetric games that decides in polynomial time whether
or , a task that was not previously known to be
decidable, together with a simple tensor-product strategy that achieves value 1
in the former case. The only previous candidate algorithm for this problem was
the Navascu\'{e}s-Pironio-Ac\'{i}n (also known as noncommutative Sum of Squares
or ncSoS) hierarchy, but no convergence bounds were known.
2. A family of games with three players and with , where it
takes doubly exponential time for the ncSoS algorithm to witness this (in
contrast with our algorithm which runs in polynomial time).
3. A family of games achieving a bias difference
arbitrarily close to the maximum possible value of (and as a consequence,
achieving an unbounded bias ratio), answering an open question of Bri\"{e}t and
Vidick.
4. Existence of an unsatisfiable phase for random (non-symmetric) XOR games:
that is, we show that there exists a constant depending
only on the number of players, such that a random -XOR game over an
alphabet of size has with high probability when the number
of clauses is above .
5. A lower bound of on the number of levels
in the ncSoS hierarchy required to detect unsatisfiability for most random
3-XOR games. This is in contrast with the classical case where the -th level
of the sum-of-squares hierarchy is equivalent to brute-force enumeration of all
possible solutions.Comment: 55 page
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